Thèmes de recherche
Domaines de Recherche:
 Mathématiques pour la cryptographie symétrique et la
théorie des codes correcteurs.
 Algébre commutative et géométrie algébrique effective.
Disciplines
L'ensemble de ma recherche s'inscrit dans les disciplines suivantes:
 Mathématiques (corps finis, polynômes sur des corps
finis, sommes exponentielles, sommes de caractères, transformée de
Fourier discrète, fonctions courbes, fonctions "spéciales" sur des
corps finis, corps cyclotomiques, theorie algébriques de nombres,
etc.) ;
 Cryptographie symétrique;
 Théorie des codes correcteurs;
 Algèbre commutative et Géométrie algébrique effective;
Contexte général de mes recherches actutelles et futures
La société moderne dépend essentiellement de la capacité à
sécuriser, stocker et transmettre de grandes quantités d’informations
numériques à haute vitesse. Par exemple, la communication par satellite,
les films à la demande, les clés USB et les téléphones portables
reposent tous sur la théorie mathématique du codage qui, grâce à elle,
garantit que les images d’origine, la parole, la musique ou n’importe
quelle donnée peuvent être parfaitement récupérées même si des
erreurs sont introduites lors du stockage ou de la transmission. De plus,
la cryptographie est omniprésente dans notre vie moderne où on la met
quotidiennement en oeuvre, chaque fois qu’on utilise Internet ou qu’on
effectue un paiement ou un retrait. Les mathématiques sont au centre de
ces réalisations. Les applications émergentes conduisent continuellement
à poser de nouveaux problèmes de codes et de cryptographie. A l’inverse,
de nouveaux développements théoriques dans ces domaines permettent de
nouvelles applications. Mes recherches actuelles et futures tentent
d’apporter des développements théoriques dans ces domaines en résolvant
des problèmes mathématiques en théorie des codes et en cryptographie
(symétrique).
Axes de ma recherche actuelle
Ma recherche se situe en mathématiques appliquées à la protection de
l'information: cryptographie et théorie des codes correcteurs d'erreurs.
Plus précisément, mes travaux actuels portent sur les applications des
méthodes algébriques et combinatoires en cryptographie symétrique et dans
la théorie des codes lineaires. Les deux principaux axes de ma recherche
actuelle sont:
 cryptographie symétrique: certains de mes travaux
actuels dans le cadre de la cryptographie symétrique se
concentrent sur l'étude structurelle et algébrique (existence,
caractérisation, construction, classification, énumération, etc.)
de fonctions définies sur des corps finis (en toute
caractéristique) satisfaisant les propriétés nécessaires à la
sécurité du chiffrements les utilisant. Par exemple, les fonctions
hautement non linéaires jouent un rôle crucial dans la protection
des systèmes cryptographiques contre certaines attaques
fondamentales telles que la cryptanalyse linéaire. Dans l'approche
algébrique j'utilise la théorie des corps finis, la transformation
de Fourier discrète, des sommes exponentielles, des outils de
l'arithmétique et de la théorie des nombres, des courbes
algébriques et des objets de la géométrie finie.
 codes correcteurs: je travaille sur les aspects
algébriques (et combinatoires) de familles de codes lineaires.
Certains travaux récents sont consacrés à la construction
algébrique de familles de codes linéaires optimaux pour diverses
applications. En particulier, la conception de codes (presque)
optimaux pour le masquage de somme directe pour protéger les
données sensibles stockées dans les registres contre les attaques
par canaux cachés et les attaques par injection de faute (qui sont
aujourd'hui des méthodes de cryptanalyse importantes sur les
implémentations de chiffrements par blocs, qui représentent
menaces), des codes optimaux pour les systèmes de stockage
distribués modernes et des codes appropriés pour le partage de
secrets et également pour le calcul sécurisé à deux parties.
Je m'interesse aussi aux aspects
algorithmiques dans les axes cidessus dans le contexte de l'algèbre
informatique.
Distinctions scientifiques et prix
 Obtention en Septembre 2020 du premier Prix George
Boole International Prize. Lauréate
du prix international George Boole 2020. Ici Interview.
 PEDR (ex. "Excellence scientifique"), Université de
Paris VIII (évaluation nationale par le CNU section 25) en
20192022.
 PEDR (ex. "Excellence scientifique"), Université de
Paris VIII (évaluation nationale par le CNU section 25) en
20142017.
Publications
Revues internationales:
(dans l’ordre chronologique inverse)
 On permutation quadrinomial with boomerang
uniformity 4 and the bestknown nonlinearity
KH. Kim, S. Mesnager, JH. Choe, DN. Lee, S. Lee and
MC. Jo. Designs, Codes and Cryptography (DCC). A
paraître.
 Explicit Values of the DDT, the BCT, the
FBCT and FDBT of the Inverse, the Gold and the
BrackenLeander Sboxes. S. Eddahmani and
S. Mesnager. Cryptography and Communications  Discrete
Structures, Boolean Functions and Sequences (CCDS). A
paraître.
 A function field approach toward good
polynomials for further results on optimal LRC
codes.R. Chen and S. Mesnager. Finite Fields and
their Applications (FFA). A paraître
 An STPbased model toward designing Sboxes
with good cryptographic properties. Z. Lu,
S. Mesnager; T. Cui, Y. Fan and M. Wang. Design,
Codes and Cryptography (DCC). A
paraître.
 Constructions of TwoDimensionala
ZComplementary Array Pairs With Large ZCZ Ratio.
H.Zhang, C. Fan and S. Mesnager. Designs, Codes and
Cryptography. A paraître.
 Survey on recent trends towards generalized
differential and boomerang uniformities.
S. Mesnager, B. Mandal and M. Msahli. Cryptography and
Communications  Discrete Structures, Boolean Functions
and Sequences (CCDS). A paraître
 Cryptanalysis of the AEAD and hash
algorithm DryGASCON. H. Liang, S. Mesnager and
M.Wang. Cryptography and Coomunications  Discrete
Structures, Boolean Functions and Sequences (CCDS). A
paraître.
 On infinite families of NarrowSense
Antiprimitive BCH Codes Admitting 3Transitive
Automorphism Groups and their Consequences. Q. Liu,
C. Ding, S. Mesnager, C. Tang and V. D. Tonchev. IEEE
Transactions on Information Theory. A
paraître
 On onedimensional linear minimal codes
over finite (commutative) ringsM. Maji,
S. Mesnager, S. Sarkar and K. Hansda. IEEE
Transactions on Information Theory. A
paraître
 Linear codes from support designs of ternary
cyclic codes. P. Tan, C. Fan, S. Mesnager and
W. Guo. Design, Codes and Cryptography, Volume 90 (3), pages
681693, 2022
 Generic constructions of (Boolean and
vectorial) bent functions and their
consequences Y. Li, H. Kan, S. Mesnager,
J. Peng, CH. Tan and L. Zhend. IEEE Transactions
on Information Theory, Volume 68 (4), pages 27352751,
2022.
 A new family of polyphase sequences with
low correlation. Z. Gu, Z. Zhou, S. Mesnager and
U. Parampalli. Cryptography and Coomunications 
Discrete Structures, Boolean Functions and Sequences
(CCDS), Volume 14, pages 135144, 2022
 Classification of the codewords of weight
16 and 18 of the ReedMuller
code $RM(n3,n)$. S. Mesnager and
A. Oblaukhov. IEEE Transactions on Information Theory,
Volume 68 (2), pages 940952, 2002
 Preimages of $p$Linearized
Polynomials over $GF(p)$. K. H. Kim,
S. Mesnager, J. H. Choe and D. N. Lee. Cryptography and
Communications  Discrete Structures, Boolean Functions
and Sequences (CCDS), Volume 14, pages 7586,
2022.
 Constructions of Optimal Uniform Widegap
Frequencyhopping Sequences. P. Li, C. Fan,
S. Mesnager, Y. Yang and Z. Zhou. IEEE Transactions on
Information Theory, Volume 68 (1), pages 692700,
2022
 Constructions of ZOptimal TypeII
Quadriphase ZComplementary Pairs.
T. Yu, M. Yang, S. Mesnager, and Y. Yang.
Discrete
Mathematics, Volume 345 (2), 2022.
 Information Leakages in Codebased Masking
: An Unifed Quantification Approach.
W. Cheng, S. Guilley, C. Carlet, J.L. Danger, and
S. Mesnager. Transactions on Cryptographic
Hardware and Embedded Systems (TCHES), IACR,
Volume 3, pages 465495, 2021.
 Complete Solution
over $GF(p$ $ n$$ )$of\; the\; equation$ X$$ p$$ k$$ +1$$ +X+a=0$.$$
KH. Kim, JH. Choe and S. Mesnager. Finite Fields and
Their Applications (FFA), Volume 76,
2021
 On correlation immune Boolean functions
with minimum Hamming weight power
of $2$.S. Mesnager and S. Su. IEEE
Transactions on Information Theory., Volume 67 (11),
pages 75017517, 2021.
 Secondary constructions of (nonweakly
regular palteaued $p$ary
functions. S. Mesnager, F. Özbudak and
A. Sınak. Turkish Journal of Mathematics,
Volume 45, pages 22952306, 2021.
 On hulls of some primitive BCH codes and
selforthogonal codes. C. Gan, C. Li, S. Mesnager
et H. Qian. Journal IEEE transactions Information
Theory, Volume 67 (10), pages
64426455,2021.
Abstract :
Selforthogonal codes are an important type of linear codes due to their wide
applications in communication and cryptography.
The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual.
It is clear that the hull is selforthogonal. The main goal of this paper is to obtain selforthogonal codes by investigating the hulls.
Let $\mathcal C_{(r,r^m1,\delta,b)}$ be the primitive BCH code over $\Bbb F_r$ of length $r^m1$
with designed distance $\delta$, where $\Bbb F_r$ is the finite field of order $r$.
In this paper, we will present Euclidean (or Hermitian) selforthogonal codes and determine their parameters by
investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes.
Several sufficient and necessary conditions
for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances.
Furthermore, some Hermitian selforthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated.
In addition, we determine the dimensions of the code $\mathcal C_{(r,r^21,\delta,1)}$ and its hull in both Hermitian and Euclidean cases for $2 \le \delta \le r^21$.
We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.
 Good polynomials for optimal LRC of low locality. R. Chen, S. Mesnager et CA Zhao.
Journal Design Codes and
Cryptography. Volume 89, pages 16391660
Abstract :
According to a magnific method due to I. Tamo and A. Barg, a class of polynomials over finite fields, called good polynomials, was introduced and used to construct optimal Locally Recoverable Codes (LRC), which have been developed and exploited in distributed storage.
An important derived algebraic problem is, for a given finite field $\mathbb {F}_q$ and a fixed integer $r$, to find a polynomial of degree $r+1$ that is constant on as many subsets of $\mathbb {F}_q$ as possible of size $r+1$. Compared to the literature on this topic, our main contribution is introducing a new parameter that measures how``good''a polynomial is in the sense of LRC. Our new approach allows us to characterize completely good polynomials of a low degree over finite fields and, next, to derive new constructions of such polynomials, leading to optimal LRC with new flexible localities.
Specifically, several good polynomials of degree at most $6$ are studied and described precisely in this paper.
 Investigation for 8bit SKINNYlike Sboxes,
analysis and applications. Y. Fan, S. Mesnager, W. Wang,
Y. Li, T. Cui et M. Wang. Journal Cryptography and
Communications Discrete Structures, Boolean Functions and
Sequences (CCDS), Volume 13, pages 617636, 2021.
Abstract :
Nowadays, ciphers have been widely used in highend
platforms, resourceconstrained, and sidechannel attacks
vulnerable environments. This motivates various Sboxes
aimed at providing a good tradeoff between security and
efficiency. For small Sboxes, the most natural approach of
constructing such Sboxes is a comprehensive search in the
space of permutations, which inevitably becomes more
challenging when the size grows. For large Sboxes (e.g.,
8bit), previous works concentrated on creations from finite
fields or smaller ones (e.g., 4bit). This paper proposes a
new algorithm with a layered structure to search for 8bit
{\SKINNY}like Sboxes. We compare our new Sbox with the
original 8bit {\SKINNY} Sbox by analyzing its security
properties. Besides, due to our searching algorithm's rules
and constraints, {\SKINNY}like Sboxes have other features
of lightweight implementation, low multiplicative
complexity, low AND depth, and an effective inverse.
Eventually, the searching algorithm outputs $224\,000$ 8bit
{\SKINNY}like Sboxes. The cipher designers can use these
new Sboxes to construct lightweight block ciphers with
easytomask property and efficient implementation
performance.
 On constructions of weightwise perfectly balanced
Boolean functions. S. Mesnager et S. Su. Journal
Cryptography and Communications Discrete Structures, Boolean
Functions and Sequences (CCDS). Volume 13, pages 951979, 2021.
Abstract :
The recent FLIP cipher is an encryption scheme described by
M\'eaux et al. at the conference EUROCRYPT 2016. It is based
on a new stream cipher model called the filter permutator
and tries to minimize some parameters (including the
multiplicative depth). In the filter permutator, the input
to the Boolean function has constant Hamming weight equal to
the weight of the secret key. As a consequence, Boolean
functions satisfying good cryptographic criteria when
restricted to the set of vectors with constant Hamming
weight play an important role in the FLIP stream cipher.
Carlet et al. have shown that for Boolean functions with
restricted input, balancedness and nonlinearity parameters
continue to play an important role with respect to the
corresponding attacks on the framework of FLIP ciphers. In
particular, Boolean functions which are uniformly
distributed over $\F_2$ on $E_{n,k}=\{x\in\F_2^n\mid
\mathrm{wt}(x)=k\}$ for every integer $k$ from $1$ to $n1$
are called weightwise perfectly balanced (WPB) functions,
where $\mathrm{wt}(x)$ denotes the Hamming weight of $x$. In
this paper, we firstly propose two methods of constructing
weightwise perfectly balanced Boolean functions in $2^k$
variables (where $k$ is a positive integer) by modifying the
support of linear and quadratic functions. Furthermore, we
derive a construction of $n$variable weightwise almost
perfectly balanced Boolean functions for any positive
integer $n$.
 Information Leakages in Codebased Masking: A
Unified Quantification Approach. W. Cheng, S. Guilley,
C. Carlet, JL Danger, et S. Mesnager. The Transactions on
Cryptographic Hardware and Embedded Sytems, volume 2021, issue
3 (TCHES 2021, issue 3), 2021.
Abstract :
In this paper, we present a unified approach to quantifying
the information leakages in the most general codebased
masking schemes. Specifically, by utilizing a uniform
representation, we highlight first that the sidechannel
resistance of all codebased masking schemes can be
quantified by an allinone framework consisting of two
easytocompute parameters (the dual distance and the number
of conditioned codewords) from a codingtheoretic
perspective. In particular, we use signaltonoise ratio
(SNR) and mutual information (MI) as two complementary
metrics, where a closedform expression of SNR and an
approximation of MI are proposed by connecting both metrics
to the two codingtheoretic parameters. Second, taking the
connection between ReedSolomon code and SSS (Shamir’s
Secret Sharing) scheme, the SSSbased masking is viewed as a
special case of generalized codebased masking. Hence as a
straightforward application, we evaluate the impact of
public points on the sidechannel security of SSSbased
masking schemes, namely the polynomial masking, and enhance
the SSSbased masking by choosing optimal public points for
it. Interestingly, we show that given a specific security
order, more shares in SSSbased masking leak more
information on secrets in an informationtheoretic sense.
Finally, our approach provides a systematic method for
optimizing the sidechannel resistance of every codebased
masking. More precisely, this approach enables us to select
optimal linear codes (parameters) for the generalized
codebased masking by choosing appropriate codes according
to the two codingtheoretic parameters. Summing up, we
provide a bestpractice guideline for the application of
codebased masking to protect cryptographic implementations.
 More permutations and involutions for
constructing bent functions. Y. Li, K. Li,
S. Mesnager et L. Qu. Journal Cryptography and
Communications Discrete Structures, Boolean
Functions and Sequences (CCDS), Volume 13 (3),
pages 459473, 2021.
Abstract :
Bent functions are extremal combinatorial objects with
several applications, such as coding theory, maximum length
sequences, cryptography, the theory of difference sets, etc.
Based on C. Carlet's secondary construction, S. Mesnager
proposed in 2014 an effective method to construct bent
functions in their bivariate representation by employing
three permutations of the finite field $\F_{2^m}$ satisfying
an algebraic property $(\mathcal{A}_{m})$. This paper is
devoted to constructing permutations that satisfy the
property $(\mathcal{A}_{m})$ and then obtaining some
explicit bent functions. Firstly, we construct one class of
involutions from vectorial functions and further obtain some
explicit bent functions by choosing some triples of these
involutions satisfying the property $(\mathcal{A}_{m})$. We
then investigate some bent functions by involutions from
trace functions and linearized polynomials. Furthermore,
based on several triples of permutations (not all
involutions) that satisfy the property $(\mathcal{A}_{m})$
constructed by D. Bartoli et al., we give some more general
results and extend most of their work. Then we also find
several general triples of permutations that can also
satisfy the property $(\mathcal{A}_{m})$.
 Fast algebraic immunity of Boolean functions and
LCD codes. S. Mesnager et C. Tang. Journal IEEE
transactions Information Theory, Volume 67 (7), pages 48284837, 2021.
Abstract :
Nowadays, the resistance against algebraic attacks and fast
algebraic attacks is considered as an important
cryptographic property for Boolean functions used in stream
ciphers. Both attacks are very powerful analysis concepts
and can be applied to symmetric cryptographic algorithms
used in stream ciphers. The notion of algebraic immunity has
received wide attention since it is a powerful tool to
measure the resistance of a Boolean function to standard
algebraic attacks. Nevertheless, an algebraic tool to handle
the resistance to fast algebraic attacks is not clearly
identified in the literature. In the current paper, we
propose a new parameter to measure the resistance of a
Boolean function to fast algebraic attack. We also introduce
the notion of fast immunity profile and show that it informs
both on the resistance to standard and fast algebraic
attacks. Further, we evaluate our parameter for two
secondary constructions of Boolean functions. Moreover, A
codingtheory approach to the characterization of perfect
algebraic immune functions is presented. Via this
characterization, infinite families of binary linear
complementary dual codes (or LCD codes for short) are
obtained from perfect algebraic immune functions. Some of
the binary LCD codes presented in this paper are optimal.
These binary LCD codes have applications in armoring
implementations against socalled sidechannel attacks (SCA)
and fault noninvasive attacks, in addition to their
applications in communication and data storage systems.
 PostQuantum Secure Inner Product Functional
Encryption Using Multivariate Public Key Cryptography.
S. K. Debnath, S. Mesnager, K. Dey et N. Kundu. Journal
Mediterranean Journal of Mathematics. Volume 18, 2021.
Abstract :
Functional encryption (FE) is an exciting new public key
paradigm that provides solutions to most of the security
challenges of cloud computing in a noninteractive manner.
In the context of FE, inner product functional encryption
(IPFE) is a widely useful cryptographic primitive. It
enables a user with secret key $usk_\mathbf{y}$ associated
to a vector $\mathbf{y}$ to retrieve only
$\langle\mathbf{x},\mathbf{y}\rangle$ from a ciphertext
encrypting a vector $\mathbf{x}$, not beyond that. In the
last few decades, several constructions of IPFE have been
designed based on traditional classical cryptosystems, which
are vulnerable to large enough quantum computers. However,
there are few quantum computer resistants i.e., postquantum
IPFE. Multivariate cryptography is one of the promising
candidates of postquantum cryptography. In this paper, we
propose for the {\em firsttime multivariate
cryptographybased} IPFE. Our work achieves nonadaptive
simulationbased security under the hardness of the MQ
problem.
 Cyclic bent functions and their applications in
sequences. K. Abdukhalikov, C. Ding, S. Mesnager, C.
Tang, et M. Xiong. Journal IEEE transactions Information
Theory, Volume 67 (6), pages 34733485, 2021.
Abstract :
Let $m$ be an even positive integer. A Boolean bent function
$f$ on $\GF{m1} \times \GF {}$ is called a \emph{cyclic
bent function} if for any $a\neq b\in \GF {m1}$ and
$\epsilon \in \GF{}$, $f(ax_1,x_2)+f(bx_1,x_2+\epsilon)$ is
always bent, where $x_1\in \GF {m1}, x_2 \in \GF {}$.
Cyclic bent functions look extremely rare. This paper
focuses on cyclic bent functions on $\GF {m1} \times \GF
{}$ and their applications. The first objective of this
paper is to establish a link between quadratic cyclic bent
functions and a special type of prequasifields, and
construct a class of quadratic cyclic bent functions from
the KantorWilliams prequasifields. The second objective is
to use cyclic bent functions to construct families of
optimal sequences. The results of this paper show that
cyclic bent functions have nice applications in several
fields such as coding theory, symmetric cryptography, and
CDMA communication.
 Solving $X^{q+1}+X+a=0$ over Finite Fields. K.
H. Kim, J. Choe et S. Mesnager. Journal Finite Fields and
Their Applications, Volume 70, 2021.
Abstract :
Solving the equation $P_a(X):=X^{q+1}+X+a=0$ over the finite
field $\GF{Q}$, where $Q=p^n, q=p^k$ and $p$ is a prime,
arises in many different contexts including finite geometry,
the inverse Galois problem [2], the construction of
difference sets with Singer parameters [8], determining
crosscorrelation between msequences [9,15] and the
construction of errorcorrecting codes [5], as well as
speeding up the index calculus method for computing discrete
logarithms on finite fields [11, 12] and on algebraic curves
[18]. Subsequently, in [3, 13, 14, 6, 4, 16, 7, 19], the
$\GF{Q}$zeros of $P_a(X)$ have been studied. It was shown
in [3] that their number is $0$, $1$, $2$ or $p^{\gcd(n,
k)}+1$. Some criteria for the number of the $\GF{Q}$zeros
of $P_a(x)$ were found in [13,14,6,16,19]. However, while
the ultimate goal is to identify all the $\GF{Q}$zeros,
even in the case $p=2$, it was solved only under the
condition $\gcd(n, k)=1$ [16]. We discuss this equation
without any restriction on p and gcd(n,k). Criteria for the
number of the FQzeros of Pa(x) are proved by a new
methodology. For the cases of one or two FQzeros, we
provide explicit ex pressions for these rational zeros in
terms of a. For the case of pgcd(n,k) +1 rational zeros, we
provide a parametrization of such a’s and express the
pgcd(n,k) + 1 rational zeros by using that parametrization.
 Further study of $2$to$1$ mappings over
$F_{2^n}$. K. Li, S. Mesnager et
L. Qu. Journal IEEE transactions Information
Theory, Volume 67 (6), pages 34863496,
2021.
Abstract :
$2$to$1$ mappings over finite fields play an important
role in symmetric cryptography, in particular in the
constructions of APN functions, bent functions, semibent
functions. Very recently, Mesnager and Qu [IEEE Trans. Inf.
Theory 65 (12): 78847895] provided a systematic study of
$2$to$1$ mappings over finite fields. In particular, they
determined all $2$to$1$ mappings of degree at most 4 over
any finite field. In addition, another research direction is
to consider $2$to$1$ polynomials with few terms. Some
results about $2$to$1$ monomials and binomials have been
obtained in [IEEE Trans. Inf. Theory 65 (12): 78847895].
Motivated by their work, in this present paper, we push
further the study of $2$to$1$ mappings, particularly, over
finite fields with characteristic $2$ (binary case being the
most interesting for applications). Firstly, we completely
determine $2$to$1$ polynomials with degree $5$ over
$\gf_{2^n}$ using the well known HasseWeil bound. Besides,
we consider $2$to$1$ mappings with few terms, mainly
trinomials and quadrinomials. Using the multivariate method
and the resultant of two polynomials, we present two classes
of $2$to$1$ trinomials, which explain all the examples of
$2$to$1$ trinomials of the form $x^k+\beta x^{\ell} +
\alpha x\in\gf_{{2^n}}[x]$ with $n\le 7$, and derive twelve
classes of $2$to$1$ quadrinomials with trivial
coefficients over $\gf_{2^n}$.
 A direct proof of APNness of the Kasami functions.
C. Carlet, K. H. Kim et S. Mesnager. Journal Design Codes and
Cryptography, 89(3), pages 441446, 2021.
Abstract :
Using recent results on solving the equation
$X^{2^k+1}+X+a=0$ over a finite field $\GF{2^n}$ provided by
the second and the third authors, we address an open
question raised by the first author in WAIFI 2014 concerning
the APNness of the Kasami functions $x\mapsto
x^{2^{2k}2^k+1}$ with $\gcd(k,n)=1$.
 A construction method of balanced rotation
symmetric Boolean functions on arbitrary even number of
variables with optimal algebraic immunity., S. Mesnager,
S. Su et H. Zhang. Journal Design Codes and Cryptography,
89(1), pp. 117, 2021.
Abstract :
Rotation symmetric Boolean functions incorporate a
superclass of symmetric functions which represent an
attractive corpus for computer investigation. These
functions have been investigated from the viewpoints of
bentness and correlation immunity and have also played a
role in the study of nonlinearity. In the literature, many
constructions of balanced oddvariable rotation symmetric
Boolean functions with optimal algebraic immunity have been
derived. While it seems that the construction of balanced
evenvariable rotation symmetric Boolean functions with
optimal algebraic immunity is very hard work to
breakthrough. In this paper, we present for the first time a
construction of balanced rotation symmetric Boolean
functions on an arbitrary even number of variables with
optimal algebraic immunity by modifying the support of the
majority function. The nonlinearity of the newly constructed
rotation symmetric Boolean functions is also derived.
 Linear codes with onedimensional hull
associated with Gaussian sums., L. Qian,
X. Cao et S. Mesnager. Journal Cryptography and
Communications Discrete Structures, Boolean
Functions and Sequences (CCDS), Volume 13, pages
225243, 2021.
Abstract :
The hull of a linear code over finite fields, the
intersection of the code and its dual, has been of interest
and extensively studied due to its wide applications. For
example, it plays a vital role in determining the complexity
of algorithms for checking permutation equivalence of two
linear codes and for computing the automorphism group of a
linear code. People are interested in pursuing linear codes
with small hulls since, for such codes, the aforementioned
algorithms are very efficient. In this field, Carlet,
Mesnager, Tang and Qi gave a systematic characterization of
LCD codes, i.e, linear codes with null hull. In 2019,
Carlet, Li and Mesnager presented some constructions of
linear codes with small hulls. In the same year, Li and Zeng
derived some constructions of linear codes with
onedimensional hull by using some specific Gaussian sums.
In this paper, we use general Gaussian sums to construct
linear codes with onedimensional hull by utilizing number
fields, which generalizes some results of Li and Zeng
[Constructions of linear codes with onedimensional hull,
IEEE Trans. Inf. Theory, vol. 65, no. 3, 2019] and also of
those presented by Carlet, Li and Mesnager [Linear codes
with small hulls in semiprimitive case, Des. Codes
Cryptogr., Des. Codes Cryptogr., vol. 87, no. 12, 2019]. We
give sufficient conditions to obtain such codes. Notably,
some codes we obtained are optimal or almost optimal
according to the Database. This is the first attempt on
constructing linear codes by general Gaussian sums which
have onedimensional hull and are optimal. Moreover, we also
develop a bound of on the minimum distances of linear codes
we constructed.
 Optimizing Inner Product Masking Scheme by A Coding
Theory Approach., W. Cheng, S. Guilley, C. Carlet, S.
Mesnager et JL. Danger, IEEE Transactions on Information
Forensics and Security, 16, pages 220235, 2021.
Abstract :
Masking is one of the most popular countermeasures to
protect cryptographic implementations against sidechannel
analysis since it is provably secure and can be deployed at
the algorithm level. To strengthen the original Boolean
masking scheme, several works have suggested using schemes
with high algebraic complexity. The Inner Product Masking
(IPM) is one of those. In this paper, we propose a unified
framework to quantitatively assess the sidechannel security
of the IPM in a codingtheoretic approach. Specifically,
starting from the expression of IPM in a coded form, we use
two defining parameters of the code to characterize its
sidechannel resistance. In order to validate the framework,
we then connect it to two leakage metrics (namely
signaltonoise ratio and mutual information, from an
informationtheoretic aspect) and one typical attack metric
(success rate, from a practical aspect) to build a firm
foundation for our framework. As an application, our results
provide ultimate explanations on the observations made by
Balasch et al. at EUROCRYPT’15 and at ASIACRYPT’17, Wang et
al. at CARDIS’16 and Poussier et al. at CARDIS’17 regarding
the parameter effects in IPM, like higher security order in
bounded moment model. Furthermore, we show how to
systematically choose optimal codes (in the sense of a
concrete security level) to optimize IPM by using this
framework. Eventually, we present a simple but effective
algorithm for choosing optimal codes for IPM, which is of
special interest for designers when selecting optimal
parameters for IPM.
 On those multiplicative subgroups of $F_{2^n}^*$.,
C. Carlet et S. Mesnager. Journal of Algebraic combinatorics, 2020
Abstract :
We study those multiplicative subgroups of $F_{2^n}^*$ which
are Sidon sets and/or sumfree sets in the group $(
F_{2^n},+)$. These Sidon and sumfree sets play an important
role relative to the exponents of APN power functions, as
shown by a paper coauthored by the first author.
 Linear codes from vectorial Boolean functions in
the context of algebraic attacks., M. Boumezbeur, S.
Mesnager et K. Guenda, Journal Discrete Mathematics,
Algorithms and Applications (DMAA), Volume 13 (3), 2021
Abstract :
In this paper we study the relationship between vectorial
(Boolean) functions and cyclic codes in the context of
algebraic attacks. We first derive a direct link between the
annihilators of a vectorial function (in univariate form)
and certain $2^{n}$ary cyclic codes (which we show that
they are LCD codes). We also present some properties of
those cyclic codes as well as their weight enumerator. In
addition we generalize the socalled algebraic complement
and study its properties.
 Letters for postquantum cryptography standard
evaluation., J. Ding, S. Mesnager et LC. Wang. Journal
Adv. Math. Commun. 14(1), 2020.
 Thresholdbased postquantum secure verifiable
multisecret sharing for distributed storage blockchain.
S. Mesnager, A. Sinak et O. Yayla. Journal MathematicsMDPI
journals, Special Issue Mathematics, MDPI Journals, Special Issue "The Cryptography of Cryptocurrency", 2020.
Abstract :
Blockchain systems store transaction data in the form of a
distributed ledger where each node stores a copy of all
data, which gives rise to storage issues. It is wellknown
that the tremendous storage and distribution of the block
data are common problems in blockchain systems. In the
literature, some types of secret sharing schemes are
employed to overcome these problems. The secret sharing
method is one of the most significant cryptographic
protocols used to ensure the privacy of the data. The main
purpose of this paper is to improve the recent distributed
storage blockchain systems by proposing an alternative
secret sharing method. We first propose a secure threshold
verifiable multisecret sharing scheme that has the
verification and private communication steps based on
postquantum latticebased hard problems. We then apply the
proposed threshold scheme to the distributed storage
blockchain (DSB) system in order to share transaction data
at each block. In the proposed DSB system, we encrypt the
data block with the AES$256$ encryption algorithm before
distributing it among nodes at each block, and both its
secret key and the hash value of the block are privately
shared among nodes simultaneously by the proposed scheme.
Thereafter, in the DSB system, the encrypted data block is
encoded by the ReedSolomon code, and it is shared among
nodes. We finally analyze the storage and recovery
communication costs and the robustness of the proposed DSB
system. We observe that our approach improves effectively
the recovery communication cost and makes it more robust
compared to the previous DSB systems. It also improves
extremely the storage cost of the traditional blockchain
systems. Furthermore, the proposed scheme brings to the DSB
system the desirable properties such as verification process
and secret communication without private channels in
addition to the known properties of the schemes used in the
previous DSB systems. Because of the flexibility on the
threshold parameter of the scheme, a diverse range of
qualified subsets of nodes in the DSB system can privately
recover the secret values.
 New characterizations and construction methods of
bent and hyperbent Boolean functions., S. Mesnager, B.
Mandal et C. Tang. Journal Discrete Mathematics, 343 (11),
112081, 2020.
Abstract :
In this paper, we first derive a necessary and sufficient
condition for a bent Boolean function by analyzing their
support set. Next, using this condition and the Pless power
moment identities, we propose a construction method of bent
functions of $2k$ variables by a suitable choice of
$2k$dimension subspace of $\mathbb F_2^{2^{2k1}2^{k1}}$.
Further, we extend our results to the socalled hyperbent
functions.
 Solving some affine equations over finite fields.,
S. Mesnager, K. H. Kim, J. H. Choe et D. N. Lee. Journal
Finite Fields and their Applications, 68, 101746, 2020.
Abstract :
Let $l$ and $k$ be two integers such that $l  k$. Define
$T_l^k(X):=X+X^{p^l}+\cdots+X^{p^{k2l}}+X^{p^{kl}}$ and
$S_l^k(X):=XX^{p^l}+\cdots+(1)^{(k/l1)}X^{p^{kl}}$,
where $p$ is any prime. This paper gives explicit
representations of all solutions in $\GF{p^n}$ to the affine
equations $T_l^{k}(X)=a$ and $S_l^{k}(X)=a$, $a\in
\GF{p^n}$. The case $p=2$ was solved very recently in
\cite{MKCL2019}. The results of this paper reveal another
solution.
 On the boomerang uniformity of quadratic
permutations., S. Mesnager, C. Tang et M. Xiong. Journal
Design Codes and Cryptography 88(10), pages 22332246, 2020.
Abstract :
At Eurocrypt'18, Cid, Huang, Peyrin, Sasaki, and Song
introduced a new tool called Boomerang Connectivity Table
(BCT) for measuring the resistance of a block cipher against
the boomerang attack which is an important cryptanalysis
technique introduced by Wagner in 1999 against block
ciphers. Next, Boura and Canteaut introduced an important
parameter related to the BCT for cryptographic Sboxes
called boomerang uniformity. The purpose of this paper is to
present a brief stateoftheart on the notion of boomerang
uniformity of vectorial Boolean functions (or Sboxes) and
provide new results. More specifically, we present a
slightly different but more convenient formulation of the
boomerang uniformity and prove some new identities.
Moreover, we focus on quadratic permutations in even
dimension and obtain general criteria by which they have
optimal BCT. {As a consequence of the new criteria}, two
previously known results can be derived, and many new
quadratic permutations with optimal BCT (optimal means that
the maximal value in the Boomerang Connectivity Table equals
the lowest known differential uniformity) can be found. In
particular, we show that the boomerang uniformity of the
binomial differentially $4$uniform permutations presented
by Bracken, Tan, and Tan equals $4$. Furthermore, we show a
link between the boomerang uniformity and the nonlinearity
for some special quadratic permutations. Finally, we present
a characterization of quadratic permutations with boomerang
uniformity $4$. With this characterization, we show that the
boomerang uniformity of a quadratic permutation with
boomerang uniformity $4$ is preserved by the extended affine
(EA) equivalence.
 Constructions of selforthogonal codes from hulls
of BCH codes and their parameters., Z. Du, C. Li, et S.
Mesnager. Journal IEEE transactions Information Theory 66(11),
pages 67746785, 2020.
Abstract :
Selforthogonal codes are an interesting type of linear
codes due to their wide applications in communication and
cryptography. It is known that selforthogonal codes are
often used to construct quantum errorcorrecting codes,
which can protect quantum information in quantum
computations and quantum communications. Let $\mathcal C$ be
an $[n, k]$ cyclic code over $\Bbb F_q$, where $\Bbb F_q$ is
the finite field of order $q$. The hull of $\mathcal C$ is
defined to be the intersection of the code and its dual. In
this paper, we will employ the defining sets of cyclic codes
to present two general characterizations of the hulls that
have dimension $k1$ or $k^\perp1$, where $k^\perp$ is the
dimension of the dual code $\mathcal C^\perp$. Several
sufficient and necessary conditions for primitive and
projective BCH codes to have $(k1)$dimensional (or
$(k^\perp1)$dimensional) hulls are also developed by
presenting lower and upper bounds on their designed
distances. Furthermore, several classes of selforthogonal
codes are proposed via the hulls of BCH codes and their
parameters are also investigated. The dimensions and minimum
distances of some selforthogonal codes are determined
explicitly. In addition, several optimal codes are obtained.
 Recent results and problems on constructions of
linear codes from cryptographic functions, N. Li et S.
Mesnager, Journal Cryptography and Communications Discrete
Structures, Boolean Functions and Sequences (CCDS) 12(5),
pages 965986, 2020.
Abstract :
Linear codes have a wide range of applications in the data
storage systems, communication systems, consumer electronics
products since their algebraic structure can be analyzed and
they are easy to implement in hardware. How to construct
linear codes with excellent properties to meet the demands
of practical systems becomes a research topic, and it is an
efficient way to construct linear codes from cryptographic
functions. In this paper, we will introduce some methods to
construct linear codes by using cryptographic functions over
finite fields and present some recent results and problems
in this area.
 Solving $x^{2^k+1}+x+a=0$ in $\GF{n}$ with
$\gcd(n,k)=1$, K. H. Kim et S. Mesnager, Journal Finite
Fields and Their Applications (FFA) 63: 101630, 2020.
Abstract :
Let $N_a$ be the number of solutions to the equation
$x^{2^k+1}+x+a=0$ in $\GF {n}$ where $\gcd(k,n)=1$. In 2004,
by Bluher \cite{BLUHER2004} it was known that possible
values of $N_a$ are only 0, 1 and 3. In 2008, Helleseth and
Kholosha \cite{HELLESETH2008} found criteria for $N_a=1$ and
an explicit expression of the unique solution when
$\gcd(k,n)=1$. In 2010 \cite{HELLESETH2010}, the extended
version of \cite{HELLESETH2008}, they also got criteria for
$N_a=0,3$. In 2014, Bracken, Tan and Tan \cite{BRACKEN2014}
presented another criterion for $N_a=0$ when $n$ is even and
$\gcd(k,n)=1$. This paper completely solves this equation
$x^{2^k+1}+x+a=0$ with only the condition $\gcd(n,k)=1$. We
explicitly calculate all possible zeros in $\GF{n}$ of
$P_a(x)$. New criteria for which $a$, $N_a$ is equal to $0$,
$1$ or $3$ are byproducts of our result.
 Minimal linear codes from characteristic functions,
S. Mesnager, Y. Qi, H. Ru et C. Tang, Journal IEEE
Transactions on Information Thepry 66(9), pages 54045413,
2020.
Abstract :
Minimal linear codes have interesting applications in secret
sharing schemes and secure twoparty computation. This paper
uses characteristic functions of some subsets of
$\mathbb{F}_q$ to construct minimal linear codes. By
properties of characteristic functions, we can obtain more
minimal binary linear codes from known minimal binary linear
codes, which generalizes results of Ding et al. [IEEE Trans.
Inf. Theory, vol. 64, no. 10, pp. 65366545, 2018]. By
characteristic functions corresponding to some subspaces of
$\mathbb{F}_q$, we obtain many minimal linear codes, which
generalizes results of [IEEE Trans. Inf. Theory, vol. 64,
no. 10, pp. 65366545, 2018] and [IEEE Trans. Inf. Theory,
vol. 65, no. 11, pp. 70677078, 2019]. Finally, we use
characteristic functions to present a characterization of
minimal linear codes from the defining set method and
present a class of minimal linear codes.
 Constructions of optimal locally recoverable codes
via Dickson polynomials, J. Liu, S. Mesnager et D. Tang.
Journal Design Codes and Cryptography (DCC) 88(9), pages
17591780, 2020
Abstract :
In 2014, Tamo and Barg have presented in a very remarkable
paper a family of optimal linear locally recoverable codes
(LRC codes) that attain the maximum possible distance (given
code length, cardinality, and locality). The key ingredients
for constructing such optimal linear LRC codes are the
socalled $r$good polynomials, where $r$ is equal to the
locality of the LRC code. In 2018, Liu et al. presented two
general methods of designing $r$good polynomials by using
function composition, which led to three new constructions
of $r$good polynomials. Next, Micheli provided a Galois
theoretical framework which allows to construct $r$good
polynomials. The wellknown Dickson polynomials form an
important class of polynomials which have been extensively
investigated in recent years in different contexts. In this
paper, we provide new methods of designing $r$good
polynomials based on Dickson polynomials. Such $r$good
polynomials provide new constructions of optimal LRC codes.
 Solving $x+x^{2^l}+\cdots+x^{2^{ml}}=a$ over
$\GF{2^n}$, S. Mesnager, K. H. Kim, J. H. Choe, D. N.
Lee et D. S. Go. Journal Cryptography and Communications
Discrete Structures, Boolean Functions and Sequences (CCDS)
12(4), pages 809817, 2020.
Abstract :
This paper presents an explicit representation for the
solutions of the equation $\sum_{i=0}^{\frac kl1}x^{2^{li}}
= a \in \GF{2^n}$ for any given positive integers $k,l$ with
$lk$ and $n$, in the closed field ${\overline{\GF{2}}}$ and
in the finite field $\GF{2^n}$. As a byproduct of our
study, we are able to completely characterize the $a$'s for
which this equation has solutions in $\GF{2^n}$.
 On the number of the rational zeros of linearized
polynomials and the secondorder nonlinearity of cubic
Boolean functions, S. Mesnager, K. H. Kim et M. S. Jo,
Journal Cryptography and Communications Discrete Structures,
Boolean Functions and Sequences (CCDS) 12(4), pages 659674,
2020
Abstract :
Determine the number of the rational zeros of any given
linearized polynomial is one of the vital problems in finite
field theory, with applications in modern symmetric
cryptosystems. But, the known general theory for this task
is much far from giving the exact number when applied to a
specific linearized polynomial. The first contribution of
this paper is a better general method to get a more precise
upper bound on the number of rational zeros of any given
linearized polynomial over arbitrary finite field. We
anticipate this method would be applied as a useful tool in
many research branches of finite field and cryptography.
Really we apply this result to get tighter estimations of
the lower bounds on the secondorder nonlinearities of
general cubic Boolean functions, which has been an active
research problem during the past decade. Furthermore, this
paper shows that by studying the distribution of radicals of
derivatives of a given Boolean function one can get a better
lower bound of the secondorder nonlinearity, through an
example of the monomial Boolean functions $g_{\mu}=Tr(\mu
x^{2^{2r}+2^r+1})$ defined over the finite field $\GF{n}$.
 On the MenezesTeskeWeng conjecture, S.
Mesnager, K. H. Kim, J. Choe et C. Tang, Journal Cryptography
and Communications Discrete Structures, Boolean Functions and
Sequences (CCDS) 12 (1), pages 1927, 2020.
Abstract :
In 2003, Alfred Menezes, Edlyn Teske and Annegret Weng
presented a conjecture on properties of the solutions of a
type of quadratic equations over the binary extension
fields, which had been confirmed by extensive experiments
but the proof was unknown until now. We prove that this
conjecture is correct. Furthermore, using this proved
conjecture, we have completely determined the null space of
a class of linearized polynomials.
 Several classes of minimal linear codes with few
weights from weakly regular plateaued function , S.
Mesnager et A. Sinak, Journal IEEE transactions Information
Theory, vol. 66, no. 4, pp. 22962310, 2020.
Abstract :
Minimal linear codes have significant applications in secret
sharing schemes and secure twoparty computation. There are
several methods to construct linear codes, one of which is
based on functions over finite fields. Recently, many
construction methods for linear codes from functions have
been proposed in the literature. In this paper, we
generalize the recent construction methods given by Tang et
al.~in [IEEE Transactions on Information Theory, 62(3),
11661176, 2016] to weakly regular plateaued functions over
finite fields of odd characteristic. We first construct
threeweight linear codes from weakly regular plateaued
functions based on the second generic construction and then
determine their weight distributions. We also give a
punctured version and subcode of each constructed code. We
note that they may be (almost) optimal codes and can be
directly employed to obtain (democratic) secret sharing
schemes, which have diverse applications in the industry. We
next observe that the constructed codes are minimal for
almost all cases and finally describe the access structures
of the secret sharing schemes based on their dual codes.
 Codebooks from generalized bent
$\mathbb{Z}_4$valued quadratic forms , Y. Qi, S.
Mesnager et C. Tang, Journal Discrete Mathematics, 343(3),
111736, 2020.
Abstract :
Codebooks with small innerproduct correlation have
applications in unitary spacetime modulations, multiple
description coding over erasure channels, direct spread code
division multiple access communications, compressed sensing,
and coding theory. It is interesting to construct codebooks
(asymptotically) achieving the Levenshtein bound. This paper
presents a class of generalized bent $\mathbb{Z}_4$valued
quadratic forms, which contains functions proposed by Heng
and Yue (Optimal codebooks achieving the Levenshtein bound
from generalized bent functions over $\mathbb{Z}_4$.
Cryptogr. Commun. 9(1), 4153, 2017). Using these
generalized bent $\mathbb{Z}_4$valued quadratic forms, we
construct optimal codebooks achieving the Levenshtein bound.
These codebooks have parameters $(2^{2m}+2^m,2^m)$ and
alphabet size $6$.
 A class of narrowsense BCH codes over
$\mathbb{F}_q$ of length $\frac{q^m1}{2}$ , X. Lin, S.
Mesnager, Y. Qi et C. Tang, Journal Design Codes and
Cryptography (DCC) 88(2), pages 413427, 2020.
Abstract :
BCH codes with efficient encoding and decoding algorithms
have many applications in communications, cryptography and
combinatorial design. This paper studies a class of linear
codes of length $ \frac{q^m1}{2}$ over $\mathbb{F}_q$ with
special trace representation, where $q$ is an odd prime
power. With the help of the inner distributions of some
subsets of association schemes of quadratic forms, we
determine the weight enumerators of these codes. From
determining some cyclotomic coset leaders $\delta_i$ of
cyclotomic cosets modulo $ \frac{q^m1}{2}$, we prove that
narrowsense BCH codes of length $ \frac{q^m1}{2}$ with
designed distance
$\delta_i=\frac{q^mq^{m1}}{2}1\frac{q^{ \lfloor
\frac{m3}{2} \rfloor+i}1}{2}$ have the corresponding trace
representation, and have the minimal distance $d=\delta_i$
and the Bose distance $d_B=\delta_i$, where $1\leq i\leq
\lfloor \frac{m+11}{6} \rfloor$.
 A Proof of the BeierleKranzLeander Conjecture
related to Lightweight Multiplication in $\mathbb{F}_{2^n}$,
S. Mesnager, K. H. Kim, D. Jo, J. Choe, M. Han et D. N, Lee,
Journal Design Codes and Cryptography (DCC), 88(1), pages
5162, 2020.
Abstract :
Lightweight cryptography is an important tool for building
strong security solutions for pervasive devices with limited
resources. Due to the stringent cost constraints inherent in
extremely large applications, the efficient implementation
of cryptographic hardware and software algorithms is of
utmost importance to realize the vision of generalized
computing. In CRYPTO 2016, Beierle, Kranz and Leander have
considered lightweight multiplication in $\mathds{F}_{2^n}$.
Specifically, they have considered the fundamental question
of optimizing finite field multiplications with one fixed
element and investigated which field representation, that is
which choice of basis, allows for an optimal implementation.
They have left open a conjecture related to an XORcount of
two. Using the theory of linear algebra, we prove in the
present paper that their conjecture is correct.
Consequently, this proved conjecture can be used as a
reference for further developing and implementing
cryptography algorithms in lightweight devices.
 On generalized hyperbent functions, S.
Mesnager, Journal Cryptography and Communications Discrete
Structures, Boolean Functions and Sequences (CCDS)12(3), pages
455468, 2020.
Abstract :
Hyperbent Boolean functions were introduced in 2001 by
Youssef and Gong (and initially proposed by Golomb and Gong
in 1999 as a component of Sboxes) to ensure the security of
symmetric cryptosystems but no cryptographic attack has been
identified until the one on the filtered LFSRs made by
Canteaut and Rotella in 2016. Hyperbent functions have
properties still stronger than the wellknown bent functions
which were introduced by Rothaus and already studied by
Dillon and next by several researchers in more than four
decades. Hyperbent functions are very rare and whose
classification is still elusive. Therefore, not only their
characterization, but also their generation are challenging
problems. Recently, an important direction in the theory of
hyperbent functions was the extension of Boolean hyperbent
functions to whose codomain is the ring of integers modulo a
power of a prime, that is, generalized hyperbent functions.
In this paper, we synthesize previous studies on generalized
hyperbent functions in a unified framework. We provide two
characterizations of generalized hyperbent functions in
terms of their digits. We establish a complete
characterization of a family of generalized hyperbent
functions defined over spreads and establish a link between
vectorial hyperbent functions found recently and that
family.
 On twotoone mappings over finite fields, S.
Mesnager et L. Qu, Journal IEEE transactions Information
Theory, 65(12), pages 78847895, 2019.
Abstract :
Twotoone ($2$to$1$) mappings over finite fields play an
important role in symmetric cryptography. In particular they
allow to design APN functions, bent functions and semibent
functions. In this paper we provide a systematic study of
twotoone mappings that are defined over finite fields. We
characterize such mappings by means of the Walsh transforms.
We also present several constructions, including an AGWlike
criterion, constructions with the form of
$x^rh(x^{(q1)/d})$, those from permutation polynomials,
from linear translators and from APN functions. Then we
present $2$to$1$ polynomial mappings in classical classes
of polynomials: linearized polynomials and monomials, low
degree polynomials, Dickson polynomials and
MullerCohenMatthews polynomials, etc. Lastly, we show
applications of $2$to$1$ mappings over finite fields for
constructions of bent Boolean and vectorial bent functions,
semibent functions, planar functions and permutation
polynomials. In all those respects, we shall review what is
known and provide several new results.
 Multiple characters transforms and generalized
Boolean functions, S. Mesnager, C. Riera et P. Stanica,
Journal Cryptography and Communications Discrete Structures,
Boolean Functions and Sequences (CCDS) 11(6), pages 12471260,
2019.
Abstract :
In this paper we investigate generalized Boolean functions
whose spectrum is flat with respect to a set of
WalshHadamard transforms defined using various complex
primitive roots of $1$. We also study some differential
properties of the generalized Boolean functions in even
dimension defined in terms of these different characters. We
show that those functions have similar properties to the
vectorial bent functions. We next clarify the case of gbent
functions in odd dimension. As a byproduct of our proofs,
more generally, we also provide several results about
plateaued functions. Furthermore, we find characterizations
of plateaued functions with respect to different characters
in terms of second derivatives and fourth moments.
 Several new classes of selfdual bent functions
derived from involutions, G. Luo, X. Cao et S. Mesnager,
Journal Cryptography and Communications Discrete Structures,
Boolean Functions and Sequences (CCDS), 1(6), pages 12611273,
2019.
Abstract :
Bent functions are a kind of Boolean function which have the
maximum Hamming distance to linear and affine functions,
they have some interesting applications in combinatorics,
coding theory, cryptography and sequences. However,
generally speaking, how to find new bent functions is a hard
work and is a hot research project during the past decades.
A subclass of bent functions that has received attention
since Dillon's seminal thesis (1974) is the subclass of
those Boolean functions that are equal to their dual (or
Fourier transform in Dillon's terminology): the socalled
self dual bent functions. In this paper, we propose a
construction of involutions from linear translators, and
provide two methods for constructing new involutions by
utilizing some given involutions. With the involutions
presented in this paper, several new classes of selfdual
bent functions are produced.
 Minimal Linear Codes with Few Weights and Their
Secret Sharing, S. Mesnager, A. Sinak, O. Yayla,
International Journal of Information Security Science, Vol.8,
No.3, pages 4452, 2019.
Abstract :
Minimal linear codes with few weights have significant
applications in secure twoparty computation and secret
sharing schemes. In this paper, we construct twoweight and
threeweight minimal linear codes by using weakly regular
plateaued functions in the wellknown construction method
based on the second generic construction. We also give
punctured codes and subcodes for some constructed minimal
codes. We finally obtain secret sharing schemes with high
democracy from the dual codes of our minimal codes.
 Linear codes with small hulls in semiprimitive
case, C. Carlet, C. Li et S. Mesnager, Journal Design
Codes and Cryptography (DCC), 87(12), pages 28132834, 2019.
Abstract :
The hull of a linear code is defined to be the intersection
of the code and its dual, and was originally introduced to
classify finite projective planes. The hull plays an
important role in determining the complexity of algorithms
for checking permutation equivalence of two linear codes and
computing the automorphism group of a linear code. It has
been shown that these algorithms are very effective in
general if the size of the hull is small. It is clear that
the linear codes with the smallest hull are LCD codes and
with the second smallest hull are those with onedimensional
hull. In this paper, we employ character sums in
semiprimitive case to construct LCD codes and linear codes
with onedimensional hull from cyclotomic fields and
multiplicative subgroups of finite fields. Some sufficient
and necessary conditions for these codes are obtained, where
prime ideal decompositions of prime $p$ in cyclotomic fields
play a key role. In addition, we show the nonexistence of
these codes in some cases.
 Further study on the maximum number of bent
components of vectorial functions, S. Mesnager, F.
Zhang, C. Tang et Y. Zhou, Journal Design Codes and
Cryptography (DCC), 87(11): 25972610, 2019.
Abstract :
In 2018, Pott et al. have studied in [IEEE Transactions on
Information Theory. Volume: 64, Issue: 1, 2018] the maximum
number of bent components of vectorial functions. They have
presented many nice results and suggested several open
problems in this context. This paper is in the continuation
of their study in which we solve two open problems raised by
Pott et al. and partially solve an open problem raised by
the same authors. Firstly, we prove that for a vectorial
function, the property of having the maximum number of bent
components is invariant under the socalled CCZ equivalence.
Secondly, we prove the nonexistence of APN plateaued
functions having the maximum number of bent components. In
particular, quadratic APN functions cannot have the maximum
number of bent components. Finally, we present some
sufficient conditions that the vectorial function defined
from $\mathbb{F}_{2^{2k}}$ to $\mathbb{F}_{2^{2k}}$ by its
univariate representation: $$ \alpha
x^{2^i}\left(x+x^{2^k}+\sum\limits_{j=1}^{\rho}\gamma^{(j)}x^{2^{t_j}}
+\sum\limits_{j=1}^{\rho}\gamma^{(j)}x^{2^{t_j+k}}\right)$$
has the maximum number of { bent components, where $\rho\leq
k$}. Further, we show that the differential spectrum of the
function $
x^{2^i}(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}})$
(where $i,t_1,t_2$ satisfy some conditions) is different
from the binomial function $F^i(x)= x^{2^i}(x+x^{2^k})$
presented in the article of Pott et al.
 Some (almost) optimally extendable linear codes,
C. Carlet, C. Li et S. Mesnager, Journal Design Codes and
Cryptography, 87(12), pages 28132834, 2019
Abstract :
Sidechannel attacks (SCA) and fault injection attacks (FIA)
are nowadays important cryptanalysis methods on the
implementations of block ciphers, which represent huge
threats. Direct sum masking (DSM) has been proposed to
protect the sensitive data stored in registers against both
SCA and FIA. It uses two linear codes $\mathcal C$ and
$\mathcal D$ whose sum is direct and equals $\Bbb F_q^n$.
The resulting security parameter is the pair $(d(\mathcal
C)1,d({\mathcal D}^\perp)1)$. For being able to protect
not only the sensitive input data stored in registers
against SCA and FIA but the whole algorithm (which is
required at least in software applications), it is necessary
to change $\mathcal C$ and $\mathcal D$ into $\mathcal
C^\prime$, which has the same minimum distance as $\mathcal
C$, and $\mathcal D^\prime$, which may have smaller dual
distance than $\mathcal D$. Precisely, $\mathcal D^\prime$
is the linear code obtained by appending on the right of its
generator matrix the identity matrix with the same number of
rows. It is then highly desired to construct linear codes
$\mathcal D$ such that $d({\mathcal D^\prime}^\perp)$ is
very close to $d({\mathcal D}^\perp)$. In such case, we say
that $\mathcal D$ is almost optimally extendable (and is
optimally extendable if $d({\mathcal D^\prime}^\perp)=
d(\mathcal D^\perp)$). In general, it is notoriously
difficult to determine the minimum distances of the codes
$\mathcal D^\perp$ and ${\mathcal D^\prime}^\perp$
simultaneously.
 Weightwise perfectly balanced functions with high
weightwise nonlinearity profil, J. Liu et S. Mesnager,
Journal Designs, Codes and Cryptography (DCC) 87(8), pages
17971813, 2019.
Abstract :
Boolean functions satisfying good cryptographic criteria
when restricted to the set of vectors with constant Hamming
weight play an important role in the recent FLIP stream
cipher~\cite{Meaux2016}. In this paper, we propose a large
class of weightwise perfectly balanced (WPB) functions,
which is $2$rotation symmetric. This new class of WPB
functions is not extended affinely equivalent to the known
constructions. We also discuss the weightwise nonlinearity
profile of these functions, and present general lower bounds
on $k$weightwise nonlinearity, where $k$ is a power of $2$.
Moreover, we exhibit a subclass of the family. By a
recursive lower bound, we show that these subclass of WPB
functions have very high weightwise nonlinearity profile
 On qary plateaued functions over $F_q$ and their
explicit characterizations, S. Mesnager, F. Ozbudak, A.
Sinak et G. Cohen, European Journal of Combinatorics 80, pages
7181, 2019
Abstract :
Plateaued and bent functions play a significant role in
cryptography, sequence theory, coding theory and
combinatorics. In 1997, Coulter and Matthews redefined bent
functions over any finite field $\F_q$ where $q$ is a prime
power, and established their properties. The objective of
this work is to redefine the notion of plateaued functions
over $\F_q$, and to present several explicit
characterizations of those functions. We first give, over
$\F_q$, the notion of $q$ary plateaued functions, which
relies on the concept of the WalshHadamard transform in
terms of canonical additive character of $\F_q$. We then
give a concrete example of $q$ary plateaued function, that
is not vectorial $p$ary plateaued function. This suggests
that the study of plateauedness is also significant for
$q$ary functions over $\F_q$. We finally characterize
$q$ary plateaued functions in terms of derivatives, Walsh
power moments and autocorrelation functions.
 On the nonlinearity of Boolean functions with
restricted input, S. Mesnager, Z. Zhou et C. Ding,
Journal Cryptography and Communications Discrete Structures,
Boolean Functions and Sequences (CCDS), 11(1) pages 6376,
2019.
Abstract :
Very recently, Carlet, M\'eaux and Rotella have studied the
main cryptographic features of Boolean functions when, for a
given number $n$ of variables, the input to these functions
is restricted to some subset $E$ of $\F^n$. Their study
includes the particular case when $E$ equals the set of
vectors of fixed Hamming weight, which is important in the
robustness of the Boolean function involved in the FLIP
stream cipher. In this paper we focus on the nonlinearity of
Boolean functions with restricted input and present new
results related to the analysis of this nonlinearity
improving the upper bound given by Carlet et al.
 Linear codes from weakly regular plateaued
functions and their secret sharing schemes, S. Mesnager,
F. Ozbudak et A. Sinak, Journal Designs, Codes and
Cryptography (DCC), Volume 87, Issue 2–3, pages 463–480, 2019.
Abstract :
Linear codes, the most significant class of codes in coding
theory, have diverse applications in secret sharing schemes,
authentication codes, communication, data storage devices
and consumer electronics. The main objectives of this paper
are twofold: to construct threeweight linear codes from
plateaued functions over finite fields, and to analyze the
constructed linear codes for secret sharing schemes. To do
the first one, we generalize the recent contribution of
Mesnager given in [Cryptography and Communications 9(1),
7184, 2017]. We first introduce the notion of (non)weakly
regular plateaued functions over $\F_p$, with $p$ an odd
prime. We next construct threeweight linear $p$ary (resp.
binary) codes from weakly regular $p$ary plateaued (resp.
Boolean plateaued) functions and determine their weight
distributions. We finally show that the constructed linear
codes can be used to construct secret sharing schemes with
``nice'' access structures. To the best of our knowledge,
the construction of linear codes from plateaued functions
over $\F_p$, with $p$ an odd prime, is studied in this paper
for the first time in the literature.
 New characterization and parametrization of LCD
codes, C. Carlet, S. Mesnager, C. Tang et Y. Qi, Journal
IEEE Transactions on Information TheoryIT, 65(1) pages 3949,
2019.
Abstract :
Linear complementary dual (LCD) cyclic codes were referred
historically to as reversible cyclic codes, which had
applications in data storage. Due to a newly discovered
application in cryptography, there has been renewed interest
in LCD codes. In particular, it has been shown that binary
LCD codes play an important role in implementations against
sidechannel attacks and fault injection attacks. In this
paper, we first present a new characterization of binary LCD
codes in terms of their orthogonal or symplectic basis.
Using such a characterization, we solve a conjecture
proposed by Galvez et al. on the minimum distance of binary
LCD codes. Next, we consider the action of the orthogonal
group on the set of all LCD codes, determine all possible
orbits of this action, derive simple closed formulas of the
size of the orbits, and present some asymptotic results on
the size of the corresponding orbits. Our results show that
almost all binary LCD codes are oddlike codes with oddlike
duals, and about half of $q$ary LCD codes have orthonormal
basis, where $q$ is a power of an odd prime.
 On $sigma$LCD codes, C. Carlet, S. Mesnager,
C. Tang et Y. Qi, Journal IEEE Transactions on Information
TheoryIT. Volume 65, Issue 3, pages 16941704, 2019.
Abstract :
Linear complementary pairs (LCP) of codes play an important
role in armoring implementations against sidechannel
attacks and fault injection attacks. One of the most common
ways to construct LCP of codes is to use Euclidean linear
complementary dual (LCD) codes. In this paper, we first
introduce the concept of linear codes with $\sigma$
complementary dual ($\sigma$LCD), which includes known
Euclidean LCD codes, Hermitian LCD codes, and Galois LCD
codes. Like Euclidean LCD codes, $\sigma$LCD codes can also
be used to construct LCP of codes. We show that, for $q >
2$, all $q$ary linear codes are $\sigma$LCD and that, for
every binary linear code $\mathcal C$, the code $\{0\}\times
\mathcal C$ is $\sigma$LCD. Further, we study deeply
$\sigma$LCD generalized quasicyclic (GQC) codes. In
particular, we provide characterizations of $\sigma$LCD GQC
codes, selforthogonal GQC codes and selfdual GQC codes,
respectively. Moreover, we provide constructions of
asymptotically good $\sigma$LCD GQC codes. Finally, we
focus on $\sigma$LCD abelian codes and prove that all
abelian codes in a semisimple group algebra are
$\sigma$LCD. The results derived in this paper extend those
on the classical LCD codes and show that $\sigma$LCD codes
allow the construction of LCP of codes more easily and with
more flexibility.
 Linear codes over $F_q$ are equivalent to LCD codes
for $q>3$, C. Carlet, S. Mesnager, C. Tang, Y. Qi et
R. Pellikaan, Journal IEEE Transactions on Information
TheoryIT, Volume 64, Issue 4, pages 30103017, 2018.
Abstract :
Linear codes with complementary duals (abbreviated LCD) are
linear codes whose intersection with their dual are trivial.
When they are binary, they play an important role in
armoring implementations against sidechannel attacks and
fault injection attacks. Nonbinary LCD codes in
characteristic 2 can be transformed into binary LCD codes by
expansion. In this paper, we introduce a general
construction of LCD codes from any linear codes. Further, we
show that any linear code over $\mathbb F_{q} (q>3)$ is
equivalent to a Euclidean LCD code and any linear code over
$\mathbb F_{q^2} (q>2)$ is equivalent to a Hermitian LCD
code. Consequently an $[n,k,d]$linear Euclidean LCD code
over $\mathbb F_q$ with $q>3$ exists if there is an
$[n,k,d]$linear code over $\mathbb F_q$ and an
$[n,k,d]$linear Hermitian LCD code over $\mathbb F_{q^2}$
with $q>2$ exists if there is an $[n,k,d]$linear code
over $\mathbb F_{q^2}$. Hence, when $q>3$ (resp.
$q>2$) $q$ary Euclidean (resp. $q^2$ary Hermitian) LCD
codes possess the same asymptotical bound as $q$ary linear
codes (resp. $q^2$ary linear codes). This gives a direct
proof that every triple of parameters $[n,k,d]$ which is
attainable by linear codes over $\mathbb F_{q}$ with
$q>3$ (resp. over $\mathbb F_{q^2}$ with $q>2$) is
attainable by Euclidean LCD codes (resp. by Hermitian LCD
codes). In particular there exist families of $q$ary
Euclidean LCD codes ($q>3$) and $q^2$ary Hermitian LCD
codes ($q>2$) exceeding the asymptotical
GilbertVarshamov bound. Further, we give a second proof of
these results using the theory of Gr\"obner bases. Finally,
we present a new approach of constructing LCD codes by
extending linear codes.
 $2$correcting Lee Codes: (Quasi)Perfect Spectral
Conditions and Some Constructions,S. Mesnager, C. Tang
et Y. Qi, Journal IEEE Transactions on Information TheoryIT,
Volume 64, Issue 4, pages 30313041, 2018.
Abstract :
Let $p$ be an odd prime. Recently, Camarero and
Mart\'{\i}nez (in ``Quasiperfect Lee codes of radius $2$
and arbitrarily large dimension", IEEE Trans. Inform.
Theory, vol. 62, no. 3, 2016) constructed some $p$ary
$2$quasiperfect Lee codes for $p\equiv \pm 5 \pmod{12}$.
In this paper, some infinite classes of $p$ary
$2$quasiperfect Lee codes for any odd prime $p$ with
flexible length and dimension are presented. More
specifically, we provide a new method for constructing
quasiperfect Lee codes. Our approach uses subsets derived
from some quadratic curves over finite fields (in odd
characteristic) to obtain two classes of $2$quasiperfect
Lee codes defined in the space $\mathbb{Z}_p^n$ for
$n=\frac{p^k+1}{2}$ $(\text{with} ~p\equiv 1, 5 \pmod{12}
\text{ and } k \text{ is any integer}, \text{ or } p\equiv
1, 5 \pmod{12} \text{ and } k \text{ is an even integer})$
and $n=\frac{p^k1}{2}$ $(\text{with }p\equiv 1, 5
\pmod{12}, k \text{ is an odd integer} \text{ and }
p^k>12)$. Our codes encompass the $p$ary ($p\equiv \pm 5
\pmod{12}$) $2$quasiperfect Lee codes constructed by
Camarero and Mart\'{\i}nez. Furthermore, we prove that the
related Cayley graphs are Ramanujan or almost Ramanujan
using Kloosterman sums. This generalizes the work of Bibak,
Kapron, and Srinivasan (in ``The Cayley graphs associated
with some quasiperfect Lee codes are Ramanujan graphs",
IEEE Trans. Inform. Theory, vol. 62, no. 11, 2016) from the
case $p\equiv 3 \pmod{4}$ and $k=1$ to the case of any odd
prime $p$ and positive integer $k$. Finally, we derive some
necessary conditions with the exponential sums of all
$2$perfect codes and $2$quasiperfect codes, and present a
heuristic algorithm for constructing $2$perfect codes and
$2$quasiperfect codes. Our results show that, in general,
the Cayley graphs associated with $2$perfect codes are
Ramanujan. From the algorithm, some new 2quasiperfect Lee
codes different from those constructed from quadratic curves
are given. The Lee codes presented in this paper have
applications in constrained and partialresponse channels,
flash memories, and decision diagrams.
 Further results on generalized bent functions and
their complete characterization, S. Mesnager, C. Tang,
Y. Qi, L. Wang, B. Wu et K. Feng , Journal IEEE Transactions
on Information TheoryIT. 64(7): 54415452, 2018.
Abstract :
This paper contributes to increase our knowledge on
generalized bent functions (including generalized bent
Boolean functions and generalized $p$ary bent functions
with odd prime $p$) by bringing new results on their
characterization and construction, in arbitrary
characteristic. More specifically, we first investigate
relations between generalized bent functions and bent
functions by the decomposition of generalized bent
functions. This enables us to completely characterize
generalized bent functions and $\mathbb Z_{p^k}$bent
functions by some affine space associated with the
generalized bent functions. We also present the relationship
between generalized bent Boolean functions with odd
variables and generalized bent Boolean functions with even
variables. Based on the wellknown MaioranaMcFarland class
of Boolean functions, we present some infinite classes of
generalized bent Boolean functions. In addition, we
introduce a class of generalized hyperbent functions that
can be seen as generalized Dillon's $PS$ functions. Finally
we solve an open problem related to the description of the
dual function of a weakly regular generalized bent Boolean
function with odd variables via the WalshHadamard transform
of their component functions, and we generalize these
results to the case of odd prime.
 Euclidean and Hermitian LCD MDS codes, C.
Carlet, S. Mesnager, C. Tang et Y. Qi, Journal Des. Codes
Cryptography 86(11), pages 26052618, 2018.
Abstract :
Linear codes with complementary duals (abbreviated LCD) are
linear codes whose intersection with their dual is trivial.
When they are binary, they play an important role in
armoring implementations against sidechannel attacks and
fault injection attacks. Nonbinary LCD codes in
characteristic 2 can be transformed into binary LCD codes by
expansion. On the other hand, being optimal codes, maximum
distance separable codes (abbreviated MDS) are of much
interest from many viewpoints due to their theoretical and
practical properties. However, little work has been done on
LCD MDS codes. In particular, determining the existence of
$q$ary $[n,k]$ LCD MDS codes for various lengths $n$ and
dimensions $k$ is a basic and interesting problem. In this
paper, we firstly study the problem of the existence of
$q$ary $[n,k]$ LCD MDS codes and solve it for the Euclidean
case. More specifically, we show that for $q>3$ there
exists a $q$ary $[n,k]$ Euclidean LCD MDS code, where $0\le
k \le n\le q+1$, or, $q=2^{m}$, $n=q+2$ and $k= 3 \text{ or
} q1$. Secondly, we investigate several constructions of
new Euclidean and Hermitian LCD MDS codes. Our main
techniques in constructing Euclidean and Hermitian LCD MDS
codes use some linear codes with small dimension or
codimension, selforthogonal codes and generalized
ReedSolomon codes.
 New constructions of optimal locally recoverable
codes via good polynomials, J. Liu, S. Mesnager et L.
Chen, Journal IEEE Transactions on Information TheoryIT,
64(2), pages 889899, 2018.
Abstract :
In recent literature, a family of optimal linear locally
recoverable codes (LRC codes) that attain the maximum
possible distance (given code length, cardinality, and
locality) is presented. The key ingredient for constructing
such optimal linear LRC codes is the socalled rgood
polynomials, where r is equal to the locality of the LRC
code. However, given a prime p, known constructions of
rgood polynomials over some extension field of Fp exist
only for some special integers r, and the problem of
constructing optimal LRC codes over small field for any
given locality is still open. In this paper, by using
function composition, we present two general methods of
designing good polynomials, which lead to three new
constructions of rgood polynomials. Such polynomials bring
new constructions of optimal LRC codes. In particular, our
constructed polynomials as well as the power functions yield
optimal (n; k; r) LRC codes over Fq for all positive
integers r as localities, where q is near the code length n.
 Complementary dual algebraic geometry codes, S.
Mesnager, C. Tang et Y. Qi, Journal IEEE Transactions on
Information TheoryIT 64(4), pages 23902397, 2018.
Abstract :
Linear complementary dual (LCD) codes is a class of linear
codes introduced by Massey in 1964. LCD codes have been
extensively studied in literature recently. In addition to
their applications in data storage, communications systems,
and consumer electronics, LCD codes have been employed in
cryptography. More specifically, it has been shown that LCD
codes can also help improve the security of the information
processed by sensitive devices, especially against socalled
sidechannel attacks (SCA) and fault noninvasive attacks.
In this paper, we are interested in the construction of
particular algebraic geometry (AG) LCD codes which could be
good candidates to be resistant against SCA. We firstly
provide a construction scheme for obtaining LCD codes from
any algebraic curve. Then, some explicit LCD codes from
elliptic curves are presented. MDS codes are of the most
importance in coding theory due to their theoretical
significance and practical interests. In this paper, all the
constructed LCD codes from elliptic curves are MDS or almost
MDS. Some infinite classes of LCD codes from elliptic curves
are optimal due to the Griesmer bound. Finally, we also
derive some explicit LCD codes from hyperelliptic curves and
Hermitian curves.
 Bent functions from involutions over $F_2^n$,
R. Coulter et S. Mesnager, Journal IEEE Transactions on
Information TheoryIT, Volume 64, Issue 4, pages 29792986,
2018.
Abstract :
Bent functions are maximally nonlinear Boolean functions.
Introduced by Rothaus and first examined by Dillon, these
important functions have subsequently been studied by many
researchers over the last four decades. Since a complete
classification of bent functions appears elusive, many
researchers concentrate on methods for constructing bent
functions. In this paper, we investigate constructions of
bent functions from involutions over finite fields in even
characteristic. We present a generic construction technique,
study its equivalence issues and show that linear
involutions (which are an important class of permutations)
over finite fields give rise to bent functions in bivariate
representations. In particular, we exhibit new constructions
of bent functions involving binomial linear involutions
whose dual functions are directly obtained without
computation. The existence of bent functions from
involutions relies heavily on solving systems of equations
over finite fields.
 On the $p$ary (Cubic)Bent and Plateaued
(Vectorial) Functions, S. Mesnager, F. Ozbudak et A.
Sinak , Journal Des. Codes Cryptography 86(8), pages
18651892, 2018.
Abstract :
Plateaued functions play a significant role in cryptography,
sequences for communications, and the related combinatorics
and designs. Comparing to their importance, those functions
have not been studied in detail in a general framework. Our
motivation is to bring further results on the
characterizations of bent and plateaued functions, and to
introduce new tools which allow us firstly a better
understanding of their structure and secondly to get methods
for handling and designing such functions. We first
characterize bent functions in terms of all even moments of
the Walsh transform, and then plateaued (vectorial)
functions in terms of the value distribution of the
secondorder derivatives. Moreover, we devote to cubic
functions the characterization of plateaued functions in
terms of the value distribution of the secondorder
derivatives, and hence this reveals nonexistence of
homogeneous cubic bent (and also (homogeneous) cubic
plateaued for some cases) functions in odd characteristic.
We use a rank notion which generalizes the rank notion of
quadratic functions. This rank notion reveals new results
about (homogeneous) cubic plateaued functions. Furthermore,
we observe nonexistence of a function whose absolute Walsh
transform takes exactly $3$ distinct values (one being
zero). We finally provide a new class of functions whose
absolute Walsh transform takes exactly $4$ distinct values
(one being zero).
 Statistical integral distinguisher with
multistructure and its application on AESlike ciphers,
T. Cui, H. Chen, S. Mesnager, L. Sun et M. Wang. Journal
Cryptography and Communications 10(5), pages 755776, 2018.
Abstract :
Integral attack is one of the most powerful tool in the
field of symmetric ciphers. In order to reduce the time
complexity of original integral one, Wang \textit{et al.}
firstly proposed a statistical integral distinguisher at
FSE'16. However, they don't consider the cases that there
are several integral properties on output and multiple
structures of data should be used at the same time. In terms
of such case, we put forward a new statistical integral
distinguisher, which enables us to reduce the data
complexity comparing to the traditional integral ones under
multiple structures. As illustrations, we use it into the
knownkey distinguishers on AESlike ciphers including AES
and the permutations of Whirlpool, PHOTON and Gr\o stl256
hash functions based on the Gilbert's work at ASIACRYPT'14.
These new distinguishers are the best ones comparing with
previous ones under knownkey setting. Moreover, we propose
a secretkey distinguisher on 5round AES under
chosenciphertext mode. Its data, time and memory
complexities are $2^{114.32}$ chosen ciphertexts, $2^{110}$
encryptions and $2^{33.32}$ blocks. This is the best
integral distinguisher on AES with secret Sbox under
secretkey setting so far.
 Classification of bent monomials, constructions of
bent multinomials and upper bounds on the nonlinearity of
vectorial functions, Y. Xu, C. Carlet, S. Mesnager et C.
Wu, Journal IEEE Transactions on Information TheoryIT, Vol.
64, Issue 1, pages 367383, 2018.
Abstract :
The paper is composed of two main parts related to the
nonlinearity of vectorial functions. The first part is
devoted to maximally nonlinear $(n,m)$functions (the
socalled bent vectorial functions) which contribute to an
optimal resistance to both linear and differential attacks
on symmetric cryptosystems. They can be used in block
ciphers at the cost of additional
diffusion/compression/expansion layers, or as building
blocks for the construction of substitution boxes (Sboxes)
and they are also useful for constructing robust codes and
algebraic manipulation detection codes. A main issue on bent
vectorial functions is to characterize bent monomial
functions $Tr_{m}^n (\lambda x^d)$ from $\mathbb{F}_{2^n}$
to $\mathbb{F}_{2^m}$ (where $m$ is a divisor of $n$)
leading to a classification of those bent monomials. We also
treat the case of functions with multiple trace terms
involving general results and explicit constructions.
Furthermore, we investigate some open problems raised by
Pasalic et al. and Muratovi\'cRibi\'c et al. in a series of
papers on vectorial functions. The second part is devoted to
the nonlinearity of $(n,m)$functions. No tight upper bound
is known when $m$ is between $frac n2$ and $n$. The covering
radius bound is the only known upper bound in this range
(the SidelnikovChabaudVaudenay bound coincides with it
when $m=n1$ and it has no sense when $m$ is less than
$n1$). Finding better bounds is an open problem since the
90s. Moreover, no bound has been found during the last 23
years which improve upon the covering radius bound for a
large part of $(n,m)$functions. We derive such upper bounds
for functions which are sufficiently unbalanced or which
satisfy some conditions. These upper bounds imply some
necessary conditions for vectorial functions to have large
nonlinearity.
 Generalized plateaued functions and admissible
(plateaued) functions, S. Mesnager, C. Tang et Y. Qi,
Journal IEEE Transactions on Information TheoryIT, Vol. 61,
Issue 10, pages 61396148, 2017.
Abstract :
Plateaued functions are very important cryptographic
functions due to their various desirable cryptographic
characteristics. We point out that plateaued functions are
more general than bent functions (that is, functions with
maximum nonlinearity). Some Boolean plateaued functions have
large nonlinearity, which provides protection against fast
correlation attacks when they are used as combiners or
filters in stream ciphers, and contributes, when they are
the component functions of the substitution boxes in block
ciphers, to protection against linear cryptanalysis. Pary
plateaued functions have attracted recently some attention
in the literature and many activities on generalized pary
functions have been carried out. This paper increases our
knowledge on plateaued functions in the general context of
generalized pary functions. We firstly introduce two new
versions of plateaued functions, which we shall call
generalized plateaued functions and admissible plateaued
functions. The generalized plateaued functions extends the
standard notion of plateaued pary functions to those whose
outputs are in the ring Zpk . Next, we study the generalized
plateaued functions and use admissible plateaued functions
to characterize the generalized plateaued functions by means
of their components. Finally, we provide for the first time
two constructions of generalized plateaued functions. In
particular, we generalize a known secondary construction of
binary generalized bent functions and derive constructions
of binary generalized plateaued functions with different
amplitude.
 Decomposing generalized bent and hyperbent
functions,T. Martinsen, W. Meidl, S. Mesnager et P.
Stanica, Journal IEEE Transactions on Information TheoryIT,
Vol 63, Issue 12, pages 78047812, 2017.
Abstract :
In this paper we introduce generalized hyperbent functions
from $\F_{2^n}$ to $\Z_{2^k}$, and investigate
decompositions of generalized (hyper)bent functions. We show
that generalized (hyper)bent functions $f$ from $\F_{2^n}$
to $\Z_{2^k}$ consist of components which are generalized
(hyper)bent functions from $\F_{2^n}$ to $\Z_{2^{k^\prime}}$
for some $k^\prime less than k$. For even $n$, most notably
we show that the ghyperbentness of $f$ is equivalent to the
hyperbentness of the components of $f$ with some conditions
on the WalshHadamard coefficients. For odd $n$, we show
that the Boolean functions associated to a generalized bent
function form an affine space of semibent functions. This
complements a recent result for even $n$, where the
associated Boolean functions are bent.
 Fast algebraic immunity of Boolean functions,
S. Mesnager et G. Cohen, Journal Advances in Mathematics of
Communications (AMC), Vol 11, No. 2, pages 373377, 2017.
Abstract :
Since 1970, Boolean functions have been the focus of a lot
of at tention in cryptography. An important topic in
symmetric ciphers concerns the cryptographic properties of
Boolean functions and constructions of Boolean functions
with good cryptographic properties, that is, good resistance
to known attacks. An important progress in cryptanalysis
areas made in 2003 was the introduction by Courtois and
Meier of algebraic attacks and fast algebraic at tacks
which are very powerful analysis concepts and can be applied
to almost all cryptographic algorithms. To study the
resistance against algebraic attacks, the notion of
algebraic immunity has been introduced. In this paper, we
use a parameter introduced by Liu and al., called fast
algebraic immunity, as a tool to measure the resistance of a
cryptosystem (involving Boolean functions) to fast algebraic
attacks. We prove an upper bound on the fast algebraic im
munity. Using our upper bound, we establish the weakness of
trace inverse functions against fast algebraic attacks
confiming a recent result of Feng and Gong.
 On constructions of bent, semibent and five
valued spectrum functions from old bent functions, S.
Mesnager et F. Zhang, Journal Advances in Mathematics of
Communications (AMC), Vol 11, No. 2, pages 339345, 2017.
Abstract :
The paper presents methods for designing functions having
many applications in particular to construct linear codes
with few weights. The former codes have several applications
in secret sharing, authentication codes, association schemes
and strongly regular graphs. We firstly provide new
secondary constructions of bent functions generalizing the
wellknown Rothaus' constructions as well as their dual
functions. From our generalization, we show that we are able
to compute the dual function of a bent function built from
Rothaus' construction. Next we present a result leading to a
new method for constructing semibent functions and few
Walsh transform values functions built from bent functions.
 On construction of bent functions involving
symmetric functions and their duals, S. Mesnager, F.
Zhang et Y. Zhou, Journal Advances in Mathematics of
Communications (AMC), Vol 11, No. 2, pages 347352, 2017.
Abstract :
In this paper, we firstly compute the dual functions of
elemen tary symmetric bent functions. Next, we derive a new
secondary construction of bent functions (given with their
dual functions) involving symmetric bent functions, leading
to a generalization of the wellknow Rothaus' construction.
 Explicit constructions of bent functions from
pseudoplanar functions, K. Abdukhalikov et S. Mesnager,
Journal Advances in Mathematics of Communications (AMC), Vol
11, No. 2, pages 293299, 2017.
Abstract :
We investigate explicit constructions of bent functions
which are linear on elements of spreads. Our constructions
are obtained from symplectic presemifields which are
associated to pseudoplanar functions. The following diagram
gives an indication of the main interconnections arising in
this paper: pseudoplanar functions  commutaive
presemifields  bent functions
 Linear codes with few weights from weakly regular
bent functions based on a generic construction, S.
Mesnager. International Journal Cryptography and
Communications (CCDS), 9(1) pages 7184, Springer, 2017
Abstract :
We contribute to the knowledge of linear codes with few
weights from special polyno mials and functions.
Substantial efforts (especially due to C. Ding) have been
directed towards their study in the past few years. Such
codes have several applications in secret sharing,
authentication codes, association schemes and strongly
regular graphs. Based on a generic construction of linear
codes from mappings and by employing weakly reg ular bent
functions, we provide a new class of linear pary codes with
three weights given with its weight distribution. The class
of codes presented in this paper is different from those
known in literature.
 A comparison of Carlet's second order nonlinearity
bounds, S. Mesnager, G. McGrew, J. Davis, D. Steele et
K. Marsten. Journal of Computer Mathematics, 94(3) pages
427436, 2017.
Abstract :
Carlet provides two bounds on the second order nonlinearity
of Boolean functions. We construct a family of Boolean
functions where the first bound (the presumed weaker bound)
is tight and the second bound is strictly worse than the
first bound. We show that the difference between the two
bounds can be made arbitrarily large.
 Bent functions linear on elements of some
classical spreads and presemifields spreads, K.
Abdukhalikov et S. Mesnager. International Journal
Cryptography and Communications (CCDS), 9(1) pages 321,
Springer, 2017.
Abstract :
Bent functions are maximally nonlinear Boolean functions
with an even number of variables. They have attracted a lot
of research for four decades because of their own sake as
interesting combinatorial objects, and also because of their
relations to coding theory, sequences and their applications
in cryptography and other domains such as design theory. In
this paper we investigate explicit constructions of bent
functions which are linear on elements of spreads. After
presenting an overview on this topic, we study bent
functions which are linear on elements of presemifield
spreads and give explicit descriptions of such functions for
known commutative presemifields. A direct connection between
bent functions which are linear on elements of the
Desarguesian spread and oval polynomials over finite fields
was proved by Carlet and the second author. Very recently,
further nice extensions have been made by Carlet in another
context. We introduce oval polynomials for semifields which
are dual to symplectic semifields. In particular, it is
shown that from a linear oval polynomial for a semifield one
can get an oval polynomial for transposed semifield.
 On the nonlinearity of Sboxes and linear codes,
J. Liu, S. Mesnager et L. Chen, Journal Cryptography and
Communications Discrete Structures, Boolean Functions and
Sequences (CCDS), 9(3) pages 345361, Springer, 2017.
Abstract :
For multioutput Boolean functions (also called Sboxes),
various measures of nonlinearity have been widely discussed
in the literature but many problems are left open in this
topic. The purpose of this paper is to present a new
approach to estimating the nonlinearity of Sboxes. A more
finegrained view on the notion of nonlinearity of Sboxes
is presented and new connections to some linear codes are
established. More precisely, we mainly study the
nonlinearity indicator (denoted by $\mathcal{N}_\mathrm{v}$)
for Sboxes from a coding theory point of view. Such a
cryptographic parameter $\mathcal{N}_\mathrm{v}$ is more
related to best affine approximation attacks on stream
ciphers. We establish a direct link between
$\mathcal{N}_\mathrm{v}$ and the minimum distance of the
corresponding linear code. We exploit that connection to
derive the first general lower bounds on
$\mathcal{N}_\mathrm{v}$ of nonaffine functions from
$\F_{2^n}$ to $\F_{2^m}$ for m dividing n. Furthermore, we
show that $\mathcal{N}_\mathrm{v}$ can be determined
directly by the weight distribution of the corresponding
linear code.
 DNA cyclic codes over rings, N. Bennenni, K.
Guenda et S. Mesnager, Journal Advances in Mathematics of
Communications (AMC), Vol 11, No. 1, pages 8398, 2017.
Abstract :
In this paper we construct new DNA cyclic codes over rings.
Firstly, we introduce a new family of DNA cyclic codes over
the ring $R=F_2[u]/(u^6)$. A direct link between the
elements of such a ring and the $64$ codons used in the
amino acids of the living organisms is established. Using
this correspondence we study the reversecomplement
properties of our codes. We use the edit distance between
the codewords which is an important combinatorial notion for
the DNA strands. Next, we define the Lee weight, the Gray
map over the ring $R$ as well as the binary image of the DNA
cyclic codes allowing the transfer of studying DNA codes
into studying binary codes. Secondly, we introduce another
new family of DNA skew cyclic codes constructed over the
ring $\tilde {R}=F_2+vF_2={0,1,v,v+1\},$ where $v^2=v$. The
codes obtained are cyclic reversecomplement over the ring
$\tilde {R}$. Further we find their binary images and
construct some explicit examples of such codes.
 Involutions
over the Galois field $F_2^n$, P. Charpin, S.
Mesnager et S. Sarkar. Journal IEEE Transactions on
Information TheoryIT, Volume 62, Issue 4, pages 111, 2016.
Abstract :
An involution is a permutation such that its inverse is
itself (i.e., cycle length 2). Due to this property
involutions have been used in many applications including
cryptography and coding theory. In this paper we provide a
systematic study of involutions that are defined over finite
field of characteristic 2. We characterize the invo lution
property of several classes of polynomials and propose
several constructions. Further we study the number of fixed
points of involu tions which is a pertinent question
related to permutations with short cycle. In this paper we
mostly have used combinatorial techniques.
 Dickson
polynomials that are involutions, P. Charpin, S.
Mesnager et S. Sarkar. Journal Contemporary Developments in
Finite Fields and Their Applications, pages 2245, World
Scientific Press, 2016.
Abstract :
Dickson polynomials which are permutations are interesting
combinatorial objects and well studied. In this paper, we
describe Dickson polynomials of the first kind in
$F_{2^n}[x]$ that are involutions over finite fields of
characteristic $2$. Such description is obtained using
modular arithmetic's tools. We give results related to the
cardinality and the number of fixed points (in the context
of cryptographic application) of this corpus. We also
present infinite classes of Dickson involutions. We study
Dickson involutions which have a minimal set of fixed
points.
 Further constructions of infinite families of bent
functions from new permutations and their duals, S.
Mesnager. International journal Cryptography and
Communications (CCDS), 8(2), pages 229246, Springer 2016.
Abstract :
A Boolean function with an even number of variables is
called bent if it is maximally nonlinear. This paper extends
the recent work of the author on bent functions ("Several
new infinite families of bent functions and their duals",
IEEEIT, 60(7), pp 43974407, 2014). We exhibit several new
infinite families of bent functions with their dual (bent)
functions. Some of them are obtained via new infinite
families of permutations that we provide with their
compositional inverses. We introduce secondarylike
constructions of permutations leading to the construction of
several families of bent functions.
 Yet another variation on minimal linear codes,
G. Cohen, S. Mesnager et H. Randriam. Journal Advances in
Mathematics of Communications (AMC), Volume 10, No. 1, pages
5361, 2016.
Abstract :
Minimal linear codes are linear codes such that the support
of every codeword does not contain the support of another
linearly independent codeword. Such codes have applications
in cryptography, e.g. to secret sharing. We pursue here
their study and construct improved asymptotically good
families of minimal linear codes. We also consider
quasiminimal, $t$minimal, and $t$quasiminimal linear
codes, which are new variations on this notion.
 Further results on semibent functions in
polynomial form, X. Cao, H. Chen et S. Mesnager, Journal
Advances in Mathematics of Communications (AMC), 10(4) pages
725741, 2016.
Abstract :
Plateaued functions have been introduced by Zheng and Zhang
in 1999 as good candidates for designing cryptographic
functions since they possess many desirable cryptographic
characteristics. Plateaued functions bring together various
nonlinear characteristics and include two important classes
of Boolean functions defined in even dimension: the
wellknown bent functions ($0$plateaued functions) and the
semibent functions ($2$plateaued functions). Bent
functions have been extensively investigated since 1976.
Very recently, the study of semibent functions has
attracted a lot of attention in symmetric cryptography. Many
intensive progresses in the design of such functions have
been made especially in recent years. The paper is devoted
to the construction of semibent functions on the finite
field $\mathbb{F}_{2^n}$ ($n=2m$) in the line of a recent
work of S. Mesnager [IEEE Transactions on Information
Theory, Vol 57, No 11, 2011]. We extend Mesnager's results
and present a new construction of infinite classes of binary
semibent functions in polynomial trace. The extension is
achieved by inserting mappings $h$ on $\mathbb{F}_{2^n}$
which can be expressed as $h(0) = 0$ and $h(uy) =
h_1(u)h_2(y)$ with $u$ ranging over the circle $U$ of unity
of $\mathbb{F}_{2^n}$, $y \in \mathbb{F}_{2^m}^{*}$ and $uy
\in \mathbb{F}_{2^n}^{*}$, where $h_1$ is a isomorphism on
$U$ and $h_2$ is an arbitrary mapping on
$\mathbb{F}_{2^m}^{*}$. We then characterize the
semibentness property of the extended family in terms of
classical binary exponential sums and binary polynomials.
 Four decades of research on bent functions, C.
Carlet et S. Mesnager. International Journal Designs, Codes
and Cryptography (DCC), Vol. 78, No. 1, pages 550, Springer
2016.
Abstract :
In this survey, we revisit the Rothaus paper and the chapter
of Dil lon's thesis dedicated to bent functions, and we
describe the main results obtained on these functions during
these last 40 years. We also cover more briefly
superclasses of Boolean functions, vectorial bent functions
and bent functions in odd characteristic.
 Variation on correlation immune Boolean and
vectorial functions, J. Liu, S. Mesnager et L. Chen.
International Journal Advances in Mathematics of
Communications (AMC), 10(4) pages 895919, 2016.
Abstract :
Correlation immune functions were introduced to protect some
shift register based stream ciphers against correlation
attacks. Mathematically, the correlation immunity of a
Boolean function is a measure of the degree to which its
outputs are uncorrelated with some subset of its inputs. For
cryptographic applications, relaxing the concept of
correlation immunity has been highlighted and proved to be
more appropriate in several cryptographic situations.
Various weakened notions of correlation immunity and
resiliency have been widely introduced for cryptographic
functions, but those notions are difficult to handle. As a
variation, we focus on the notion of $\varphi$correlation
immunity which is closely related to (fast) correlation
attacks on stream ciphers based on nonlinear combiner model.
In particular, we exhibit new connections between
$\varphi$correlation immunity and $\epsilon$almost
resiliency, which are two distinct approaches for
characterizing relaxed resiliency. We also extend the
concept of $\varphi$correlation immunity introduced by
Carlet et al. in 2006 for Boolean functions to vectorial
functions and study the main cryptographic parameters of
$\varphi$correlation immune functions. Moreover, we provide
new primary constructions of $\varphi$resilient functions
with good designed immunity profile. Specially, we propose a
new recursive method to construct $\varphi$resilient
functions with high nonlinearity, high algebraic degree, and
monotone increasing immunity profile.
 Optimal codebooks from binary codes meeting the
Levenshtein bound, C. Xiang, C. Ding et S. Mesnager.
International Journal IEEE Transactions on Information
TheoryIT 61(12), pages 65266535, 2015.
Abstract :
In this paper, a generic construction of codebooks based on
binary codes is introduced. With this generic construction,
a few previous constructions of optimal codebooks are
extended, and a new class of codebooks almost meeting the
Levenshtein bound is presented. Exponentially many codebooks
meeting or amost meeting the Levenshtein bound from binary
codes are obtained in this paper. The codebooks constructed
in this paper have alphabet size 4. As a byproduct, three
bounds on the parameters of binary codes are derived.
 Bent vectorial functions and linear codes from
opolynomials, S. Mesnager. International Journal
Designs, Codes and Cryptography (DCC) 77(1), pages 99116,
2015.
Abstract :
The main topics and interconnections arising in this paper
are symmetric cryptography (Sboxes), coding theory (linear
codes) and finite projective geometry (hyperovals). The
paper describes connections between the two main areas of
information theory on the one side and finite geometry on
the other side. Bent vectorial functions are maximally
nonlinear multioutput Boolean functions. They contribute to
an optimal resistance to both linear and differential
attacks of those symmetric cryptosystems in which they are
involved as substitution boxes (Sboxes). We firstly exhibit
new connections between bent vectorial functions and the
hyperovals of the projective plane, extending the recent
link between bent Boolean functions and the hyperovals. Such
a link provides several new classes of optimal vectorial
bent functions. Secondly, we exhibit surprisingly a
connection between the hyperovals of the projective plane in
even characteristic and qary simplex codes. To this end, we
present a general construction of classes of linear codes
from opolynomials and study their weight distribution
proving that all of them are constant weight codes. We show
that the hyperovals of $PG_{2}(2^m)$ from finite projective
geometry provide new minimal codes (used in particular in
secret sharing schemes, to model the access structures) and
give rise to multiples of $2^r$ary ($r$ being a divisor of
m) simplex linear codes (whose duals are the perfect
$2^r$ary Hamming codes) over an extension field $GF 2^r$ of
$\GF 2$.
 Bent functions from spreads, S. Mesnager,
Journal of the American Mathematical Society (AMS),
Contemporary Mathematics (Proceedings of the 11th
International conference on Finite Fields and their
Applications Fq11), Volume 632, pages 295316, 2015.
Abstract :
Bent functions are optimal combinatorics objects. Since the
introduction of these functions, substantial efforts have
been directed towards their study in the last three decades.
In this paper, we are interested firstly in bent functions
on $\GF n$ whose restriction to $\frac{n}2$spreads are
constant. The study of such bent functions motivates the
clarification of connections between various subclasses of
the class of partial bent functions and relations to the
class of hyperbent functions. We investigate their logic
relations and state results giving more insight. We also
draw a Venn diagram which explains the relations between
these classes. Secondly, we present in a synthetic way the
most important progresses obtained about the bent functions
on $\GF n$ whose restrictions to $\frac{n}2$spreads are
linear. Finally, we present our advances obtained about the
bent functions on $\GF n$ whose restrictions to
$\frac{n}2$spreads are affine.
 Several new infinite families of bent functions and
their duals, S. Mesnager, IEEE Transactions on
Information TheoryIT, Vol. 60, No. 7, pages 43974407, 2014
Abstract :
Bent functions are optimal combinatorial objects. Since the
introduction of these functions, substantial efforts have
been directed towards their study in the last three decades.
A complete classification of bent functions is elusive and
looks hopeless today, therefore, not only their
characterization, but also their generation are challenging
problems. The paper is devoted to the construction of bent
functions. Firstly we provide several new effective
constructions of bent functions, selfdual bent functions
and antiselfdual bent functions. Secondly, we provide
seven new infinite families of bent functions by explicitly
calculating their dual functions.
 Sphere coverings and Identifying Codes, D.
Auger, G. Cohen et S. Mesnager, Journal Designs, Codes and
Cryptography, Volume 70, Issues 12, pages 37, 2014.
Abstract :
In any connected, undirected graph $G=(V,E)$, the {\it
distance} $d(x,y)$ between two vertices $x$ and $y$ of $G$
is the minimum number of edges in a path linking $x$ to $y$
in $G$. A {\it sphere} in $G$ is a set of the form $S_r(x) =
\{ y \in V : d(x,y)=r \},$ where $x$ is a vertex and $r$ is
a nonnegative integer called the {\it radius} of the sphere.
We first address in this paper the following question : What
is the minimum number of spheres with fixed radius $r \geq
0$ required to cover all the vertices of a finite,
connected, undirected graph $G$ ? We then turn our attention
to the Hamming Hypercube of dimension $n$, and we show that
the minimum number of spheres {\it with any radii} required
to cover this graph is either $n$ or $n+1$, depending on the
parity of $n$. We also relate the two above problems to
other questions in combinatorics, in particular to
identifying codes.
 On constructions of semibent functions from bent
functions, G. Cohen et S. Mesnager, Journal Contemporary
Mathematics 625, Discrete Geometry and Algebraic
Combinatorics, Americain Mathematical Society, Pages 141154,
2014.
Abstract :
Plateaued functions are significant in cryptography as they
possess various desirable cryptographic properties. Two
important classes of plateaued functions are those of bent
functions and semibent functions, due to their
combinatorial and algebraic properties. Constructions of
bent functions have been extensively investigated. However
only few constructions of semibent functions have been
proposed in the literature. In general, finding new
constructions of bent and semibent functions is not a
simple task. The paper is devoted to the construction of
semibent functions with even number of variables. We show
that bent functions give rise to primary and secondarylike
constructions of semibent functions.
 An efficient characterization of a family of
hyperbent functions with multiple trace terms, J. P.
Flori et S. Mesnager, Journal of Mathematical Cryptology. Vol
7 (1), pages 4368, 2013.
Abstract :
The connection between exponential sums and algebraic
varieties has been known for at least six decades. Recently,
Lisoněk exploited it to reformulate the CharpinGong
characterization of a large class of hyperbent functions in
terms of numbers of points on hyperelliptic curves. As a
consequence, he obtained a polynomial time and space
algorithm for certain subclasses of functions in the
CharpinGong family. In this paper, we settle a more
general framework, together with detailed proofs, for such
an approach and show that it applies naturally to a distinct
family of functions proposed by Mesnager. Doing so, a
polynomial time and space test for the hyperbentness of
functions in this family is obtained as well. Nonetheless, a
straightforward application of such results does not provide
a satisfactory criterion for explicit generation of
functions in the Mesnager family. To address this issue, we
show how to obtain a more efficient test leading to a
substantial practical gain. We finally elaborate on an open
problem about hyperelliptic curves related to a family of
Boolean functions studied by Charpin and Gong.
 Hyperbent functions via Dillonlike exponents,
S. Mesnager et J. P. Flori, IEEE Transactions on Information
TheoryIT. Vol. 59 No. 5, pages 3215 3232, 2013.
Abstract :
This paper is devoted to hyperbent functions with multiple
trace terms (including binomial functions) via Dillonlike
exponents. We show how the approach developed by Mesnager to
extend the CharpinGong family, which was also used by Wang
\etal to obtain another similar extension, fits in a much
more general setting.To this end, we first explain how the
original restriction for CharpinGong criterion can be
weakened before generalizing the Mesnager approach to
arbitrary Dillonlike exponents. Afterward, we tackle the
problem of devising infinite families of extension degrees
for which a given exponent is valid and apply these results
not only to reprove straightforwardly the results of
Mesnager and Wang et. al, but also to characterize the
hyperbentness of several new infinite classes of Boolean
functions. We go into full details only for a few of them,
but provide an algorithm (and the corresponding software) to
apply this approach to an infinity of other new families.
Finally, we propose a reformulation of such
characterizations in terms of hyperelliptic curves and use
it to actually build hyperbent functions in cases which
could not be attained through naive computations of
exponential sums.
 Further results on Niho bent functions, L.
Budaghyan, C. Carlet, T. Helleseth, A. Kholosha et S.
Mesnager, IEEE Transactions on Information TheoryIT. Vol 58,
No 11, pages 69796985, 2012.
Abstract :
Computed is the dual of the Niho bent function consisting of
$2^r$ exponents that was found by Leander and Kholosha. The
algebraic degree of the dual is calculated and it is shown
that this new bent function is not of the Niho type.
Finally, three infinite classes of Niho bent functions are
analyzed for their relation to the completed
MaioranaMcFarland class. This is done using the criterion
based on secondorder derivatives of a function.
 On Semibent Boolean Functions, C. Carlet et S.
Mesnager, IEEE Transactions on Information Theory, Vol 58, No
5, pages: 32873292, 2012.
Abstract :
We show that any Boolean function, in even dimension, equal
to the sum of a Boolean function g$ which is constant on
each element of a spread and of a Boolean function $h$ whose
restrictions to these elements are all linear, is semibent
if and only if g and h are both bent. We deduce a large
number of infinite classes of semibent functions in
explicit bivariate (resp. univariate) polynomial form.
 Semibent functions from Dillon and Niho exponents,
Kloosterman sums and Dickson polynomials. S. Mesnager,
IEEE Transactions on Information Theory, Vol 57, No 11, pages
74437458, 2011.
Abstract :
Kloosterman sums have recently become the focus of much
research, most notably due to their applications in
cryptography and coding theory. In this paper, we
extensively investigate the link between the semibentness
property of functions in univariate forms obtained via
Dillon and Niho functions and Kloosterman sums. In
particular, we show that zeros and the value four of binary
Kloosterman sums give rise to semibent functions in even
dimension with maximum degree. Moreover, we study the
semibentness property of functions in polynomial forms with
multiple trace terms and exhibit criteria involving Dickson
polynomials.
 On Dillon's class H of bent functions, Niho bent
functions and opolynomials, C. Carlet et S. Mesnager,
Journal of Combinatorial TheoryJCTserie A 118, pages
2392–2410, 2011.
Abstract :
One of the classes of bent Boolean functions introduced by
John Dillon in his thesis is family $H$. While this class
corresponds to a nice original construction of bent
functions in bivariate form, Dillon could exhibit in it only
functions which already belonged to the wellknown
MaioranaMcFarland class. We first notice that $H$ can be
extended to a slightly larger class that we denote by ${\cal
H}$. We observe that the bent functions constructed via Niho
power functions, for which four examples are known due to
Dobbertin et al. and to LeanderKholosha, are the univariate
form of the functions of class ${\cal H}$. Their
restrictions to the vector spaces $\omega\GF {n/2}$,
$\omega\in \GF n^\star$, are linear. We also characterize
the bent functions whose restrictions to the $\omega\GF
{n/2}$ s are affine. We answer the open question raised by
Dobbertin et al. in JCT A 2006 on whether the duals of the
Niho bent functions introduced in the paper are affinely
equivalent to them, by explicitely calculating the dual of
one of these functions. We observe that this Niho function
also belongs to the MaioranaMcFarland class, which brings
us back to the problem of knowing whether $H$ (or ${\cal
H}$) is a subclass of the MaioranaMcFarland completed
class. We then show that the condition for a function in
bivariate form to belong to class ${\cal H}$ is equivalent
to the fact that a polynomial directly related to its
definition is an opolynomial (also called oval polynomial,
a notion from finite geometry). Thanks to the existence in
the literature of 8 classes of nonlinear opolynomials, we
deduce a large number of new cases of bent functions in
${\cal H}$, which are potentially affinely inequivalent to
known bent functions (in particular, to MaioranaMcFarland's
functions).
 Bent and Hyperbent Functions in polynomial form
and Their Link With Some Exponential Sums and Dickson
Polynomials. S. Mesnager, IEEE Transactions on
Information Theory, Vol. 57, No. 9, pages 59966009, 2011.
Abstract :
Bent functions are maximally nonlinear Boolean functions
with an even number of variables. They were introduced by
Rothaus in 1976. For their own sake as interesting
combinatorial objects, but also because of their relations
to coding theory (ReedMuller codes) and applications in
cryptography (design of stream ciphers), they have attracted
a lot of research, specially in the last 15 years. The class
of bent functions contains a subclass of functions,
introduced by Youssef and Gong in 2001, the socalled
hyperbent functions, whose properties are still stronger
and whose elements are still rarer than bent functions. Bent
and hyperbent functions are not classified. A complete
classification of these functions is elusive and looks
hopeless. So, it is important to design constructions in
order to know as many of (hyper)bent functions as possible.
This paper is devoted to the constructions of bent and
hyperbent Boolean functions in polynomial forms. We survey
and present an overview of the constructions discovered
recently. We extensively investigate the link between the
bentness property of such functions and some exponential
sums (involving Dickson polynomials) and give some
conjectures that lead to constructions of new hyperbent
functions.
 A New Class of Bent and HyperBent Boolean
Functions in Polynomial Forms. S. Mesnager, Journal
Designs, Codes and Cryptography. Volume 59, No. 13, pages
265279 (2011).
Abstract :
Bent functions are maximally nonlinear Boolean functions and
exist only for functions with even number of inputs. This
paper is a contribution to the construction of bent
functions over $\GF{n}$ ($n=2m$) having the form $f(x) = \tr
{o(s_1)} (a x^ {s_1}) + \tr {o(s_2)} (b x^{s_2})$ where
$o(s_i$) denotes the cardinality of the cyclotomic class of
2 modulo $2^n1$ which contains $s_i$ and whose coefficients
$a$ and $b$ are, respectively in $F_{2^{o(s_1)}}$ and
$F_{2^{o(s_2)}}$. Many constructions of monomial bent
functions are presented in the literature but very few are
known even in the binomial case. We prove that the exponents
$s_1=2^{m}1$ and $s_2={\frac {2^n1}3}$, where $a\in\GF{n}$
($a\not=0$) and $b\in\GF[4]{}$ provide a construction of
bent functions over $\GF{n}$ with optimum algebraic degree.
For $m$ odd, we give an explicit characterization of the
bentness of these functions, in terms of the Kloosterman
sums. We generalize the result for functions whose exponent
$s_1$ is of the form $r(2^{m}1)$ where $r$ is coprime with
$2^m+1$. The corresponding bent functions are also
hyperbent. For $m$ even, we give a necessary condition of
bentness in terms of these Kloosterman sums.
 On the construction of bent vectorial functions,
C. Carlet et S. Mesnager, Journal of Information and Coding
Theory: Algebraic and Combinatorial Coding Theory, Vol 1, No.
2, pages 133148 (2010).
Abstract :
This paper is devoted to the constructions of bent vectorial
functions, that is, maximally nonlinear multioutput Boolean
functions. Such functions contribute to an optimal
resistance to both linear and differential attacks of those
cryptosystems in which they are involved as substitution
boxes (Sboxes). We survey, study more in details and
generalize the known primary and secondary constructions of
bent functions, and we introduce new ones.
 Improving the Lower Bound on the Higher Order
Nonlinearity of Boolean Functions With Prescribed Algebraic
Immunity. S. Mesnager, IEEE Transactions on Information
TheoryIT Vol. 54, No. 8, pages 36563662 (2008).
Abstract :
The recent algebraic attacks have received a lot of
attention in cryptographic literature. The algebraic
immunity of a Boolean function quantifies its resistance to
the standard algebraic attacks of the pseudorandom
generators using it as a nonlinear filtering or combining
function. Very few results have been found concerning its
relation with the other cryptographic parameters or with the
rthorder nonlinearity. As recalled by Carlet at CRYPTO'06,
many papers have illustrated the importance of the r
thorder nonlinearity profile (which includes the
firstorder nonlinearity). The role of this parameter
relatively to the currently known attacks has been also
shown for block ciphers. Recently, two lower bounds
involving the algebraic immunity on the rthorder
nonlinearity have been shown by Carlet . None of them
improves upon the other one in all situations. In this
paper, we prove a new lower bound on the rthorder
nonlinearity profile of Boolean functions, given their
algebraic immunity, that improves significantly upon one of
these lower bounds for all orders and upon the other one for
low orders.
 On the number of resilient Boolean functions.
S. Mesnager, Journal of Number Theory and its Applications,
Vol. 5, pages 139153, 2008.
Abstract :
Boolean functions are very important primitives of symmetric
cryptosystems. To increase the security of such
cryptopsystems, these Boolean functions have to fit several
security criteria. In particular, they have to be
$m$resilient, that is, to be balanced and $m$correlation
immune. This class of Boolean function has been widely
studied by cryptographers. Nevertheless, the problem of
counting the number of $m$resilient $n$variables Boolean
functions is still challenging. In this paper, we propose a
new approach to this question. We reword this question in
that to count integer solutions of a system of linear
inequalities. This allows us to deduce two representation
formulas for the number of $m$resilient $n$variables
Boolean functions.
 Improving the Upper Bounds on the Covering Radii of
Binary ReedMuller Codes, C. Carlet et S. Mesnager, IEEE
Transactions on Information Theory 53 (1), pages 162173
(2007).
Abstract :
By deriving bounds on character sums of Boolean functions
and by using the characterizations, due to Kasami , of those
elements of the ReedMuller codes whose Hamming weights are
smaller than twice and a half the minimum distance, we
derive an improved upper bound on the covering radius of the
ReedMuller code of order 2, and we deduce improved upper
bounds on the covering radii of the ReedMuller codes of
higher orders
 Test of epimorphism for finitely generated
morphisms between affine algebras over Computational rings.
S. Mesnager, Journal of Algebra and Applications, Vol 4 (4),
pages 115 (2005).
Abstract :
In this paper, based on a characterization of epimorphisms
of $R$algebras given by Roby [15], we bring an algorithm
testing whether a given ﬁnitely generated morphism $f :
A> B$, where A and B are ﬁnitely presented aﬃne algebras
over the same Nœtherian commutative ring $R$, is an
epimorphism of $R$algebras or not. We require two computa
tional conditions on $R$, which we call a computational
ring.
 Construction of the integral closure of an affine
domain in a finite field extension of its quotient field.
S. Mesnager, Journal of Pure and Applied Algebra, Vol 194,
pages 311327 (2004).
Abstract :
The construction of the normalization of an affine domain
over a field is a classical problem solved since sixteen's
by Stolzenberg (1968) and Seidenberg (19701975) thanks to
classical algebraic methods and more recently by Vasconcelos
(19911998) and de Jong (1998) thanks to homological
methods. The aim of this paper is to explain how to use such
a construction to obtain effectively the integral closure of
such a domain in any finite extension of its quotient field,
thanks to Dieudonn\'e characterization of such an integral
closure. As application of our construction, we explain how
to obtain an effective decomposition of a quasifinite and
dominant morphism from a normal affine irreducible variety
to an affine irreducible variety as a product of an open
immersion and a finite morphism, conformly to the classical
Grothendieck's version of Zariski's main theorem.
 On resultant criteria and formulas for the
inversion of a polynomial map. S. Mesnager,
Communications in Algebra 29 (8), pages 33273339 (2001).
Abstract :
About the inversion of a polynomial map $F : K^2 \mapsto
K^2$ over an arbitrary field $K$, it is natural to consider
the following questions: (1) Can we find a necessary and
sufficient criterion in terms of resultants for $F$ to be
invertible with polynomial inverse such that, this criterion
gives an explicit formula to compute the inverse of $F$ in
this case ? (2) Can we find a necessary and sufficient
condition in terms of resultants for $F$ to be invertible
with rational inverse such that, this criterion gives an
explicit formula to compute the inverse of $F$ in this case
? MacKay and Wang [5] gave a partial answer to question (1),
by giving an explicit expression of the inverse of $F$, when
$F$ is invertible without constant terms. on the other
hand,Adjamagbo and Essen \cite{AdjamagboEssen} have fully
answered questions (2) and have furnished a necessary and
sufficient criterion which relies on the existence of some
constants $\lambda_1$, $\lambda_2$ in $K^\star$. We improve
this result by giving an explicit relation between
$\lambda_1$, $\lambda_2$ and constants of the Theorem of
MacKay and Wang [5]. Concerning question (2), Adjamagbo and
Boury [2] give a criterion for rational maps which relies on
the existence of two polynomials $\lambda_1$, $\lambda_2$.
We also improve this result, by expliciting the relations
between these $\lambda_1$,$\lambda_2$ and the coefficients
of $F$. This improvement enables us, first to give an
explicit proof of the corresponding Theorem of
Abhyankhar[1], and secondly, to give a counter example where
these $\lambda_1$,$\lambda_2$ are not in $K^\star$, contrary
to a claim of Yu [6].
Actes de conférences internationales:
(dans l’ordre chronologique inverse)
 On constructions of binary locally
repairable codes with locality two and multiple repair
alternatives via autocorrelation spectra of Boolean
functions. D. Tang, J. Liu and
S. Mesnager. Proceedings of the 12th International
Workshop on Coding and Cryptography (WCC 2022), Rostock,
Germany, 2022.
 A suitable proposal of Sboxes
(inverselike) for the AES, their analysis and
performances. S. Eddahmani and
S. Mesnager. Proceedings of the International Conference
on Security and Privacy ICSP 2021, India, pages 4963,
2021.
 Infinie Classes of sixweight linear codes
derived from weakly regular plateaued
functions. S. Mesnager and A. Sinak.
Proceedings of the IEEE International Conference
on Information and Cryptology (ISCTURKEY), Ankara,
Turley, pages 93100, 2020.
 Further results on bentnegabent Boolean
functions. S. Mesnager, B. ben Moussat et Z. Zhuo,
Proceedings of International Conference on Security and
Privacy (ICSP 2020), 2020, Inde.
Abstract :
Bent functions are optimal combinatorial objects having a
lot of applications in particular in cryptography. Since
their introduction, substantial efforts have been directed
towards their study in the last three decades. In this
paper, we investigate two families of functions possessing
properties related to bentness: the socalled negabent and
bentnegabent functions, and derive several results on their
constructions and characterizations.
 Infinite Classes of sixweight linear codes
derived from weakly regular plateaued functions. S.
Mesnager et A. Sinak, the 13th International Conference on
Information Security and Cryptology 2020 with the IEEE Turkey
Section Support, Turquie 2020.
Abstract :
The construction of linear codes with few weights from
cryptographic functions over finite fields has been widely
studied in the literature since linear codes have a wide
range of applications in practical systems. In this paper,
to construct new linear codes with few weights, we
generalize the recent construction method presented by Xu,
Qu and Luo at SETA 2020 for weakly regular plateaued
functions over the finite fields of odd characteristics. We
derive sixweight minimal linear codes from the subset of
the preimage of weakly regular plateaued unbalanced
functions. We also construct sixweight linear codes with
flexible parameters from weakly regular bent and plateaued
functions by choosing two different subsets of the preimage
of these functions.
 Privacy as a Service: Anonymisation of NetFlow
Traces. A. Aloui, M. Msahli, T. Abdessalem, S. Mesnager
et S. Bressan, Proceedings of ICEBE 2019, pages 561571, 2019,
Chine.
Abstract :
Effective data anonymisation is the key to unleash ing the
full potential of big data analytics while preserving pri
vacy. An organization needs to be able to share and
consolidate the data it collects across its departments and
in its network of collaborating organizations. Some of the
data collected and the crossreferences made in its
aggregation is private. Effective data anonymisation
attempts to maintain the confidentiality and privacy of the
data while maintaining its utility for the purpose of
analytics. Preventing reidentification is also of
particular importance. The main purpose of this paper is to
provide a definition of an original data anonymisation
paradigm in order to render the reidentification of related
users impossible. Here, we consider the case of a NetFlow
Log. The solution includes a privacy risk analysis process
the result of which is the classification of the data based
on privacy levels. We use a dynamic Kanonymity paradigm
while taking into consideration the privacy risk assessment
output. Finally, we empirically evaluate the performance and
data partition of the proposed solution.
 Threeweight minimal linear codes and their
applications. S. Mesnager, A. Sinak et O. Yayla,
Proceedings of the Second International Workshop on
Cryptography and its Applications (IWCA 2019).
Abstract :
Minimal linear codes have important applications in secret
sharing schemes and secure twoparty computation. In this
paper, we first construct linear codes with three weights
from weakly regular plateaued functions based on the second
generic construction and determine their weight
distributions. We next give punctured version of each
constructed codes. We finally observe that the constructed
codes in this paper are minimal for almost all cases, which
confirms that the secret sharing schemes based on their dual
codes have the nice access structures.
 Strongly regular graphs from weakly regular
plateaued functions. S. Mesnager et A. Sinak,
Proceedings of 2019 Ninth International Workshop on Signal
Design and its Applications in Communications (IWSDA), Chine
2019.
Abstract :
This paper presents the first construction of strongly
regular graphs and association schemes from weakly regular
plateaued functions over finite fields of odd
characteristic. Indeed, we generalize the construction
method of strongly regular graphs from weakly regular bent
functions given by Chee et al. in [Journal of Algebraic
Combinatorics, 34(2), 251266, 2011] to weakly regular
plateaued functions. In this framework, we construct
strongly regular graphs with three types of parameters from
weakly regular plateaued functions with some homogeneous
conditions. We also construct a family of association
schemes of class p from weakly regular pary plateaued
functions.
 Further study of $2$to$1$ mappings over
$F_{2^n}$. K. Li, S. Mesnager et L. Qu, Proceedings of
2019 Ninth International Workshop on Signal Design and its
Applications in Communications (IWSDA), Chine 2019.
Abstract :
2to1 mappings over finite fields play important roles in
symmetric cryptography, such as APN functions, bent
functions, semibent functions and so on. Very recently,
Mesnager and Qu [9] provided a systematic study of 2to 1
mappings over finite fields. Particularly, they determined
all 2to1 mappings of degree $\leq4 over any finite
fields. In addition, another research direction is to
consider 2to 1 polynomials with few terms. Some results
about 2to1 monomials and binomials can be found in [9].
Motivated by their work, in this present paper, we continue
to study 2to1 mappings, particularly, over finite fields
with characteristic 2. Firstly, we determine 2to1
polynomials with degree 5 over $F_{2^n}$ completely by
HasseWeil bound. Besides, using the multivariate method and
the resultant of two polynomials, we present three classes
of 2to1 trinomials and four classes of 2to1
quadrinomials over $F_{2^n}$.
 Constructions of optimal locally recoverable codes
via Dickson polynomials. J. Liu, S. Mesnager, et D.
Tang, Proceedings of The Eleventh International Workshop on
Coding and Cryptography} (WCC 2019), SaintMalo, France.
Abstract :
In 2014, Tamo and Barg have presented in a very remarkable
paper a family of optimal linear locally recoverable codes
(LRC codes) that attain the maximum possible distance (given
code length, cardinality, and locality). The key ingredient
for constructing such optimal linear LRC codes is the
socalled $r$good polynomials, where $r$ is equal to the
locality of the LRC code. In 2018, Liu et al. have presented
two general methods of designing $r$good polynomials by
using function composition, which lead to three new
constructions of $r$good polynomials. Next, Micheli has
provided a Galois theoretical framework which allows to
produce $r$good polynomials. The wellknown Dickson
polynomials form an important class of polynomials which
have been extensively investigated in recent years under
different contexts. In this paper, we provide new methods of
designing $r$good polynomials based on Dickson polynomials.
Such $r$good polynomials provide new constructions of
optimal LRC codes.
 On good polynomials over finite fields for optimal
locally recoverable codes. S. Mesnager, Proceedings of
the international Conference on Codes, Cryptology and
Information Security C2SI 2019, Maroc, pages 257268, 2019.
Abstract :
[This is an extended abstract of the paper
[LiuMesnagerChen2018] A locally recoverable (LRC) code is
a code that enables a simple recovery of an erased symbol by
accessing only a small number of other symbols. LRC codes
currently form one of the rapidly developing topics in
coding theory because of their applications in distributed
and cloud storage systems. In 2014, Tamo and Barg have
presented in a very remarkable paper a family of LRC codes
that attain the maximum possible (minimum) distance (given
code length, cardinality, and locality). The key ingredient
for constructing such optimal linear LRC codes is the
socalled $r$good polynomials, where $r$ is equal to the
locality of the LRC code. In this extended abstract, we
review and discuss good polynomials over finite fields for
constructing optimal LRC codes.
 On Plateaued Functions, Linear Structures
and Permutation Polynomials. S. Mesnager, K.
Kaytannci et F. Ozbudak, Proceedings of the international
Conference on Codes, Cryptology and Information Security C2SI
2019, Maroc, pages 217235, 2019.
Abstract :
We obtain concrete upper bounds on the algebraic immunity of
a class of highly nonlinear plateaued functions without
linear structures than the one was given recently in 2017,
Cusick. Moreover, we extend Cusick's class to a much bigger
explicit class and we show that our class has better
algebraic immunity by an explicit example. We also give a
new notion of linear translator, which includes the
Frobenius linear translator given in 2018, Cepak, Pasalic
and Muratovi\'{c}Ribi\'{c} as a special case. We find some
applications of our new notion of linear translator to the
construction of permutation polynomials. Furthermore, we
give explicit classes of permutation polynomials over
$\mathbb{F}_{q^n}$ using some properties of $\mathbb{F}_q$
and some conditions of 2011, Akbary, Ghioca and Wang.
 Characterizations of partially bent and plateaued
functions over finite fields. S. Mesnager, F. Ozbudak et
A. Sinak, Proceedings of International Workshop on the
Arithmetic of Finite Fields, WAIFI 2018, Bergen, 2018.
Abstract :
Plateaued and partially bent functions over finite fields
have significant applications in cryptography, sequence
theory, coding theory, design theory and combinatorics. They
have been extensively studied due to their various desirable
cryptographic properties. In this paper, we study on
characterizations of partially bent and plateaued
(vectorial) functions over finite fields, with the aim of
clarifying their structure. We first redefine the notion of
partially bent functions over any finite field $\F_q$, with
$q$ a prime power, and then provide a few characterizations
of these functions in terms of their derivatives, Walsh
power moments and autocorrelation functions. We next
characterize partially bent (vectorial) functions over
$\F_p$, with $p$ a prime, by means of their secondorder
derivatives and Walsh power moments. We finally characterize
plateaued functions over $\F$
 Construction of Some Codes Suitable for Both Side
Channel and Fault Injection Attacks. C. Carlet, C.
Guneri, S. Mesnager et F. Ozbudak, Proceedings of
International Workshop on the Arithmetic of Finite Fields,
WAIFI 2018, Bergen, 2018.
Abstract :
Using algebraic curves over finite fields, we construct some
codes suitable for being used in the countermeasure called
Direct Sum Masking which allows, when properly implemented,
to protect the whole cryptographic block cipher algorithm
against side channel attacks and fault injection attacks,
simultaneously. These codes address a problem which has its
own interest in coding theory.
 A new class of threeweight linear codes from
weakly regular plateaued functions. S. Mesnager, F.
Ozbudak et A. Sinak, Proceedings of The Tenth International
Workshop on Coding and Cryptography (WCC 2017).
SaintPetersburg, Russie, 2017
Abstract :
Linear codes with few weights have many applications in
secret sharing schemes, authentication codes, communication
and strongly regular graphs. In this paper, we consider
linear codes with three weights in arbitrary characteristic.
To do this, we generalize the recent contribution of
Mesnager given in [Cryptography and Communications 9(1),
7184, 2017]. We first present a new class of binary linear
codes with three weights from plateaued Boolean functions
and their weight distributions. We next introduce the notion
of (weakly) regular plateaued functions in odd
characteristic p and give concrete examples of these
functions. Moreover, we construct a new class of
threeweight linear pary codes from weakly regular
plateaued functions and determine their weight
distributions. We finally analyse the constructed linear
codes for secret sharing schemes.
 Preserving Privacy in Distributed System (PPDS)
Protocol: Security analysis. A. Aloui, M. Msahli, T.
Abdessalem, S. Bressan et S. Mesnager, Proceedings of 36th
IEEE International Performance Computing and Communications
Conference}, (IPCCC 2017), San Diego, USA.
Abstract :
Within the diversity of existing Big Data and data
processing solutions, meeting the requirements of privacy
and security is becoming a real need. In this paper we
tackle the security analysis of a new protocol of data
processing in distributed system (PPDS). This protocol is
composed of three phases: authentication, node head
selection and data linking. This paper deals with its formal
validation done using HLPSL language via AVISPA. We provide
also its security analysis. Some performance analysis based
on its proof of concept are also given in this paper.
 New bent functions from permutations and linear
translators. S. Mesnager, P. Ongan et F. Ozbudak,
Proceedings of the international Conference on Codes,
Cryptology and Information Security (C2SI2017), pages
282297, Springer 2017.
Abstract :
Starting from the secondary construction originally
introduced by Carlet ["On Bent and Highly Nonlinear
Balanced/Resilient Functions and Their Algebraic
Immunities", Applied Algebra, Algebraic Algorithms and
ErrorCorrecting Codes, 2006], that we shall call \Car
let`s secondary construction", Mesnager has showed how one
can construct several new primary constructions of bent
functions. In particular, she has showed that three tuples
of permutations over the finite field F2m such that the
inverse of their sum equals the sum of their inverses give
rise to a construction of a bent function given with its
dual. It is not quite easy to find permutations satisfying
such a strong condition (Am). Nevertheless, Mesnager has
derived several candidates of such permutations in 2015, and
showed in 2016 that in the case of involutions, the problem
of construction of bent functions amounts to solve
arithmetical and algebraic problems over finite fields. This
paper is in the line of those previous works. We present new
families of permutations satisfying (Am) as well as new
infinite families of permutations constructed from
permutations in both lower and higher dimensions. Our
results involve linear translators and give rise to new
primary constructions of bent functions given with their
dual. And also, we show that our new families are not in the
class of MaioranaMcFarland in general.
 Explicit Characterizations for Plateauedness of
pary (Vectorial) Functions. C. Carlet, S. Mesnager, F.
Ozbudak et A. Sinak. Proceedings of the international
Conference on Codes, Cryptology and Information Security
(C2SI2017) pages 328345, Springer 2017.
Abstract :
Plateaued (vectorial) functions have an important role in
the sequence and cryptography frameworks. Given their
importance, they have not been studied in detail in general
framework. Several researchers found recently results on
their characterizations and introduced new tools to
understand their structure and to design such functions In
this work, we mainly extend some of the observations made in
characteristic 2 and given in [C. Carlet, IEEE T INFORM
THEORY 61(11), 2015] to arbitrary characteristic. We first
extend to arbitrary characteristic the characterizations of
plateaued (vectorial) Boolean functions by the
autocorrelation functions, next their characterizations in
terms of the secondorder derivatives, and finally their
characterizations via the moments of the Walsh transform.
 On constructions of bent functions from
involutions. S. Mesnager. Proceedings of 2016 IEEE
International Symposium on Information Theory, (ISIT 2016),
Barcelone, Espagne, 2016.
Abstract :
Bent functions are maximally nonlinear Boolean functions.
They are important functions introduced by Rothaus and
studied firstly by Dillon and next by many researchers for
four decades. Since the complete classification of bent
functions seems elusive, many researchers turn to design
constructions of bent functions. In this paper, we show that
linear involutions (which are an important class of
permutations) over finite fields give rise to bent functions
in bivariate representations. In particular, we exhibit new
constructions of bent functions involving binomial linear
involutions whose dual functions are directly obtained
without computation. The existence of bent functions from
involutions heavily relies on solving systems of equations
over finite fields.
 Partially homomorphic encryption schemes over
finite fields. J. Liu, S. Mesnager et L. Chen.
Proceedings of the Sixth International Conference on Security,
Privacy and Applied Cryptographic Engineerin (Space 2016),
pages 109123, Springer, 2016.
Abstract :
Homomorphic encryption scheme enables computation in the
encrypted do main, which is of great importance because of
its wide and growing range of applications. The main issue
with the known fully (or partially) homomorphic encryption
schemes is the high computational complexity and large
communication cost required for their exe cution. In this
work, we study symmetric partially homomorphic encryption
schemes over finite fields, establishing relationships
between homomorphisms over finite fields with qary
functions. Our proposed partially homomorphic encryption
schemes have perfect secrecy and resist cipheronly attacks
to some extent.
 A Scalable and Systolic Architectures of Montgomery
Modular Multiplication for Public Key Cryptosystems Based on
DSPs. A. Mrabet, N. ElMrabet, R. Lashermes, JB.
Rigaud, B. Bouallegue, S. Mesnager et M. Machhout. Proceedings
of the Sixth International Conference on Security, Privacy and
Applied Cryptographic Engineering (Space 2016) pages 138156,
Springer, 2016.
Abstract :
Inversion can be used in Elliptic Curve Cryptography systems
and pairingbased cryptography, which are becoming popular
for Public Key Cryptosystems. For the same security level,
ECC and pairing use much smaller key length than RSA but
need modular inversion. In ECC when points are represented
in socalled affine coordinates, the addition of two points
involves a field inversion. Some pairing require one
inversion over Fp in order to perform the final
exponentiation. Usually, inversions are avoided in Elliptic
Curve Cryptography as they are expensive. For example,
inversions in affine coordinates are transform into
multiplication in Jacobian or projective coordinates. In
order to improve performance of Public Key Cryptosystems, we
present an improved algorithm for prime field modular
inversion. We demonstrate that the affine coordinates can be
more efficient than projective or jacobian for the scalar
multiplication.
 Secret sharing schemes with general access
structures, J. Liu, S. Mesnager et L. Chen, proceedings
of the "11th International Conference on Information Security
and Cryptology" Inscrypt 2015 (IACR), Volume 9589, LNCS,
Springer, 2016.
Abstract :
Secret sharing schemes with general monotone access
structures have been widely discussed in the literature. But
in some scenarios, nonmonotone access structures may have
more practical significance. In this paper, we shed a new
light on secret sharing schemes realizing general (not
necessarily monotone) access structures. Based on an attack
model for secret sharing schemes with general access
structures, we redefine perfect secret sharing schemes,
which is a generalization of the known concept of perfect
secret sharing schemes with monotone access structures.
Then, we provide for the first time two constructions of
perfect secret sharing schemes with general access
structures. The first construction can be seen as a
democratic scheme in the sense that the shares are generated
by the players themselves. Our second construction
significantly enhance the efficiency of the system, where
the shares are distributed by the trusted center (TC).
 On existence (based on an arithmetical problem)
and constructions of bent functions. S. Mesnager, G.
Cohen et D. Madore. Proceedings of the fifteenth International
Conference on Cryptography and Coding, Oxford, United Kingdom,
IMACC 2015, Pages 319, LNCS, Springer, Heidelberg, 2015.
Abstract :
Bent functions are maximally nonlinear Boolean functions.
They are wonderful creatures introduced by O. Rothaus in the
1960's and studied firstly by J. Dillon since 1974. Using
some involutions over finite fields, we present new
constructions of bent functions in the line of recent
Mesnager's works. One of the constructions is based on an
arithmetical problem. We discuss existence of such bent
functions using Fermat hypersurface and LangWeil
estimations.
 On the diffusion property of iterated functions.
J. Liu, S. Mesnager et L. Chen. Proceedings of the fifteenth
International Conference on Cryptography and Coding, Oxford,
United Kingdom, IMACC 2015, Pages 239253, LNCS, Springer,
Heidelberg, 2015.
Abstract :
For vectorial Boolean functions, the behavior of iteration
has consequence in the diffusion property of the system. We
present a study on the diffusion property of iterated
vectorial Boolean functions. The measure that will be of
main interest here is the notion of the degree of
completeness, which has been suggested by the NESSIE
project. We provide the first (to the best of our knowledge)
two constructions of $(n,n)$functions having perfect
diffusion property and optimal algebraic degree. We also
obtain the complete enumeration results for the constructed
functions.
 Bent and semibent functions via linear
translators. N. Kocak, S. Mesnager et F. Ozbudak.
Proceedings of the fifteenth International Conference on
Cryptography and Coding, Oxford, United Kingdom, IMACC 2015,
Pages 205224, LNCS, Springer, Heidelberg, 2015.
Abstract :
This paper is dealing with two important subclasses of
plateaued functions in even dimension: bent and semibent
functions. In the first part of the paper, we construct
mainly bent and semibent functions in MaioranaMcFarland
class using Boolean functions having linear structures
(linear translators) systematically. Although most of these
results are rather direct applications of some recent
results, using linear structures (linear translators) allows
us to have certain flexibilities to control extra properties
of these plateaued functions. In the second part of the
paper, using the results of the first part and exploiting
these flexibilities, we modify many secondary constructions.
Therefore, we obtain new secondary constructions of bent and
semibent functions not belonging to MaioranaMcFarland
class. Instead of using bent (semibent) functions as
ingredients, our secondary constructions use only Boolean
(vectorial Boolean) functions with linear structures (linear
translators) which are very easy to choose. Moreover, all of
them are very explicit and we also determine the duals of
the bent functions in our constructions. We show how these
linear structures should be chosen in order to satisfy the
corresponding conditions coming from using derivatives and
quadratic/cubic functions in our secondary constructions.
 Results on characterizations of plateaued
functions in arbitrary characteristic. S. Mesnager, F.
Ozbudak et A. Sinak, Proceedings of BalkanCryptSec 2015, LNCS
9540, pages 1730, 2015.
Abstract :
Bent and plateaued functions play a signicant role in
cryptography since they can possess various desirable
cryptographic characteristics. We provide the
characterizations of bent and plateaued functions in
arbitrary characteristic in terms of their secondorder
directional dierences. Moreover, we present a new
characterization of plateaued functions in arbitrary
characteristic in terms of fourth power moments of their
Walsh transforms. Furthermore, we give a new proof of the
characterization of vectorial bent functions in arbitrary
characteristic. Finally, we also present new
characterizations of vectorial splateaued functions in
arbitrary characteristic.
 On involutions of finite fields. P. Charpin, S.
Mesnager et S. Sarkar, Proceedings of 2015 IEEE International
Symposium on Information Theory, ISIT 2015, HongKong, 2015.
Abstract :
In this paper we study involutions over a finite field of
order $2^n$. We present some classes, several constructions
of involutions and we study the set of their fixed points.
 Cyclic codes and algebraic immunity of Boolean
functions. S. Mesnager et G. Cohen, Proceedings of the
IEEE Information Theory Workshop (ITW) 2015, Jerusalem,
Israel, 2015.
Abstract :
Since 2003, algebraic attacks have received a lot of
attention in the cryptography literature. In this context,
algebraic immunity quantifies the resistance of a Boolean
function to the standard algebraic attack of the
pseudorandom generators using it as a nonlinear Boolean
function. A high value of algebraic immunity is now an
absolutely necessary cryptographic criterion for a
resistance to algebraic attacks but is not sufficient,
because of more general kinds of attacks socalled Fast
Algebraic Attacks. In view of these attacks, the study of
the set of annihilators of a Boolean function has become
very important. We show that studying the annihilators of a
Boolean function can be translated into studying the
codewords of a linear code. We then explain how to exploit
that connection to evaluate or estimate the algebraic
immunity of a cryptographic function. Direct links between
the theory of annihilators used in algebraic attacks and
coding theory are established using an atypical univariate
approach.
 Variations on Minimal Linear Codes. G. Cohen
et S. Mesnager. Proceedings of the 4th International Castle
Meeting on coding theory and Application. Series: CIM Series
in Mathematical Sciences, Vol. 3, SpringerVerlag, pages
125131, 2015.
Abstract :
Minimal linear codes are linear codes such that the support
of every codeword does not contain the support of another
linearly independent codeword. Such codes have applications
in cryptography, e.g. to secret sharing. We pursue here
their study and construct asymptotically good families of
minimal linear codes. We also push further the study of
quasiminimal and almostminimal linear codes, relaxations
of the minimal linear codes.
 Characterizations of plateaued and bent functions
in characteristic p. S. Mesnager. Proceedings of the 8th
International Conference on SEquences and Their Applications
(SETA 2014), Melbourne, Australie, LNCS, Springer, pages
7282, 2014.
Abstract :
We characterize bent functions and plateaued functions in
terms of moments of their Walsh transforms. We introduce in
any characteristic the notion of directional difference and
establish a link between the fourth moment and that notion.
We show that this link allows to identify bent elements of
particular families. Notably, we characterize bent functions
of algebraic degree $3$.
 On semibent functions and related plateaued
functions over the Galois field $F_{2^n}$. S. Mesnager.
Proceedings "Open Problems in Mathematics and Computational
Science", LNCS, Srpinger, pages 243273, 2014
Abstract :
Plateaued functions have been introduced in 1999 by Zheng
and Zhang as good candidates for designing cryptographic
functions since they possess desirable various cryptographic
characteristics. They are defined in terms of the
WalshHadamard spectrum. Plateaued functions bring together
various nonlinear characteristics and include two important
classes of Boolean functions defined in even dimension: the
wellknown bent functions and the semibent functions. Bent
functions (including their constructions) have been
extensively investigated for more than 35 years. Very
recently, the study of semibent functions has attracted the
attention of several researchers. Many progresses in the
design of such functions have been made. The paper is
devoted to certain plateaued functions. The focus is
particularly on semibent functions defined over the Galois
field $\GF n$ ($n$ even). We review what is known in this
framework and investigate constructions.
 A note on linear codes and algebraic immunity of
Boolean functions, S. Mesnager. Proceedings of the 21st
International Symposium on Mathematical Theory of Networks and
Systems (MTNS 2014), Invited session "Coding Theory: Coding
for Security", pages 923927, Groningen, the Netherlands, 2014
Abstract :
Since 2003, Algebraic Attacks have received a lot of
attention in the cryptography literature. In this context,
algebraic immunity quantifies the resistance of a Boolean
function to the standard algebraic attack of the
pseudorandom generators using it as a nonlinear Boolean
function. A high value of algebraic immunity is now an
absolutely necessary cryptographic criterion for a
resistance to algebraic attacks but is not sufficient,
because of a more general kind of attacks so called Fast
Algebraic Attacks. In view of these attacks, the study of
the set of annihilators of a Boolean function has become
very important. We show that studying the annihilators of a
Boolean function can be translated in studying the codewords
of a linear code. We then explain how to exploit that
connection to evaluate or estimate the algebraic immunity of
a cryptographic function.
 Implementation of Faster Miller over
BarretoNaehrig Curves in Jacobian Cordinates, A. Mrabet
Amine, B. Bouallegue, M. Machhout, N. EL Mrabet et S.
Mesnager, Proceedings of GSCIT 2014IEEE, pages 16, 2014.
Abstract :
Few years ago, cryptography based on elliptic curves was
increasingly used in the field of security. It has also
gained a lot of importance in the academic community and
industry. This is particularly due to the high level of
security that it offers with relatively small size of the
keys, in addition to its ability to the construction of
original protocols which are characterized by high
efficiency. Moreover, it is a technique of great interest
for hardware and software implementation. Pairingfriendly
curves are important for speeding up the arithmetic
calculation of pairing on elliptic curves such as the
BarretoNaehrig (BN) curves that arguably constitute one of
the most versatile families. In this paper, the proposed
architecture is designed for field programmable gate array
(FPGA) platforms. We present implementation results of the
Miller’s algorithm of the optimal ate pairing targeting the
128bit security level using such a curve BN defined over a
256bit prime field. And we present also a fast formulas for
BN ellipticcurve addition and doubling. Our architecture is
able to compute the Miller’s algorithm in just 638337 of
clock cycles.
 On Minimal and AlmostMinimal Linear Codes, G.
Cohen et S. Mesnager, Proceedings of the 21st International
Symposium on Mathematical Theory of Networks and Systems (MTNS
2014), Session "Théorie des codes", pages 928931 Groningen,
Pays bas, 2014.
Abstract :
Minimal linear codes are such that the support of every
codeword does not contain the support of another linearly
independent codeword. Such codes have applications in
cryptography, e.g. to secret sharing and secure twoparty
computations. We pursue here the study of minimal codes and
construct infinite families with asymptotically nonzero
rates. We also introduce a relaxation to almost minimal
codes, where a fraction of codewords is allowed to violate
the minimality constraint. Finally, we construct new minimal
codes based on hyperovals.
 Semibent functions from oval polynomials, S.
Mesnager, Proceedings of Fourteenth International Conference
on Cryptography and Coding, Oxford, United Kingdom, IMACC
2013, LNCS 8308, pages. 115. Springer, Heidelberg, 2013.
Abstract :
Although there are strong links between finite geometry and
coding theory (it has been proved since 1960's that all
these connections between the two areas are important from
theoretical point of view and for applications), the
connections between finite geometry and cryptography remains
little studied. In 2011, Carlet and Mesnager have showed
that projective finite geometry can also be useful in
constructing significant cryptographic primitives such as
plateaued Boolean functions. Two important classes of
plateaued Boolean functions are those of bent functions and
of semibent functions, due to their algebraic and
combinatorial properties. In this paper, we show that oval
polynomials (which are closely related to the hyperovals of
the projective plane) give rise to several new constructions
of infinite classes of semibent Boolean functions in even
dimension.
 On Minimal and quasiminimal linear codes, G.
Cohen, S. Mesnager et A. Patey, Proceedings of Fourteenth
International Conference on Cryptography and Coding, Oxford,
United Kingdom, IMACC 2013, LNCS 8308, pages 8598. Springer,
Heidelberg, 2013.
Abstract :
Minimal linear codes are linear codes such that the support
of every codeword does not contain the support of another
linearly independent codeword. Such codes have applications
in cryptography, e.g. to secret sharing. We here study
minimal codes, give new bounds and properties and exhibit
families of minimal linear codes. We also introduce and
study the notion of quasiminimal linear codes, which is a
relaxation of the notion of minimal linear codes, where two
nonzero codewords have the same support if and only if they
are linearly dependent.
 On hyperbent functions via Dillonlike exponents,
S. Mesnager et J.P. Flori, ISIT 2012IEEE Internaional
Symposium on Information Theory, IMT, Cambridge, MA, USA,
2012.
Abstract :
This paper is devoted to hyperbent functions with multiple
trace terms (including binomial functions) via Dillonlike
exponents. We show how the approach developed by Mesnager to
extend the Charpin–Gong family and subsequently extended by
Wang et al. fits in a much more general setting. To this
end, we first explain how the original restriction for
Charpin–Gong criterion can be weakened before generalizing
the Mesnager approach to arbitrary Dillonlike exponents.
Afterward, we tackle the problem of devising infinite
families of extension degrees for which a given exponent is
valid and apply these results not only to reprove
straightforwardly the results of Mesnager and Wang et al.,
but also to characterize the hyperbentness of new infinite
classes of Boolean functions.
 Semibent functions with multiple trace terms and
hyperelliptic curves, S. Mesnager, Proceeding of
International Conference on Cryptology and Information
Security in Latin America, Latincrypt 2012, LNCS 7533,
Springer, pages 1836, 2012.
Abstract :
Semibent functions with even number of variables are a
class of important Boolean functions whose Hadamard
transform takes three values. Semibent functions have been
extensively studied due to their applications in
cryptography and coding theory. In this paper we are
interested in the property of semibentness of Boolean
functions defined on the Galois field $\GF n$ (n even) with
multiple trace terms obtained via Niho functions and two
Dillonlike functions (the first one has been studied by the
author and the second one has been studied very recently by
Wang et al. using an approach introduced by the author). We
subsequently give a connection between the property of
semibentness and the number of rational points on some
associated hyperelliptic curves. We use the hyperelliptic
curve formalism to reduce the computational complexity in
order to provide an efficient test of semibentness leading
to substantial practical gain thanks to the current
implementation of point counting over hyperelliptic curves.
 Niho Bent Functions and Subiaco Hyperovals, T.
Helleseth, A. Kholosha et S. Mesnager, Proceedings of the
10th International Conference on Finite Fields and Their
Applications (Fq'10), Contemporary Math., AMS, 2012. Vol 579,
pages 91101, 2012.
Abstract :
In this paper, the relation between binomial Niho bent
functions discovered by Dobbertin et al. and opolynomials
that give rise to Subiaco class of hyperovals is found. This
allows to expand the original class of bent functions in the
case when $m \equiv 2 (mod 4)$. These results provide an
interesting connection between Hadamard and cyclic
difference sets.
 Dickson polynomials, hyperelliptic curves and
hyperbent functions,J.P. Flori et S. Mesnager,
Proceedings of 7th International conference SEquences and
Their Applications, SETA 2012, Waterloo, Canada. LNCS 7780,
pages 4052, Springer, 2012.
Abstract :
In this paper, we study the action of Dickson polynomials on
subsets of finite fields of even characteristic related to
the trace of the inverse of an element and provide an
alternate proof of a not so wellknown result. Such
properties are then applied to the study of a family of
Boolean functions and a characterization of their
hyperbentness in terms of exponential sums recently
proposed by Wang et al. Finally, we extend previous works of
Lisonek and Flori and Mesnager to reformulate this
characterization in terms of the number of points on
hyperelliptic curves and present some numerical results
leading to an interesting problem.
 On Dillon’s class H of Niho bent functions and
opolynomials, C. Carlet et S. Mesnager, Symposium on
Artificial Intelligence and Mathematics (ISAIM 2012), Fort
Lauderdale, Floride, USA, 2012.
Abstract :
This extended abstract is a reduced version of the paper
(Carlet and Mesnager 2011). We refer to this paper for the
proofs and for complements.
 Binary Kloosterman sums with value 4, J.P.
Flori, S. Mesnager et G. Cohen. Proceedings of Thirteenth
International Conference on Cryptography and Coding, Oxford,
Angleterre, IMACC 2011, LNCS 7089 pages 6178, Springer, 2011.
Abstract :
Kloosterman sums have recently become the focus of much
research, most notably due to their applications in
cryptography and their relations to coding theory. Very
recently Mesnager has showed that the value 4 of binary
Kloosterman sums gives rise to several infinite classes of
bent functions, hyperbent functions and semibent functions
in even dimension. In this paper we analyze the different
strategies used to find zeros of binary Kloosterman sums to
develop and implement an algorithm to find the value 4 of
such sums. We then present experimental results showing that
the value 4 of binary Kloosterman sums gives rise to bent
functions for small dimensions, a case with no mathematical
solution so far.
 Sphere coverings and Identifying Codes, D.
Auger, G. Cohen et S. Mesnager, Proceeding of 3rd
International Castle Meeting on coding theory and Application
(3ICMTA), Barcelone, Espagne, 2011.
Abstract :
In any connected, undirected graph $G=(V,E)$, the {\it
distance} $d(x,y)$ between two vertices $x$ and $y$ of $G$
is the minimum number of edges in a path linking $x$ to $y$
in $G$. A {\it sphere} in $G$ is a set of the form $S_r(x) =
\{ y \in V : d(x,y)=r \},$ where $x$ is a vertex and $r$ is
a nonnegative integer called the {\it radius} of the sphere.
We first address in this paper the following question : What
is the minimum number of spheres with fixed radius $r \geq
0$ required to cover all the vertices of a finite,
connected, undirected graph $G$ ? We then turn our attention
to the Hamming Hypercube of dimension $n$, and we show that
the minimum number of spheres {\it with any radii} required
to cover this graph is either $n$ or $n+1$, depending on $n
\mod 2$. We also relate the two above problems to other
questions in combinatorics, in particular to identifying
codes.
 On the Dual of Bent Functions with 2^r Niho
Exponents, C. Carlet, T. Helleseth, A. Kholosha et S.
Mesnager, IEEE International Symposium on Information Theory,
ISIT 2011, pages 703707, SaintPetersturg, Russie,
Juilletaout 2011.
Abstract :
Computed is the dual of the Niho bent function consisting of
$2^r$ exponents that was found by Leander and Kholosha. The
algebraic degree of the dual is calculated and it is shown
that this new bent function is not of the Niho type. This
note is a followup of the recent paper by Carlet and
Mesnager.
 Generalized witness sets, G. Cohen et S.
Mesnager, Proceeding 1st International Conference on Data
Compression, Communication and Processing CCP 2011, Italie,
2124 juin 2011.
Abstract :
Given a set C of qary ntuples and c in C, how many symbols
of c suffice to distinguish it from the other elements in C?
This is a generalization of an old combinatorial problem, on
which we present (asymptotically tight) bounds and
variations.
 On the link of some semibent functions with
Kloosterman sums, S. Mesnager et G. Cohen, Proceeding of
International Workshop on Coding and Cryptology, IWCC 2011,
LNCS 6639, pages 263272, Springer, Heidelberg ,2011.
Abstract :
We extensively investigate the link between the
semibentness property of some functions in polynomial forms
and Kloosterman sums.
 On a conjecture about binary strings distribution,
J. P. Flori, H. Randriambololona, G. Cohen et S. Mesnager,
Proceedings of 6th International conference SEquences and
Their Applications, SETA 2010, Paris, France, SETA 2010, LNCS
6338, pages 346358. Springer, Heidelberg (2010).
Abstract :
It is a diﬃcult challenge to ﬁnd Boolean functions used in
stream ciphers achieving all of the necessary criteria and
the research of such functions has taken a signiﬁcant delay
with respect to crypt analyses. Very recently, an inﬁnite
class of Boolean functions has been proposed by Tu and Deng
having many good cryptographic properties under the
assumption that the following combinatorial conjecture about
binary strings is true: Conjecture. Let $S_{t,k}$ be the
following set: $S_{t,k}=\{(a,b) \in \left(\Zk\right)^2  a +
b = t and w(a) + w(b) < k}$. Then the size of $S_{t,k}$
is less or equal to $2^{k1}$. The main contribution of the
present paper is the reformulation of the problem in terms
of carries which gives more insight on it than simple
counting arguments. Successful applications of our tools
include explicit formulas of the cardinality of $S_{t,k}$
for numbers whose binary expansion is made of one block, a
proof that the conjecture is asymptotical ly true and a
proof that a family of numbers (whose binary expansion has a
high number of 1's and isolated 0's) reaches the bound of
the conjecture. We also conjecture that the numbers in that
family are the only ones reaching the bound.
 Recent Results on Bent and Hyperbent Functions and
Their Link With Some Exponential Sums, S. Mesnager, IEEE
Information Theory Workshop (ITW 2010), Dublin, Iralande,
AoutSeptembre 2010.
Abstract :
Bent functions are maximally nonlinear Boolean functions
with an even number of variables. They were introduced by
Rothaus in 1976. For their own sake as interesting
combinatorial objects, but also because of their relations
to coding theory (ReedMuller codes) and applications in
cryptography (design of stream ciphers), they have attracted
a lot of research, specially in the last 15 years. The class
of bent functions contains a subclass of functions,
introduced by Youssef and Gong in 2001, the socalled
hyperbent functions whose properties are still stronger and
whose elements are still rarer than bent functions. Bent and
hyperbent functions are not classified. A complete
classification of these functions is elusive and looks
hopeless. So, it is important to design constructions in
order to know as many of (hyper)bent functions as possible.
This paper is devoted to the constructions of bent and
hyperbent Boolean functions in polynomial forms. We survey
and present an overview of the constructions discovered
recently. We extensively investigate the link between the
bentness property of such functions and some exponential
sums (involving Dickson polynomials)
 Hyperbent Boolean Functions with Multiple Trace
Terms, S. Mesnager, Proceedings of International
Workshop on the Arithmetic of Finite Fields, WAIFI 2010, LNCS
6087, pages. 97113. Springer, Heidelberg (2010).
Abstract :
Bent functions are maximally nonlinear Boolean functions
with an even number of variables. These combinatorial
objects, with fascinating properties, are rare. The class of
bent functions contains a subclass of functions the
socalled hyperbent functions whose properties are still
stronger and whose elements are still rarer. In fact,
hyperbent functions seem still more difficult to generate
at random than bent functions and many problems related to
the class of hyperbent functions remain open. (Hyper)bent
functions are not classified. A complete classification of
these functions is elusive and looks hopeless. In this
paper, we contribute to the knowledge of the class of
hyperbent functions on finite fields $\GF n$ (where $n$ is
even) by studying a subclass $\mathfrak {F}_n$ of the
socalled Partial Spreads class $PS^$ (such functions are
not yet classified, even in the monomial case). Functions of
$\mathfrak {F}_n$ have a general form with multiple trace
terms. We describe the hyperbent functions of $\mathfrak
{F}_n$ and we show that the bentness of those functions is
related to the Dickson polynomials. In particular, the link
between the Dillon monomial hyperbent functions of
$\mathfrak {F}_n$ and the zeros of some Kloosterman sums has
been generalized to a link between hyperbent functions of
$\mathfrak {F}_n$ and some exponential sums where Dickson
polynomials are involved. Moreover, we provide a possibly
new infinite family of hyperbent functions. Our study
extends recent works of the author and is a complement of a
recent work of Charpin and Gong on this topic.
 A new family of hyperbent Boolean functions in
polynomial form, S. Mesnager, Proceedings of Twelfth
International Conference on Cryptography and Coding.
Cirencester, Angleterre, IMACC 2009, LNCS 5921, pages 402417.
Springer, Heidelberg (2009).
Abstract :
Bent functions are maximally nonlinear Boolean functions and
exist only for functions with even number of inputs. These
combinatorial objects, with fascinating properties, are
rare. The class of bent functions contains a subclass of
functions the socalled hyperbent functions whose
properties are still stronger and whose elements are still
rarer. (Hyper)bent functions are not classified. A complete
classification of these functions is elusive and looks
hopeless. So, it is important to design constructions in
order to know as many of (hyper)bent functions as possible.
Few constructions of hyperbent functions defined over the
Galois field $\GF{n}$ ($n = 2m$) are proposed in the
literature. The known ones are mostly monomial functions.\\
This paper is devoted to the construction of hyperbent
functions. We exhibit an infinite class over $\GF{n}$
($n=2m$, $m$ odd) having the form $f(x) = \tr {o(s_1)} (a x^
{s_1}) + \tr {o(s_2)} (b x^{s_2})$ where $o(s_i$) denotes
the cardinality of the cyclotomic class of $2$ modulo
$2^n1$ which contains $s_i$ and whose coefficients $a$ and
$b$ are, respectively in $\GF{{o(s_1)}}$ and
$\GF{{o(s_2)}}$. We prove that the exponents
$s_1={3(2^m1)}$ and $s_2={\frac {2^n1}3}$, where
$a\in\GF{n}$ ($a\not=0$) and $b\in\GF[4]{}$ provide a
construction of hyperbent functions over $\GF{n}$ with
optimum algebraic degree. We give an explicit
characterization of the bentness of these functions, in
terms of the Kloosterman sums and the cubic sums involving
only the coefficient $a$.
 A new class of Bent Boolean functions in polynomial
forms, S. Mesnager, Proceedings of international
Workshop on Coding and Cryptography, WCC 2009, pages 518,
Ullensvang, Norvége.
Abstract :
Bent functions are maximally nonlinear Boolean functions and
exist only for functions with even number of inputs. This
paper is a contribution to the construction of bent
functions over $\GF{n}$ ($n=2m$) having the form $f(x) = \tr
{o(s_1)} (a x^ {s_1}) + \tr {o(s_2)} (b x^{s_2})$ where
$o(s_i$) denotes the cardinality of the cyclotomic class of
2 modulo $2^n1$ which contains $s_i$ and whose coefficients
$a$ and $b$ are, respectively in $F_{2^{o(s_1)}}$ and
$F_{2^{o(s_2)}}$. Many constructions of monomial bent
functions are presented in the literature but very few are
known even in the binomial case. We prove that the exponents
$s_1=2^{\frac n2}1$ and $s_2={\frac {2^n1}3}$, where
$a\in\GF{n}$ ($a\not=0$) and $b\in\GF[4]{}$ provide a
construction of bent functions over $\GF{n}$ with optimum
algebraic degree. For $m$ odd, we give an explicit
characterization of the bentness of these functions, in
terms of the Kloosterman sums. For $m$ even, we give a
necessary condition in terms of these Kloosterman sums.
 Secret Sharing Schemes Based on Selfdual Codes,
S.T. Dougherty, S. Mesnager et P. Solé. IEEE Information
Theory Workshop (ITW 2008), Porto, Portugal 59 Mai 2008.
Abstract :
Secret sharing is an important topic in cryptography and has
applications in information security. We use selfdual codes
to construct secretsharing schemes. We use combinatorial
properties and invariant theory to understand the access
structure of these secretsharing schemes. We describe two
techniques to determine the access structure of the scheme,
the first arising from design properties in codes and the
second from the Jacobi weight enumerator, and invariant
theory.
 On immunity profile of Boolean functions, C.
Carlet, P. Guillot. et S. Mesnager, Proceedings of SEquences
and Their Applications, SETA 2006, Pékin, Chine. Lecture Notes
in Computer Science, pages 364375, 2006, Springer.
Abstract :
The notion of resilient function has been recently weakened
to match more properly the features required for Boolean
functions used in stream ciphers. We introduce and we study
an alternate notion of almost resilient function. We show
that it corresponds more closely to the requirements that
make the cipher more resistant to precise attacks.
 On the Walsh support of Boolean functions, C.
Carlet et S. Mesnager. Proceedings of the first workshop on
Boolean functions: Cryptography and Applications, BFCA'05,
Rouen, France, Mars 2005, pages 6582.
Abstract :
In this paper, we study, in relationship with covering
sequences, the structure of those subsets of $\V {n}$ which
can be the Walsh supports of Boolean functions.
 NonLinearity and Security of Self Synchronizing
Stream Ciphers, P. Guillot et S. Mesnager. International
Symposium on Nonlinear Theory and its Applications, NOLTA
2005, Bruges, Belgique, Octobre 2005.
Abstract :
Several chaos based ciphers has been proposed that exploit
the ergodic property of chaotic orbits. As chaotic systems
are unstable and have sensitive dependence on initial
conditions, the main difficulty for the receiver is to
reproduce the chaotic signal that has been generated by the
sender in order to correctly decrypt the message. This is
performed by a self synchronizing device. In discrete
cryptography, the closest scheme is the so called self
synchronizing stream cipher (SSSC). After recalling general
security models for assessing cryptographic algorithms, we
present SSSC scheme and two examples of cryptanalysis. In
order to resist to theses attacks, the ciphering function
must satisfy high non linearity properties which are
presented.
 Improving the upper bounds on the covering radii of
ReedMuller codes, C. Carlet et S. Mesnager, IEEE
International Symposium on Information Theory, ISIT 2005,
Australie, Septembre 2005.
Abstract :
By deriving bounds on character sums of Boolean functions
and by using the characterizations, due to Kasami and
Tokura, of those elements of the ReedMuller codes whose
Hamming weights are smaller than twice the minimum distance,
we derive an improved upper bound on the covering radius of
the ReedMuller code of order 2, and we deduce improved
upper bounds on the covering radii of the ReedMuller codes
of higher orders.
 Test of monomorphism for finitely generated
morphisms between affine schemes. S. Mesnager,
Proceedings of the sixth workshop on Computer Algebra in
Scientific Computing, CASC'04, Euler International
Mathematical Institute, SaintPétersbourg, Russie, Juilllet
2004, pages 348357.
Abstract :
In this paper, we give algorithmic criterion for morphisms
of finite type between affine schemes to be a monomorphism.
As side results, this paper also contains an algorithmic
test for separability and an algorithmic criterion for
``radiciality'' in the sense of Grothendieck.
Livres et chapitres de livres:
(dans l’ordre chronologique inverse)
 "Linear codes from functions"S. Mesnager,
Chapitre 20 in A
Concise Encyclopedia of Coding Theory CRC Press/Taylor
and Francis Group (Publisher) London, New York, 2021 (94
pages)
 "Direct Sum Masking as a Countermeasure to
SideChannel and Fault Injection Attacks"C. Carlet, S.
Guiley et S. Mesnager, Chapitre dans Security and Privacy in
the Internet of Things 2019 : 148166, 2019.
 "Construction of Efficient Codes for HighOrder
Direct Sum Masking" C. Carlet, S. Guilley, C. Guneri, S.
Mesnager et F. Ozbudak, Chapitre dans Security and Privacy in
the Internet of Things 2019 : 108128, 2019.
 Livre
"Bent functions: fundamentals and results", S.
Mesnager, Springer Swiss, 2016.
 Livre "Arithmetic of Finite Fields", Ç.K. Koç,
S. Mesnager et E. Savaş, 5th International Workshop, WAIFI
2014, Volume 9061, pages 1213, Springer, 2015.
 Livre "Corps finis et théorie des codes", S.
Mesnager, Pearson Education, 2007 (En Français).
Présidente de comités de programmes
dans des conférences internationales
 Coprésidente et organisatrice (avec
Claude Carlet) of the International Castle Meeting
on Coding Theory and Applications" 2023, Florence,
Italie.
 Coprésidente et organisatrice (avec
Zhengchun Zhou) de "Workshop on the Arithmetic of
Finite Fields", international conference WAIFI
2022, Chengdu, Chine.
 Coprésidente of International Conference
on Security and Privacy (ICSP 2021), Jamshedpur
Inde, Novembre 1617, 2021.
 Coprésidente et organisatrice de
"International conference in Finite Fields and
Their Applications", Paris, France, 1317 Juin
2022.
 Coprésidente (avec Kojima Tetsuya et
Kwang Soon Kim) of the international conference
IWSDA 2021 (The 10th International Workshop on
Signal Design and its Applications in
Communications), Aout 15, Angleterre 2022.
 Coorganisatrice (avec Hugues
Randriambololona and Gilles Zemor) de la
conférence "Codes and combinatorics" le 45
Juillet 2016 à Telecom Paris, Paris, France.
 Présidente du comité de programme de la
conférence ICCC 2015 ,International
Conference on Coding and Cryptography,
Alger, Algerie, 25 Novembre 2015.
 Coprésidente (avec Ilias Kosterias et
Kenza Guenda) du comité de programme de la Session
"Computational aspects and mathematical methods
for finite field and their applications in
information theory" dans la conférence
internationale ACA 2015 ,
International Conference on Applications of
Computer Algebra, Kalamata, Grece, 2023
Juillet 2015.
 Coprésidente (avec Erkay Savas) du
comité de programme de la conférence WAIFI 2014,
International Workshop on the Arithmetic of Finite
Fields, Gebze, Turquie, 2628 Septembre 2014.
Participations à des comités
de programmes dans des conférences
internationales
 Membre du comité de
programme du congrès International
12th International Workshop on Coding
and Cryptography (WCC 2022), Mars 711
2022, Rostock, Allemagne.
 Membre du comité de
programme du congrès International
The 6th International Workshop on
Boolean Functions and their
Applications (BFA 2021), Granada,
Espagne, Septembre 610, 2021.
 Membre du comité de
programme de "International
Conference International
Workshop on Trusted Smart
Contracts" (WTSC 2021),
Grenada, Mars 2021.
 Membre du
comité de programme de
"International Conference
on Security and Privacy
"(ICSP 2020), Indee,
0506, Novembre 2020.
 Membre du
comité de programme de
"5th International
Conference on Computer
and Communication
System", Shanghai,
Chine, Mai 1517,
2020.
 Membre du
comité de programme
de la 11eme
conference
internationale on
SEquences and Their
Applications (SETA
2020),
SaintPetersburg,
Russie, 2225
Septembre 2020.
 Membre
du comité de
programme de la
conference
internationale
"Workshop on
Trusted Smart
Contracts" WTSC20
, Fevrier 14,
2020.
 Membre
du comité de
programme du ,
International
Workshop on the
Arithmetic of
Finite Fields "
(WAIFI 2020)
Rennes, France,
Juillet 68, 2020.

Membre du
comité de
programme du
5eme,
International
Workshop on
Boolean
Functions and
their
Applications"
(BFA 2020)
Granada,
Espagne, Mai
2529, 2020.

Membre du
comité de
programme
congrès
International
"Workshop on
Trusted Smart
Contracts"
WTSC19, 2019.

Membre du
comité de
programme du
3eme congrès
international
C2SI2019 "
International
Conference on
Codes,
Cryptology and
Information
Security",
Rabat, Maroc,
Avril 2224,
2019.

Membre du
comité de
programme de
C2 codes et
cryptographie,
Octobre 2018.

Membre du
comité de
programme du,
10th
International
Workshop on
Coding and
Cryptography
(WCC 2017)
St Petersburg,
Russie, 1822
Septembre,
2017.

Membre du
comité de
programme du
congré
international
Castle Meeting
on Coding
Theory and
Applications",
5ICMCTA
,"5th
International
Castle Meeting
on Coding
Theory and
Applications"
Estonie,
AoutSeptembre
2017.

Membre du
comité de
programme du,
2sd
International
Conference
"Codes,
Cryptology and
Information
Security"
Rabat, Maroc,
1012, Avril
2017.

Membre du
comité de
programme de
9th
International
Conference on
SEquences and
Their
Applications
(SETA 2016),
Chengdu, Chine
914 Octobre
2016.

Membre du
comité de
programme de International
Workshop on
the Arithmetic
of Finite
Fields (WAIFI
2016),
Ghent,
Belgique,
1316 Juillet
2016.
 Membre
du comité de
programme de 2sd
International
Conference on
Cryptography
and its
Applications
ICCA 2016 UST,
Oran, Algerie
2627 Avril
2016.
 Membre
du comité de
programme du
congré
international
9th
International
Workshop on
Coding and
Cryptography
(WCC 2015)
Paris, France
1317 Avril
2015.

Membre du
comité de
programme du
congré
international
SETA
2014, "8th
International
Conference on
SEquences and
Their
Applications"
Melbourne,
Australie,
2428 novembre
2014.
 Membre
du comité de
programme de International
Workshop on
the Arithmetic
of Finite
(WAIFI 2014)
Fields, Gebze,
Turquie, 2628
Septembre
2014.
 Membre
du comité de
programme du
congré
international
Castle Meeting
on Coding
Theory and
Applications",
4ICMCTA
, "4th
International
Castle Meeting
on Coding
Theory and
Applications"
Pamela,
Portugal 1518
Septembre
2014.
 Membre
du comité de
programme du
congré
international
WCC
2013, "8th
International
Workshop on
Coding and
Cryptography"
Bergen,
Norvége 1519
Avril 2013.
 Membre
du comité de
programme du
congré
international
WCC
2011, "7th
International
Workshop on
Coding and
Cryptography"
Paris, France,
1115 Avril
2011.
 Membre
du comité de
programme du
congré
international
SETA
2010, "6th
International
Conference on
SEquences and
Their
Applications"
Paris, France,
1217
septembre
2010.
 Membre
du comité de
programme du
congré
international
Africacrypt
2009, "2sd
African
International
Conference on
Cryptology "
Gammarth,
Tunisie, 2125
juin 2009.
Membre de
directeurs de comités
 International
Workshop on the
Arithmetic of Finite
Fields
Editrice
dans des journaux
internationaux

Editrice en
Chef (avec
Jintai Ding)
du journal
international
Advances
in Mathematics
of
Communications
(AMC)
Publié par
AIMS (American
Institute of
Mathematical
Sciences).
 Editrice
en Chef du
journal
international
International
Journal of
Information
and Coding
Theory"
(IJOCT).
 Editrice
au journal
international
IEEE
Transactions
on Information
Theory
(IEEEIT).
 Editrice
dans le
journal
international
Cryptography
and
Communications
Discrete
Structures,
Boolean
Functions and
Sequences
(CCDS)
Publié par
Springer.

Editrice dans
le journal
international
RAIR
ITA
(Theoretical
Informatics
and
Applications)
Publié par
Cambridge
University
Press.

Editrice dans
le journal
international
Computer
Mathematics:
Computer
Systems Theory
(IJCMTCOM)
publié par
Taylor
Francis.
Editrice
pour des
Special Issues
dans des
journaux
internationaux

International
Journal
IEEEInformation
Theory:
Special Issue
dedicated to
V. I.
Levenshtein,
2021.

International
Journal
Cryptography
and
Communications
Discrete
Structures,
Boolean
Functions, and
Sequences
(CCDS):
Special Issue:
"Contemporary
interactions
between codes,
cryptographic
functions
and/or
sequences,
20212022.

International
Journal of
mathematics:
Special Issue
"The
Cryptography
of
Cryptocurrency",
20202021.

International
Journal of
Computer
Mathematics
(IJCMCST):
Special Issue:
"Mathematics
of
Cryptography
and Coding in
the Quantum
Era",
20202021.
Exposés
Conférences
internationales
(dans l’ordre chronologique inverse)

On
constructions
of weightwise
perfectly
balanced
functions,
Conférence
Internationale
Workshop on
Boolean
Functions and
Their
Applications(BFA
2020)

On Strongly
Regular Graphs
from Weakly
Regular
Plateaued
Functions,
Conférence
Internationale
International
Conference“
The 9th
International
Workshop on
Signal Design
and its
Applications
in
Communication"(IWSDA’19)
en Chine,
Octobre 2019.

Constructions
of optimal
locally
recoverable
codes via
Dickson
polynomials,
Conférence
internationale
"The 14th
international
conference on
Finite Fields
and their
Applications"
(Fq14) à
Vancouver,
Canada Juin
2019.

Constructions
of optimal
locally
recoverable
codes via
Dickson
polynomials,
Conférence
internationale
"The Eleventh
International
Workshop on
Coding and
Cryptography"
(WCC 2019) à
Saint Malo,
France, Avril
2019.

Generalized
plateaued
functions and
admissible
(plateaued)
functions,
Conférence
internationale
Workshop on
Boolean
Functions and
Their
Applications(BFA
2017) à
Solstrand,
Norvège ,
Juillet 2017.

On the
nonlinearity
of Boolean
functions with
restricted
input,
Conférence
internationale
Finite field
and their
Applications
Fq13 à
GaetaItalie,
Juin 2017.

On
constructions
of bent
functions from
involutions,
IEEE
International
Symposium on
Information
Theory (ISIT
2016) à
Barcelone,
Espagne,
Juillet 2016.

On
construction
of bent
functions
involving
symmetric
functions and
their duals,
Conference
Internationale
"Workshop on
Mathematics in
Communications
(WMC 2016),
Santander,
Espagne,
Juillet 2016.

Fast algebraic
immunity of
Boolean
functions,
Conference
Internationale
"Workshop on
Mathematics in
Communications
(WMC 2016),
Santander,
Espagne,
Juillet 2016.

Explicit
constructions
of bent
functions from
pseudoplanar
functions,
Conference
Internationale
"Workshop on
Mathematics in
Communications
(WMC 2016),
Santander,
Espagne,
Juillet 2016.

On
constructions
of bent,
semibent and
five valued
spectrum
functions from
old bent
functions,
Conference
Internationale
"Workshop on
Mathematics in
Communications
(WMC 2016),
Santander,
Espagne,
Juillet 2016.

On the
diffusion
property of
iterated
functions,
International
Conference on
Cryptography
and Coding,
Oxford, United
Kingdom,
Decembre 2015.

On pary bent
functions from
(maximal)
partial
spreads,
Conférence
internationale
Finite field
and their
Applications
Fq12, New
York, Juillet
2015.

Dickson
Polynomials
that are
Involutions,
Conférence
internationale
Finite field
and their
Applications
Fq12, New
York, Juillet
2015.

On involutions
of finite
fields,
Conférence
internationale
ISIT 2015
International
Symposium on
Information
Theory
HongKong,
Chine, Juin
2015.

Cyclic codes
and Algebraic
immunity of
Boolean
functions,
Conférence
internationale
IEEE Workshop
Information
Theory (ITW
2015),
Jérusalem,
Israel, Avril
2015.

Characterizations
of plateaued
and bent
functions in
characteristic
p,
Conférence
internationale
8th
International
Conference on
SEquences and
Their
Applications
(SETA 2014),
Melbourne,
Australie,
Novembre 2014.

Semibent
functions from
oval
polynomials.
Conférence
internationale
Cryptography
and Coding
IMACC 2013,
Oxford,
Angleterre,
Decembre 2013.

Bent functions
from spreads.
Conférence
internationale
Finite Fields
and their
Applications,
Fq11,
Magdebourg,
Allemagne,
Juillet 2013.

Semibent
functions with
multiple trace
terms and
hyperelliptic
curves.
Conférence
internationale,
Cryptology and
Information
Security in
Latin America
(Latincrypt)
2012 Santiago,
Chili, Octobre
2012.

Bent and
hyperbent
functions via
Dillonlike
exponents.
Conférence
internationale,
Yet Another
Conference on
Cryptography
(YACC 2012)
2012. Iles de
Porquerolles,
France,
Septembre
2012.

On hyperbent
functions via
Dillonlike
exponents.
Conférence
internationale,
ISIT 2012.
IEEE
International
Symopsium on
Infomation
Theory à IMT,
Boston, USA,
Juillet 2012.

Dickson
polynomials,
hyperelliptic
curves and
hyperbent
functions.
Conférence
internationale,
SETA (The 7th
international
conference on
SEquences and
Their
Applications)
à Waterloo
(Canada), juin
2012.

New semibent
functions with
multiple trace
terms.
Conférence
internationale
sur
invitation,
Workshop
Information
Theory and
Applications
(ITA 2012) à
San Diego
(USA), Février
2012.

Identifying
and Covering
by Spheres.
25eme
Conférence
internationale
on
Combinatorics,
Cryptography,
and Computing
(MCCCC), Las
Vegas (USA),
Octobre 2011.

Sphere
coverings and
Identifying
Codes.
Conférence
internationale
Castle Meeting
on coding
theory and
Application
(3ICMTA),
Cardona
(Espagne),
Septembre
2011.

On the link of
some semibent
functions with
Kloosterman
sums.
Conférence
internationale
sur
invitation,
Workshop of
International
Workshop on
Coding and
Cryptology
(IWCC 2011) à
Qingdao
(Chine), Mai
2011.

On the link of
some semibent
functions in
polynomial
forms with
exponential
sums.
Conférence
internationale
sur
invitation,
Workshop
Information
Theory and
Applications
(ITA 2011) à
San Diego
(USA), Février
2011.

Recent Results
on Bent and
Hyperbent
Functions and
Their Link
With Some
Exponential
Sums.
Conférence
internationale
(conférence
invitée)
Information
Theory
Workshop (ITW
2010) à Dublin
(Irlande),
Séptembre
2010.

Hyperbent
Boolean
Functions with
Multiple Trace
Terms.
Conférence
internationale,
Workshop on
the Arithmetic
of Finite
Fields (WAIFI
2010) à
Istanbul
(Turquie),
Juin 2010.

A new family
of hyperbent
Boolean
functions in
polynomial
form.
Conférence
internationale,Twelfth
International
Conference on
Cryptography
and Coding
(IMACC 2009) à
Cirencester
(Angleterre),
Décembre 2009.
 A
new class of
Bent Boolean
functions in
polynomial
forms.
Conférence
internationale
Workshop on
Coding and
Cryptography
(WCC 2009) à
Ullensvang
(Norvége), Mai
2009
 On
the number of
resilient
Boolean
functions.
Conférence
internationale,
Symposium on
Algebraic
Geometry and
its
Applications
(SAGA 2007) à
Papeete
(Tahiti), Mai
2007.
 On
immunity
profile of
Boolean
functions.
Conférence
internationale,
SEquences and
Their
Applications
(SETA 2006) à
Pékin (Chine),
Séptembre
2006.
 On
the Walsh
support of
Boolean
functions.
Conférence
internationale,
Boolean
Functions,
Cryptography
and
Applications
(BFCA 2005) à
Rouen
(France), Mars
2005.
Conférences
internationales
invitées
(dans l’ordre
chronologique
inverse)

Confèrence
invitée
intitulé
"Reader’s
digest of
“16year
achievements
on Boolean
functions and
open problems"
à
"International
conference
"The 4th
International
Workshop on
Boolean
Functions and
their
Applications"
(BFA 2020)",
Invitation de
Lilya
Budaghyan et
Tor Helleseth.

Conference
invitée" the
Applied
Algebra and
Geometry" UK
research
network à
l’université
d’Oxford.
Invitation de
Heather
Harrington
(Université
d’Oxford),
Angelterre,
Oxford,
Decembre 2019.

International
Conference“
The 9th
International
Workshop on
Signal Design
and its
Applications
in
Communications
" (IWSDA’19),
Octobre 2019,
Chine 2019.
Invitation de
Tor Helleseth
(University de
Bergen,
Norway), Zheng
Ma (Southwest
Jiaotong
University,
China),
HongYeop Song
(Yonsei
University,
Korea) et
Hideyuki Torii
(Kanagawa
Institute of
Technology,
Japan).

Conférence
"The 4th
International
Workshop on
Boolean
Functions and
their
Applications"
(BFA 2019),
Florence,
Italie, Juin
2019.
Invitation de
Lilya
Budaghyan, et
Tor Helleseth

Conférence
Internationale
CanaDam,
Discrete
mathematics,Vancouver,
Canada. Mai
2019.
Invitation par
les
organisateurs.

Conférence
Internationale
on Codes,
Cryptology And
Information
Security`
(C2SI), Rabat,
Maroc, Avril
2019.
Invitation par
les
organisateurs.

Workshop
international
"Contemporary
Coding Theory"
at Oberwolfach
(Germany),
March 2019.
Invitation of
Camilla
Hollanti
(University
Aalto),
Joachim
Rosenthal
(University of
Zurich), et
Marcus
Greferath
(University
Aalto).

Workshop
international
en codage
algebrique et
securité à
Dagstuhl
(Allemagne),
Decembre 2018.
Invitation de
Martin Bossert
(Universität
Ulm, DE),
Eimear Byrne
(University
College
Dublin, IE) et
Antonia
WachterZeh
(TU München,
DE).

Conference
Internationale
SETA 2018
(Sequences and
Their
Applications)
à Hong Kong,
Octobre 2018.

International
conference BFA
2018 (Boolean
Functions and
Applications)
en Novège Juin
2018.

Conference
Internationale
on Group,
Group Ring and
Related topics
(GGRRT 2017) à
Khorfakkan,
UAE, Novembre
2017.

Instructional
Workshop in
Cryptology à
New Delhi,
Inde, Octobre
2017.

Conférence
internationale
"Yet Another
Conference on
Cryptography"
(YACC 2016),
Iles de
Porquerolles,
France, Juin
2016.

International
Conference on
Cryptography
and Coding,
Oxford, United
Kingdom,
Decembre 2015.
Invitation de
Jens Groth.

Conférence
internationale
en fonctions
booléennes BFA
2014
"International
Workshop on
Boolean
Functions and
Their
Applications"
à Rosendal
(Norvège).
Invitation de
Lilya
Budaghyan, Tor
Helleseth et
Alexander
Kholosha.
 Conférence
internationale
en codage,
"The 21th
international
symposium on
Mathematical
Theory of
Networks and
Systems"
(MTNS2014),
Session
"Théorie des
codes" à
Groningen
(Pays Bas),
Juillet 2014.
Invitation de
Heide
GluesingLuerssen,
Joachim
Rosenthal et
Margreta
Kuijper.

Conférence
internationale
"Workshop on
Polynomials
over Finite
Fields:
Functional and
Algebraic
Properties"
Barcelone
(Espagne).
Invitation de
Joachim von
zur Gathen,
Jaime
Gutierrez,
Alina Ostafe,
Daniel Panario
et Alev
Topuzoglu.

International
seminar in
Coding Theory,
Dagstuhl
(Allemagne),
Aout 2013.
Invitation de
HansAndrea
Loeliger,
Emina Soljanin
et Judy L.
Walker.

International
Conférence
Trends in
coding theory,
Monté Verita
(Switzerland),
Octobre 2012.
Invitation de
Elisa Gorla,
Joachim
Rosenthal et
Amin
Shokrollahi.

International
Workshop on
finite fields
character sums
end
polynomials,
Strobl
(Autriche),
Septembre
2012.
Invitation des
oragnisateurs
de la
conférence.

International
workshop on
coding based
crypto (Ecrytp
2012), Lyngby
(Denemark) en
mai 2012.
Invitation de
Tom Høholdt.

International
Workshop
Information
Theory and
Applications
(ITA 2012) à
San Diego
(USA) en
Février 2012.
Invitation de
Alexander
Vardy.

International
seminar in
Coding Theory
à Dagstuhl,
Allemagne en
Novembre 2011.
Invitation de
Joachim
Rosenthal et
Amin
Shokrollahi.

International
Workshop on
Coding and
Cryptology
(IWCC 2011) à
Qingdao
(Chine) en Mai
2011.
Invitation de
Xian Hequn.

International
Workshop
Information
Theory and
Applications
(ITA 2011) à
San Diego
(USA) en
Février 2011.
Invitation de
Alexander
Vardy.

International
Information
Theory
Workshop (ITW
2010) à Dublin
(Irlande) en
Séptembre
2010.
Invitation de
Marcus
Greferath.
Conférences
nationales,
séminaires
(dans l’ordre
chronologique
inverse)

Seminaire AGAA
à l'
Université de
Paris 8/ Paris
13 (en visio
conference),
Mai 2020,
France.

Seminaire à l'
Université de
York, Février
2020,
Angleterre.
Invitation du
Professeur
Delaram
Kahrobaei.

Séminaire à
l’université
de Guangzhou,
Octobre 2019,
Chine.
Invitation du
professeur
Yuyin Yu.

Séminaire à
l’université
de Sun Yatsen
à Guangzhou,
Octobre 2019,
Chine.
Invitation du
professeur
ChangAn Zhao.

Séminaire
international
en codage
"Contemporary
Coding
Theory", Mars
2019 à
Oberwolfach
(Allemagne).
Invitation de
Camilla
Hollanti
(University
Aalto),
Joachim
Rosenthal
(University of
Zurich), et
Marcus
Greferath
(University
Aalto).

Seminaire à
l'INRIA Lyon,
France,
Janvier 2019.

Seminaire en
mathématiques
à l'université
de Porto,
Portugal,
Juillet 2018.

Seminaire en
mathématiques
à l'université
de Zurich,
Suisse,
Décembre 2017.

Seminaire
d'algébre et
théorie des
nombres à
l'Université
d'Aalto,
Finlande,
Février 2017.

Seminaire
protection de
l'information
à l'université
Paris 8,
Novembre 2016.

Seminaire de
mathmatiques
pour la
cryptographie
et theorie des
codes à
Telecom
Paristech,
France ,
Septembre
2016.

Seminaire de
mathmatiques
pour la
cryptographie
et theorie des
codes à
l'Academie des
Sciences,
Pékin, Chine,
Septembre
2016.

Seminaire de
mathmatiques
pour la
cryptographie
et theorie des
codes à
l'université
de Tianjin et
université de
Nankai, Chine,
Septembre
2016.

Seminaire en
mathématiques
discrètes à
Télécom
ParisTech,
Paris, France,
Septembre
2016.

Seminaire en
mathématiques
discrètes à
l'université
Paul Sabatier
(Institut de
maths IMT),
Toulouse,
France, Avril
2016.

Séminaire
"Combinatoire
et
algorithmique"
à l'université
de Rouen,
France,
Février 2016.

Séminaire à
HongKong
université
science et
technologie,
HongKong,
Chine, Juin
2015

Séminaire
d'Algèbre et
Géometrie à
l'université
de Versailles,
France, Avril
2015.
 Journée
thématique
Cryptographie
à l'université
de Cergy
(France),
Avril 2015.
Invitation de
Valerie Nachef
et Emmanuel
Volte.

Séminaire
Mathématiques
discrètes à
l'université
de Nanjing
(Chine),
Décembre 2014.
Invitation de
Xiwang Cao.

Séminaire
Cryptographie
à l'université
de Xuzhou
(Chine),
Décembre 2014.
Invitation de
Fengrong
Zhang.

Séminaire
"Algébre",
Département de
Mathtématiques
à l'université
UAE, Octobre
2014.

Séminaire
Combinatoire à
l'université
Paris XIII,
Mai 2014.

Séminaire à
l'université
Paris VI, Mai
2014.

Séminaire
Boole à
l'université
Paris VI,
France, Juin
2013.

Séminaire UCD
School of
Mathematical
Sciences,
Dublin,
Iralande,
Février 2012.

Séminaire
Boole à
l'institut
Henri
Poincaré,
Paris V,
France,
Janvier 2012.

Seminaire
Therie de
l'infomation,
Telecom
ParisTech,
France,
Decembre 2011.

Tutorial
(conférence
invitée),
journées
Codage et
Cryptographie
(C2) à St
Pierre
d'Oléron,
Avril 2011.

Séminaire
Arithmétique
et théorie de
l’information
(ATI) à
l'lnstitut de
Mathématiques
de Luminy,
France,
Février 2011.

Séminaire
Mathematiques
pour le
traitement de
l'information
et de l'image
(MTII) à
Université
Paris VIII,
Janvier 2011.

Séminaire
Boole à
l'institut
Henri
Poincaré,
Paris, France,
Mai 2010.

Séminaire
Mathematiques
pour le
traitement de
l'information
et de l'image
(MTII) à
Université
Paris VIII,
France, Juin
2009.

Séminaire I3S
à
SophiaAntipolis,
Nice, France,
Avril 2009.

Séminaire
Codes,
Cryptographie
et
Algorithmique
à l'ENSTA,
Paris, France,
Octobre 2005.

Séminaire de
combinatoire
algébrique de
l'université
de Paris 13,
France, Avril
2005.

Séminaire de
Cryptographie
de
l'université
de Rennes,
Rennes,
France, Avril
2005.

Séminaire pour
la sécurité de
l'information
de
l'université
de Paris VIII,
France, Juin
2003.

Séminaire de
géométrie
algébrique de
l'université
de Rennes I,
Rennes,
France, Avril
2002.

Forum des
jeunes
mathématiciennes
et
informaticiennes
à l'Institut
Henri
Poincaré,
Paris, France,
Mars 2002.
Invitations
internationales
par des
chercheurs

Invitation en
Octobre 2017
par
Professeurs
Shri Kant,
Shanta
Laishram et
Subhamoy
Maitra, New
Delhi, Inde.

Invitation en
Aout et
Septembre 2017
par
Professeurs Qi
Wang (Southern
University of
Science and
Technology,
Shenzhen,
Chine),
Yongzhuang
Wei, Minquan
Cheng et
Dianhua Wu
(University of
Guilin and
Guangxi Normal
University,
Chine),
Yanfeng Qi
(University of
School of
Science,
Hangzhou
Dianzi
University,
Hangzhou,
Chine),
Longjiang Qu
(National
University of
Defense
Technology,
Changsha,
Chine) et
Maosheng Xiong
(HongKong
university of
science and
technology,
HongKong).

Invitation en
Fevrier 2017
par
Professeurs
Marcus
Greferath et
Camilla
Hollanti dans
le departement
de
mathematiques
de
l'Université
d'Aalto,
Finlande.

Invitation en
Septembre 2016
par
Professeurs
Dongdai Lin,
Keqin Feng et
Baofeng Wu à
l'Acadamie des
Sciences,
Chine.

Invitation en
Septembre 2016
par
Professeurs
Francoise
Soulier,
Fangwei Fu et
Jian Liu à
l'univerité de
Tianjin et
université de
Nankai, Chine.
 Invitation
en Juillet
2016 par
professeur
Zhengchun
Zhou,
département de
mathématiques,
l'université
de Southwest
Jiaotong,
Chungdu,
Chine.
 Invitation
en Juin 2015
par professeur
Cunsheng Ding,
HongKong
université
science et
technologie,
HongKong,
Chine.
 Invitation
en octobre
2014 par
professeur
Kanat
Abdukhalikov,
departement of
mathematics,
El Ain, UAE.
 Invitation
en Septembre
2014 par
professeur
Ferruh
Özbudak,
Middle East
Technical
University,
Ankara,
Turquie.
 Invitation
en octobre
2013 par
professeur
Janos Korner,
Université de
Rome, Italie.
 Invitation
en novembre
2010 par
professeur
Simon Litsyn,
Université de
Tel Aviv,
Israel.
 Invitation
en septembre
2010 par
professeur
Marcus
Greferath.,
College
Dublin,
Irlande.