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 discrètes (corps finis, polynômes sur des corps finis, sommes exponentielles, sommes de caractères, transformée de Fourier discrète, fonctions courbes, fonctions "spéciales" sur des corps finis, etc.) ;
• Cryptographie symétrique;
• Théorie des codes correcteurs;
• Algèbre commutative et Géométrie algébrique effective;

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 correcteurs d'erreurs. Les deux principaux axes de ma recherche actuelle sont:
• Les fonctions hautement non-linéaires: ces fonctions sont d'une importance essentielle dans la cryptographie symétrique pour éviter certaines attaques fondamentales contre les chiffrements tels que la cryptanalyse linéaire. En particulier, je m'intéresse beaucoup aux constructions et caractérisations des fonctions courbes (ou plus généralement les fonctions plateaux) que sont des objets combinatoires fascinants qui jouent un role important dans plusieurs domaines (cryptographie, codage, théorie de séquences, etc). Dans l'approche algébrique j'utilise des corps finis, 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 d'erreurs: je travaille sur les aspects algébriques et combinatoires des codes correcteurs d'erreurs pour le canal classique. Les méthodes algébriques impliquent la construction de bons codes pour des diverses applications (stockage, partage du secret, etc).

• Je m'interesse aussi aux aspects algorithmiques dans les axes ci-dessus dans le contexte de l'algèbre informatique.

Publications

Revues internationales:

(dans l’ordre chronologique inverse)
1. Generalized plateaued functions and admissible (plateaued) functions, S. Mesnager, C. Tang et Y. Qi, Journal IEEE Transactions on Information Theory-IT, Vol. 61, Issue 10, pages 6139-6148, 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. P-ary plateaued functions have attracted recently some attention in the literature and many activities on generalized p-ary functions have been carried out. This paper increases our knowledge on plateaued functions in the general context of generalized p-ary 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 p-ary 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.
2. New constructions of optimal locally recoverable codes via good polynomials, J. Liu, S. Mesnager et L. Chen, Journal IEEE Transactions on Information Theory-IT. A paraître.
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 so-called r-good polynomials, where r is equal to the locality of the LRC code. However, given a prime p, known constructions of r-good 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 r-good 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.
3. Complementary dual algebraic geometry codes, S. Mesnager, C. Tang et Y. Qi, Journal IEEE Transactions on Information Theory-IT. A paraître.
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 so-called side-channel attacks (SCA) and fault non-invasive 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.
4. Bent functions from involutions over $F_2^n$, R. Coulter et S. Mesnager, Journal IEEE Transactions on Information Theory-IT. A paraître.
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.
5. 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 Theory-IT. A paraître.
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 so-called 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 (S-boxes) 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\'c-Ribi\'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 Sidelnikov-Chabaud-Vaudenay bound coincides with it when $m=n-1$ and it has no sense when $m$ is less than $n-1$). 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.
6. Decomposing generalized bent and hyperbent functions,T. Martinsen, W. Meidl, S. Mesnager et P. Stanica, Journal IEEE Transactions on Information Theory-IT, Vol 63, Issue 12, pages 7804-7812, 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 g-hyperbentness of $f$ is equivalent to the hyperbentness of the components of $f$ with some conditions on the Walsh-Hadamard 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.
7. On the $p$-ary (Cubic)Bent and Plateaued (Vectorial) Functions, S. Mesnager, F. Ozbudak et A. Sinak , Journal DCC, 2017. A paraître.
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 second-order derivatives. Moreover, we devote to cubic functions the characterization of plateaued functions in terms of the value distribution of the second-order derivatives, and hence this reveals non-existence 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 non-existence 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).
8. On q-ary plateaued functions over $F_q$ and their explicit characterizations, S. Mesnager, F. Ozbudak, A. Sinak et G. Cohen, European Journal of Combinatorics. A paraître.
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 Walsh-Hadamard 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 plateaued-ness 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.
9. Fast algebraic immunity of Boolean functions, S. Mesnager et G. Cohen, Journal Advances in Mathematics of Communications (AMC), Vol 11, No. 2, pages 373-377, 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.
10. On constructions of bent, semi-bent 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 339-345, 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 well-known 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 semi-bent functions and few Walsh transform values functions built from bent functions.
11. 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 347-352, 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 well-know Rothaus' construction.
12. Explicit constructions of bent functions from pseudo-planar functions, K. Abdukhalikov et S. Mesnager, Journal Advances in Mathematics of Communications (AMC), Vol 11, No. 2, pages 293-299, 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 pseudo-planar functions. The following diagram gives an indication of the main interconnections arising in this paper: pseudo-planar functions - commutaive presemifields - bent functions
13. 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 71-84, 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 p-ary codes with three weights given with its weight distribution. The class of codes presented in this paper is different from those known in literature.
14. 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 427-436, 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.
15. 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 3-21, 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.
16. On the nonlinearity of S-boxes and linear codes, J. Liu, S. Mesnager et L. Chen, Journal Cryptography and Communications- Discrete Structures, Boolean Functions and Sequences (CCDS), 9(3) pages 345-361, Springer, 2017.
Abstract :
For multi-output Boolean functions (also called S-boxes), 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 S-boxes. A more fine-grained view on the notion of nonlinearity of S-boxes 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 S-boxes 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 non-affine 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.
17. DNA cyclic codes over rings, N. Bennenni, K. Guenda et S. Mesnager, Journal Advances in Mathematics of Communications (AMC), Vol 11, No. 1, pages 83-98, 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 reverse-complement 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 reverse-complement over the ring $\tilde {R}$. Further we find their binary images and construct some explicit examples of such codes.
18. Involutions over the Galois field $F_2^n$, P. Charpin, S. Mesnager et S. Sarkar. Journal IEEE Transactions on Information Theory-IT, Volume 62, Issue 4, pages 1-11, 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.
19. Dickson polynomials that are involutions, P. Charpin, S. Mesnager et S. Sarkar. Journal Contemporary Developments in Finite Fields and Their Applications, pages 22-45, 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.
20. 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 229-246, 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", IEEE-IT, 60(7), pp 4397-4407, 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 secondary-like constructions of permutations leading to the construction of several families of bent functions.
21. 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 53-61, 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 quasi-minimal, $t$-minimal, and $t$-quasi-minimal linear codes, which are new variations on this notion.
22. Further results on semi-bent functions in polynomial form, X. Cao, H. Chen et S. Mesnager, Journal Advances in Mathematics of Communications (AMC), 10(4) pages 725-741, 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 well-known bent functions ($0$-plateaued functions) and the semi-bent functions ($2$-plateaued functions). Bent functions have been extensively investigated since 1976. Very recently, the study of semi-bent 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 semi-bent 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 semi-bent 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 semi-bentness property of the extended family in terms of classical binary exponential sums and binary polynomials.
23. Four decades of research on bent functions, C. Carlet et S. Mesnager. International Journal Designs, Codes and Cryptography (DCC), Vol. 78, No. 1, pages 5-50, 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 super-classes of Boolean functions, vectorial bent functions and bent functions in odd characteristic.
24. 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 895-919, 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.
25. Optimal codebooks from binary codes meeting the Levenshtein bound, C. Xiang, C. Ding et S. Mesnager. International Journal IEEE Transactions on Information Theory-IT 61(12), pages 6526-6535, 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.
26. Bent vectorial functions and linear codes from o-polynomials, S. Mesnager. International Journal Designs, Codes and Cryptography (DCC) 77(1), pages 99-116, 2015.
Abstract :
The main topics and interconnections arising in this paper are symmetric cryptography (S-boxes), 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 multi-output 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 (S-boxes). 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 q-ary simplex codes. To this end, we present a general construction of classes of linear codes from o-polynomials 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 $\GF 2$.
27. 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 295-316, 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 hyper-bent 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.
28. Several new infinite families of bent functions and their duals, S. Mesnager, IEEE Transactions on Information Theory-IT, Vol. 60, No. 7, pages 4397-4407, 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, self-dual bent functions and anti-self-dual bent functions. Secondly, we provide seven new infinite families of bent functions by explicitly calculating their dual functions.
29. Sphere coverings and Identifying Codes, D. Auger, G. Cohen et S. Mesnager, Journal Designs, Codes and Cryptography, Volume 70, Issues 1-2, pages 3-7, 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.
30. On constructions of semi-bent functions from bent functions, G. Cohen et S. Mesnager, Journal Contemporary Mathematics 625, Discrete Geometry and Algebraic Combinatorics, Americain Mathematical Society, Pages 141-154, 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 semi-bent functions, due to their combinatorial and algebraic properties. Constructions of bent functions have been extensively investigated. However only few constructions of semi-bent functions have been proposed in the literature. In general, finding new constructions of bent and semi-bent functions is not a simple task. The paper is devoted to the construction of semi-bent functions with even number of variables. We show that bent functions give rise to primary and secondary-like constructions of semi-bent functions.
31. An efficient characterization of a family of hyper-bent functions with multiple trace terms, J. P. Flori et S. Mesnager, Journal of Mathematical Cryptology. Vol 7 (1), pages 43-68, 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 Charpin--Gong characterization of a large class of hyper-bent 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 Charpin--Gong 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 hyper-bentness 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.
32. Hyper-bent functions via Dillon-like exponents, S. Mesnager et J. P. Flori, IEEE Transactions on Information Theory-IT. Vol. 59 No. 5, pages 3215- 3232, 2013.
Abstract :
This paper is devoted to hyper-bent functions with multiple trace terms (including binomial functions) via Dillon-like exponents. We show how the approach developed by Mesnager to extend the Charpin--Gong 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 Charpin--Gong criterion can be weakened before generalizing the Mesnager approach to arbitrary Dillon-like 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 hyper-bentness 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 hyper-bent functions in cases which could not be attained through naive computations of exponential sums.
33. Further results on Niho bent functions, L. Budaghyan, C. Carlet, T. Helleseth, A. Kholosha et S. Mesnager, IEEE Transactions on Information Theory-IT. Vol 58, No 11, pages 6979-6985, 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 Maiorana-McFarland class. This is done using the criterion based on second-order derivatives of a function.
34. On Semi-bent Boolean Functions, C. Carlet et S. Mesnager, IEEE Transactions on Information Theory, Vol 58, No 5, pages: 3287-3292, 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$whose restrictions to these elements are all linear, is semi-bent if and only if g and h are both bent. We deduce a large number of infinite classes of semi-bent functions in explicit bivariate (resp. univariate) polynomial form. 35. Semi-bent functions from Dillon and Niho exponents, Kloosterman sums and Dickson polynomials. S. Mesnager, IEEE Transactions on Information Theory, Vol 57, No 11, pages 7443-7458, 2011. Abstract : Kloosterman sums have recently become the focus of much research, most notably due to their applications in cryptography and coding theory. In this paper, we extensively investigate the link between the semi-bentness property of functions in univariate forms obtained via Dillon and Niho functions and Kloosterman sums. In particular, we show that zeros and the value four of binary Kloosterman sums give rise to semi-bent functions in even dimension with maximum degree. Moreover, we study the semi-bentness property of functions in polynomial forms with multiple trace terms and exhibit criteria involving Dickson polynomials. 36. On Dillon's class H of bent functions, Niho bent functions and o-polynomials, C. Carlet et S. Mesnager, Journal of Combinatorial Theory-JCT-serie A 118, pages 2392–2410, 2011. Abstract : One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family$H$. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well-known Maiorana-McFarland class. We first notice that$H$can be extended to a slightly larger class that we denote by${\cal H}$. We observe that the bent functions constructed via Niho power functions, for which four examples are known due to Dobbertin et al. and to Leander-Kholosha, are the univariate form of the functions of class${\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}$s are affine. We answer the open question raised by Dobbertin et al. in JCT A 2006 on whether the duals of the Niho bent functions introduced in the paper are affinely equivalent to them, by explicitely calculating the dual of one of these functions. We observe that this Niho function also belongs to the Maiorana-McFarland class, which brings us back to the problem of knowing whether$H$(or${\cal H}$) is a subclass of the Maiorana-McFarland completed class. We then show that the condition for a function in bivariate form to belong to class${\cal H}$is equivalent to the fact that a polynomial directly related to its definition is an o-polynomial (also called oval polynomial, a notion from finite geometry). Thanks to the existence in the literature of 8 classes of nonlinear o-polynomials, we deduce a large number of new cases of bent functions in${\cal H}$, which are potentially affinely inequivalent to known bent functions (in particular, to Maiorana-McFarland's functions). 37. Bent and Hyper-bent Functions in polynomial form and Their Link With Some Exponential Sums and Dickson Polynomials. S. Mesnager, IEEE Transactions on Information Theory, Vol. 57, No. 9, pages 5996-6009, 2011. Abstract : Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were introduced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (Reed-Muller codes) and applications in cryptography (design of stream ciphers), they have attracted a lot of research, specially in the last 15 years. The class of bent functions contains a subclass of functions, introduced by Youssef and Gong in 2001, the so-called hyper-bent functions, whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyper-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. This paper is devoted to the constructions of bent and hyper-bent Boolean functions in polynomial forms. We survey and present an overview of the constructions discovered recently. We extensively investigate the link between the bentness property of such functions and some exponential sums (involving Dickson polynomials) and give some conjectures that lead to constructions of new hyper-bent functions. 38. A New Class of Bent and Hyper-Bent Boolean Functions in Polynomial Forms. S. Mesnager, Journal Designs, Codes and Cryptography. Volume 59, No. 1-3, pages 265-279 (2011). Abstract : Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. This paper is a contribution to the construction of bent functions over$\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$2^n-1$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$s_2={\frac {2^n-1}3}$, 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 co-prime with$2^m+1$. The corresponding bent functions are also hyper-bent. For$m$even, we give a necessary condition of bentness in terms of these Kloosterman sums. 39. 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 133-148 (2010). Abstract : This paper is devoted to the constructions of bent vectorial functions, that is, maximally nonlinear multi-output 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 (S-boxes). We survey, study more in details and generalize the known primary and secondary constructions of bent functions, and we introduce new ones. 40. Improving the Lower Bound on the Higher Order Nonlinearity of Boolean Functions With Prescribed Algebraic Immunity. S. Mesnager, IEEE Transactions on Information Theory-IT Vol. 54, No. 8, pages 3656-3662 (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 rth-order nonlinearity. As recalled by Carlet at CRYPTO'06, many papers have illustrated the importance of the r th-order nonlinearity profile (which includes the first-order 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 rth-order 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 rth-order 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. 41. On the number of resilient Boolean functions. S. Mesnager, Journal of Number Theory and its Applications, Vol. 5, pages 139-153, 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. 42. Improving the Upper Bounds on the Covering Radii of Binary Reed-Muller Codes, C. Carlet et S. Mesnager, IEEE Transactions on Information Theory 53 (1), pages 162-173 (2007). Abstract : By deriving bounds on character sums of Boolean functions and by using the characterizations, due to Kasami , of those elements of the Reed-Muller 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 Reed-Muller code of order 2, and we deduce improved upper bounds on the covering radii of the Reed-Muller codes of higher orders 43. Test of epimorphism for finitely generated morphisms between affine algebras over Computational rings. S. Mesnager, Journal of Algebra and Applications, Vol 4 (4), pages 1-15 (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. 44. 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 311-327 (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 (1970-1975) thanks to classical algebraic methods and more recently by Vasconcelos (1991-1998) 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 quasi-finite 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. 45. On resultant criteria and formulas for the inversion of a polynomial map. S. Mesnager, Communications in Algebra 29 (8), pages 3327-3339 (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{Adjamagbo-Essen} 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]. 46. (dans l’ordre chronologique inverse) 47. A new class of three-weight 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). Saint-Petersburg, 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), 71-84, 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 three-weight linear p-ary codes from weakly regular plateaued functions and determine their weight distributions. We finally analyse the constructed linear codes for secret sharing schemes. 48. 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. 49. 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 (C2SI-2017), pages 282-297, 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 Error-Correcting Codes, 2006], that we shall call \Car- lets 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 Maiorana-McFarland in general. 50. Explicit Characterizations for Plateaued-ness of p-ary (Vectorial) Functions. C. Carlet, S. Mesnager, F. Ozbudak et A. Sinak. Proceedings of the international Conference on Codes, Cryptology and Information Security (C2SI-2017) pages 328-345, 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 second-order derivatives, and finally their characterizations via the moments of the Walsh transform. 51. 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. 52. 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 109-123, 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 q-ary functions. Our proposed partially homomorphic encryption schemes have perfect secrecy and resist cipher-only attacks to some extent. 53. A Scalable and Systolic Architectures of Montgomery Modular Multiplication for Public Key Cryptosystems Based on DSPs. A. Mrabet, N. El-Mrabet, R. Lashermes, J-B. Rigaud, B. Bouallegue, S. Mesnager et M. Machhout. Proceedings of the Sixth International Conference on Security, Privacy and Applied Cryptographic Engineering (Space 2016) pages 138-156, Springer, 2016. Abstract : Inversion can be used in Elliptic Curve Cryptography systems and pairing-based 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 so-called 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. 54. 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, non-monotone 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). 55. 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 3-19, 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 Lang-Weil estimations. 56. 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 239-253, 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. 57. Bent and semi-bent 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 205-224, LNCS, Springer, Heidelberg, 2015. Abstract : This paper is dealing with two important subclasses of plateaued functions in even dimension: bent and semi-bent functions. In the first part of the paper, we construct mainly bent and semi-bent functions in Maiorana-McFarland 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 semi-bent functions not belonging to Maiorana-McFarland class. Instead of using bent (semi-bent) 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. 58. Results on characterizations of plateaued functions in arbitrary characteristic. S. Mesnager, F. Ozbudak et A. Sinak, Proceedings of BalkanCryptSec 2015, LNCS 9540, pages 17-30, 2015. Abstract : Bent and plateaued functions play a signi cant 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 second-order directional di erences. 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 s-plateaued functions in arbitrary characteristic. 59. On involutions of finite fields. P. Charpin, S. Mesnager et S. Sarkar, Proceedings of 2015 IEEE International Symposium on Information Theory, ISIT 2015, Hong-Kong, 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. 60. 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 pseudo-random 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 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 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. 61. 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, Springer-Verlag, pages 125-131, 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 quasi-minimal and almost-minimal linear codes, relaxations of the minimal linear codes. 62. 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 72-82, 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$3$. 63. On semi-bent 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 243-273, 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 Walsh-Hadamard spectrum. Plateaued functions bring together various nonlinear characteristics and include two important classes of Boolean functions defined in even dimension: the well-known bent functions and the semi-bent functions. Bent functions (including their constructions) have been extensively investigated for more than 35 years. Very recently, the study of semi-bent 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 semi-bent functions defined over the Galois field$\GF n$($n$even). We review what is known in this framework and investigate constructions. 64. 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 923-927, 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 pseudo-random 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. 65. Implementation of Faster Miller over Barreto-Naehrig Curves in Jacobian Cordinates, A. Mrabet Amine, B. Bouallegue, M. Machhout, N. EL Mrabet et S. Mesnager, Proceedings of GSCIT 2014-IEEE, pages 1-6, 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. Pairing-friendly curves are important for speeding up the arithmetic calculation of pairing on elliptic curves such as the Barreto-Naehrig (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 128-bit security level using such a curve BN defined over a 256-bit prime field. And we present also a fast formulas for BN elliptic-curve addition and doubling. Our architecture is able to compute the Miller’s algorithm in just 638337 of clock cycles. 66. On Minimal and Almost-Minimal 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 928-931 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 two-party computations. We pursue here the study of minimal codes and construct infinite families with asymptotically non-zero 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. 67. Semi-bent functions from oval polynomials, S. Mesnager, Proceedings of Fourteenth International Conference on Cryptography and Coding, Oxford, United Kingdom, IMACC 2013, LNCS 8308, pages. 1-15. 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 semi-bent 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 semi-bent Boolean functions in even dimension. 68. On Minimal and quasi-minimal 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 85-98. 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 quasi-minimal linear codes, which is a relaxation of the notion of minimal linear codes, where two non-zero codewords have the same support if and only if they are linearly dependent. 69. On hyper-bent functions via Dillon-like exponents, S. Mesnager et J.P. Flori, ISIT 2012-IEEE Internaional Symposium on Information Theory, IMT, Cambridge, MA, USA, 2012. Abstract : This paper is devoted to hyper-bent functions with multiple trace terms (including binomial functions) via Dillon-like 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 Dillon-like 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. 70. Semi-bent 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 18-36, 2012. Abstract : Semi-bent functions with even number of variables are a class of important Boolean functions whose Hadamard transform takes three values. Semi-bent functions have been extensively studied due to their applications in cryptography and coding theory. In this paper we are interested in the property of semi-bentness of Boolean functions defined on the Galois field$\GF n$(n even) with multiple trace terms obtained via Niho functions and two Dillon-like 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 semi-bentness 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 semi-bentness leading to substantial practical gain thanks to the current implementation of point counting over hyperelliptic curves. 71. Niho Bent Functions and Subiaco Hyperovals, T. Helleseth, A. Kholosha et S. Mesnager, Proceedings of the 10-th International Conference on Finite Fields and Their Applications (Fq'10), Contemporary Math., AMS, 2012. Vol 579, pages 91-101, 2012. Abstract : In this paper, the relation between binomial Niho bent functions discovered by Dobbertin et al. and o-polynomials 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. 72. Dickson polynomials, hyperelliptic curves and hyper-bent functions,J.P. Flori et S. Mesnager, Proceedings of 7-th International conference SEquences and Their Applications, SETA 2012, Waterloo, Canada. LNCS 7780, pages 40-52, 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 well-known result. Such properties are then applied to the study of a family of Boolean functions and a characterization of their hyper-bentness 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. 73. On Dillon’s class H of Niho bent functions and o-polynomials, 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. 74. 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 61-78, 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, hyper-bent functions and semi-bent 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. 75. 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. 76. 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 703-707, Saint-Petersturg, Russie, Juillet-aout 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 follow-up of the recent paper by Carlet and Mesnager. 77. Generalized witness sets, G. Cohen et S. Mesnager, Proceeding 1st International Conference on Data Compression, Communication and Processing CCP 2011, Italie, 21-24 juin 2011. Abstract : Given a set C of q-ary n-tuples 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. 78. On the link of some semi-bent functions with Kloosterman sums, S. Mesnager et G. Cohen, Proceeding of International Workshop on Coding and Cryptology, IWCC 2011, LNCS 6639, pages 263-272, Springer, Heidelberg ,2011. Abstract : We extensively investigate the link between the semi-bentness property of some functions in polynomial forms and Kloosterman sums. 79. On a conjecture about binary strings distribution, J. P. Flori, H. Randriambololona, G. Cohen et S. Mesnager, Proceedings of 6-th International conference SEquences and Their Applications, SETA 2010, Paris, France, SETA 2010, LNCS 6338, pages 346-358. 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$2^{k-1}$. 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. 80. Recent Results on Bent and Hyper-bent Functions and Their Link With Some Exponential Sums, S. Mesnager, IEEE Information Theory Workshop (ITW 2010), Dublin, Iralande, Aout-Septembre 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 (Reed-Muller 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 so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer than bent functions. Bent and 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. This paper is devoted to the constructions of bent and hyper-bent 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) 81. Hyper-bent Boolean Functions with Multiple Trace Terms, S. Mesnager, Proceedings of International Workshop on the Arithmetic of Finite Fields, WAIFI 2010, LNCS 6087, pages. 97-113. 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 so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer. In fact, hyper-bent functions seem still more difficult to generate at random than bent functions and many problems related to the class of hyper-bent 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 hyper-bent functions on finite fields$\GF n$(where$n$is even) by studying a subclass$\mathfrak {F}_n$of the so-called 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 hyper-bent 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 hyper-bent functions of$\mathfrak {F}_n$and the zeros of some Kloosterman sums has been generalized to a link between hyper-bent functions of$\mathfrak {F}_n$and some exponential sums where Dickson polynomials are involved. Moreover, we provide a possibly new infinite family of hyper-bent functions. Our study extends recent works of the author and is a complement of a recent work of Charpin and Gong on this topic. 82. A new family of hyper-bent Boolean functions in polynomial form, S. Mesnager, Proceedings of Twelfth International Conference on Cryptography and Coding. Cirencester, Angleterre, IMACC 2009, LNCS 5921, pages 402-417. 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 so-called hyper-bent 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 hyper-bent 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 hyper-bent 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$2^n-1$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$s_1={3(2^m-1)}$and$s_2={\frac {2^n-1}3}$, where$a\in\GF{n}$($a\not=0$) and$b\in\GF[4]{}$provide a construction of hyper-bent 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$a$. 83. A new class of Bent Boolean functions in polynomial forms, S. Mesnager, Proceedings of international Workshop on Coding and Cryptography, WCC 2009, pages 5-18, 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$2^n-1$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$s_2={\frac {2^n-1}3}$, 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. 84. Secret Sharing Schemes Based on Self-dual Codes, S.T. Dougherty, S. Mesnager et P. Solé. IEEE Information Theory Workshop (ITW 2008), Porto, Portugal 5-9 Mai 2008. Abstract : Secret sharing is an important topic in cryptography and has applications in information security. We use self-dual codes to construct secret-sharing schemes. We use combinatorial properties and invariant theory to understand the access structure of these secret-sharing 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. 85. 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 364-375, 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. 86. 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 65-82. 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.
87. Non-Linearity 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.
88. Improving the upper bounds on the covering radii of Reed-Muller 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 Reed-Muller codes whose Hamming weights are smaller than twice the minimum distance, we derive an improved upper bound on the covering radius of the Reed-Muller code of order 2, and we deduce improved upper bounds on the covering radii of the Reed-Muller codes of higher orders.
89. 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, Saint-Pétersbourg, Russie, Juilllet 2004, pages 348-357.
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.
90. Livres:

(dans l’ordre chronologique inverse)
91. Livre "Bent functions: fundamentals and results", S. Mesnager, Springer Verlag. Springer 2016.
92. Livre "Arithmetic of Finite Fields", Ç.K. Koç, S. Mesnager et E. Savaş, 5th International Workshop, WAIFI 2014, Volume 9061, pages 1--213, Springer, 2015.
93. Livre "Corps finis et théorie des codes", S. Mesnager, Pearson Education, 2007 (En Français).
1. Présidente du comité de programme de la conférence ICCC 2015 ,International Conference on Coding and Cryptography, Alger, Algerie, 2-5 Novembre 2015.
2. Co-présidente (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, 20-23 Juillet 2015.
3. Co-présidente (avec Erkay Savas) du comité de programme de la conférence WAIFI 2014, International Workshop on the Arithmetic of Finite Fields, Gebze, Turquie, 26-28 Septembre 2014.
1. Membre du comité de programme du, 10th International Workshop on Coding and Cryptography (WCC 2017) St Petersburg, Russie, 18-22 Septembre, 2017.
2. 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, Aout-Septembre 2017.
3. Membre du comité de programme du, 2sd International Conference "Codes, Cryptology and Information Security" Rabat, Maroc, 10-12, Avril 2017.
4. Membre du comité de programme de 9th International Conference on SEquences and Their Applications (SETA 2016), Chengdu, Chine 9-14 Octobre 2016.
5. Membre du comité de programme de International Workshop on the Arithmetic of Finite Fields (WAIFI 2016), Ghent, Belgique, 13-16 Juillet 2016.
6. Membre du comité de programme de 2sd International Conference on Cryptography and its Applications ICCA 2016 UST, Oran, Algerie 26-27 Avril 2016.
7. Membre du comité de programme du congré international 9th International Workshop on Coding and Cryptography (WCC 2015) Paris, France 13-17 Avril 2015.
8. Membre du comité de programme du congré international SETA 2014, "8th International Conference on SEquences and Their Applications" Melbourne, Australie, 24-28 novembre 2014.
9. Membre du comité de programme de International Workshop on the Arithmetic of Finite (WAIFI 2014) Fields, Gebze, Turquie, 26-28 Septembre 2014.
10. 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 15-18 Septembre 2014.
11. Membre du comité de programme du congré international WCC 2013, "8th International Workshop on Coding and Cryptography" Bergen, Norvége 15-19 Avril 2013.
12. Membre du comité de programme du congré international WCC 2011, "7th International Workshop on Coding and Cryptography" Paris, France, 11-15 Avril 2011.
13. Membre du comité de programme du congré international SETA 2010, "6th International Conference on SEquences and Their Applications" Paris, France, 12-17 septembre 2010.
14. Membre du comité de programme du congré international Africacrypt 2009, "2sd African International Conference on Cryptology " Gammarth, Tunisie, 21-25 juin 2009.
1. International Workshop on the Arithmetic of Finite Fields
1. Editrice en Chef du journal international International Journal of Information and Coding Theory" (IJOCT).
2. Editrice au journal international IEEE Transactions on Information Theory (IEEE-IT).
3. Editrice dans le journal international Advances in Mathematics of Communications (AMC) -Publié par AIMS (American Institute of Mathematical Sciences).
4. Editrice dans le journal international Cryptography and Communications- Discrete Structures, Boolean Functions and Sequences (CCDS) -Publié par Springer.
5. Editrice dans le journal international RAIRO ITA (Theoretical Informatics and Applications) -Publié par Cambridge University Press.

Exposés

Conférences internationales

(dans l’ordre chronologique inverse)
1. On constructions of bent functions from involutions, IEEE International Symposium on Information Theory (ISIT 2016) à Barcelone, Espagne, Juillet 2016.
2. On construction of bent functions involving symmetric functions and their duals, Conference Internationale "Workshop on Mathematics in Communications (WMC 2016), Santander, Espagne, Juillet 2016.
3. Fast algebraic immunity of Boolean functions, Conference Internationale "Workshop on Mathematics in Communications (WMC 2016), Santander, Espagne, Juillet 2016.
4. Explicit constructions of bent functions from pseudo-planar functions, Conference Internationale "Workshop on Mathematics in Communications (WMC 2016), Santander, Espagne, Juillet 2016.
5. On constructions of bent, semi-bent and five valued spectrum functions from old bent functions, Conference Internationale "Workshop on Mathematics in Communications (WMC 2016), Santander, Espagne, Juillet 2016.
6. On the diffusion property of iterated functions, International Conference on Cryptography and Coding, Oxford, United Kingdom, Decembre 2015.
7. On p-ary bent functions from (maximal) partial spreads, Conférence internationale Finite field and their Applications Fq12, New York, Juillet 2015.
8. Dickson Polynomials that are Involutions, Conférence internationale Finite field and their Applications Fq12, New York, Juillet 2015.
9. On involutions of finite fields, Conférence internationale ISIT 2015 International Symposium on Information Theory Hong-Kong, Chine, Juin 2015.
10. Cyclic codes and Algebraic immunity of Boolean functions, Conférence internationale IEEE Workshop Information Theory (ITW 2015), Jérusalem, Israel, Avril 2015.
11. 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.
12. Semi-bent functions from oval polynomials. Conférence internationale Cryptography and Coding IMACC 2013, Oxford, Angleterre, Decembre 2013.
13. Bent functions from spreads. Conférence internationale Finite Fields and their Applications, Fq11, Magdebourg, Allemagne, Juillet 2013.
14. Semi-bent functions with multiple trace terms and hyperelliptic curves. Conférence internationale, Cryptology and Information Security in Latin America (Latincrypt) 2012 Santiago, Chili, Octobre 2012.
15. Bent and hyper-bent functions via Dillon-like exponents. Conférence internationale, Yet Another Conference on Cryptography (YACC 2012) 2012. Iles de Porquerolles, France, Septembre 2012.
16. On hyper-bent functions via Dillon-like exponents. Conférence internationale, ISIT 2012. IEEE International Symopsium on Infomation Theory à IMT, Boston, USA, Juillet 2012.
17. Dickson polynomials, hyperelliptic curves and hyper-bent functions. Conférence internationale, SETA (The 7th international conference on SEquences and Their Applications) à Waterloo (Canada), juin 2012.
18. New semi-bent functions with multiple trace terms. Conférence internationale sur invitation, Workshop Information Theory and Applications (ITA 2012) à San Diego (USA), Février 2012.
19. Identifying and Covering by Spheres. 25eme Conférence internationale on Combinatorics, Cryptography, and Computing (MCCCC), Las Vegas (USA), Octobre 2011.
20. Sphere coverings and Identifying Codes. Conférence internationale Castle Meeting on coding theory and Application (3ICMTA), Cardona (Espagne), Septembre 2011.
21. On the link of some semi-bent functions with Kloosterman sums. Conférence internationale sur invitation, Workshop of International Workshop on Coding and Cryptology (IWCC 2011) à Qingdao (Chine), Mai 2011.
22. On the link of some semi-bent functions in polynomial forms with exponential sums. Conférence internationale sur invitation, Workshop Information Theory and Applications (ITA 2011) à San Diego (USA), Février 2011.
23. Recent Results on Bent and Hyper-bent Functions and Their Link With Some Exponential Sums. Conférence internationale (conférence invitée) Information Theory Workshop (ITW 2010) à Dublin (Irlande), Séptembre 2010.
24. Hyper-bent Boolean Functions with Multiple Trace Terms. Conférence internationale, Workshop on the Arithmetic of Finite Fields (WAIFI 2010) à Istanbul (Turquie), Juin 2010.
25. A new family of hyper-bent Boolean functions in polynomial form. Conférence internationale,Twelfth International Conference on Cryptography and Coding (IMACC 2009) à Cirencester (Angleterre), Décembre 2009.
26. A new class of Bent Boolean functions in polynomial forms. Conférence internationale Workshop on Coding and Cryptography (WCC 2009) à Ullensvang (Norvége), Mai 2009
27. On the number of resilient Boolean functions. Conférence internationale, Symposium on Algebraic Geometry and its Applications (SAGA 2007) à Papeete (Tahiti), Mai 2007.
28. On immunity profile of Boolean functions. Conférence internationale, SEquences and Their Applications (SETA 2006) à Pékin (Chine), Séptembre 2006.
29. On the Walsh support of Boolean functions. Conférence internationale, Boolean Functions, Cryptography and Applications (BFCA 2005) à Rouen (France), Mars 2005.
30. (dans l’ordre chronologique inverse)
31. International Conference on Group, Group Ring and Related topics (GGRRT 2017) à Khorfakkan, UAE, Novembre 2017.
32. Instructional Workshop in Cryptology à New Delhi, Inde, Octobre 2017.
33. Conférence internationale"Yet Another Conference on Cryptography" (YACC 2016), Iles de Porquerolles, France, Juin 2016.
34. International Conference on Cryptography and Coding, Oxford, United Kingdom, Decembre 2015. Invitation de Jens Groth.
35. 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.
36. 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 Gluesing-Luerssen, Joachim Rosenthal et Margreta Kuijper.
37. 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.
38. International seminar in Coding Theory, Dagstuhl (Allemagne), Aout 2013. Invitation de Hans-Andrea Loeliger, Emina Soljanin et Judy L. Walker.
39. International Conférence Trends in coding theory, Monté Verita (Switzerland), Octobre 2012. Invitation de Elisa Gorla, Joachim Rosenthal et Amin Shokrollahi.
40. International Workshop on finite fields character sums end polynomials, Strobl (Autriche), Septembre 2012. Invitation des oragnisateurs de la conférence.
41. International workshop on coding based crypto (Ecrytp 2012), Lyngby (Denemark) en mai 2012. Invitation de Tom Høholdt.
42. International Workshop Information Theory and Applications (ITA 2012) à San Diego (USA) en Février 2012. Invitation de Alexander Vardy.
43. International seminar in Coding Theory à Dagstuhl, Allemagne en Novembre 2011. Invitation de Joachim Rosenthal et Amin Shokrollahi.
44. International Workshop on Coding and Cryptology (IWCC 2011) à Qingdao (Chine) en Mai 2011. Invitation de Xian Hequn.
45. International Workshop Information Theory and Applications (ITA 2011) à San Diego (USA) en Février 2011. Invitation de Alexander Vardy.
46. International Information Theory Workshop (ITW 2010) à Dublin (Irlande) en Séptembre 2010. Invitation de Marcus Greferath.
47. (dans l’ordre chronologique inverse)
48. Seminaire en mathématiques à l'université de Zurich, Suisse, Décembre 2017.
49. Seminaire d'algébre et théorie des nombres à l'Université d'Aalto, Finlande, Février 2017.
50. Seminaire protection de l'information à l'université Paris 8, Novembre 2016.
51. Seminaire de mathmatiques pour la cryptographie et theorie des codes à Telecom Paristech, France , Septembre 2016.
52. Seminaire de mathmatiques pour la cryptographie et theorie des codes à l'Academie des Sciences, Pékin, Chine, Septembre 2016.
53. Seminaire de mathmatiques pour la cryptographie et theorie des codes à l'université de Tianjin et université de Nankai, Chine, Septembre 2016.
54. Seminaire en mathématiques discrètes à Télécom ParisTech, Paris, France, Septembre 2016.
55. Seminaire en mathématiques discrètes à l'université Paul Sabatier (Institut de maths IMT), Toulouse, France, Avril 2016.
56. Séminaire "Combinatoire et algorithmique" à l'université de Rouen, France, Février 2016.
57. Séminaire à Hong-Kong université science et technologie, Hong-Kong, Chine, Juin 2015
58. Séminaire d'Algèbre et Géometrie à l'université de Versailles, France, Avril 2015.
59. Journée thématique Cryptographie à l'université de Cergy (France), Avril 2015. Invitation de Valerie Nachef et Emmanuel Volte.
60. Séminaire Mathématiques discrètes à l'université de Nanjing (Chine), Décembre 2014. Invitation de Xiwang Cao.
61. Séminaire Cryptographie à l'université de Xuzhou (Chine), Décembre 2014. Invitation de Fengrong Zhang.
62. Séminaire "Algébre", Département de Mathtématiques à l'université UAE, Octobre 2014.
63. Séminaire Combinatoire à l'université Paris XIII, Mai 2014.
64. Séminaire à l'université Paris VI, Mai 2014.
65. Séminaire Boole à l'université Paris VI, France, Juin 2013.
66. Séminaire UCD School of Mathematical Sciences, Dublin, Iralande, Février 2012.
67. Séminaire Boole à l'institut Henri Poincaré, Paris V, France, Janvier 2012.
68. Seminaire Therie de l'infomation, Telecom Paris-Tech, France, Decembre 2011.
69. Tutorial (conférence invitée), journées Codage et Cryptographie (C2) à St Pierre d'Oléron, Avril 2011.
70. Séminaire Arithmétique et théorie de l’information (ATI) à l'lnstitut de Mathématiques de Luminy, France, Février 2011.
71. Séminaire Mathematiques pour le traitement de l'information et de l'image (MTII) à Université Paris VIII, Janvier 2011.
72. Séminaire Boole à l'institut Henri Poincaré, Paris, France, Mai 2010.
73. Séminaire Mathematiques pour le traitement de l'information et de l'image (MTII) à Université Paris VIII, France, Juin 2009.
74. Séminaire I3S à Sophia-Antipolis, Nice, France, Avril 2009.
75. Séminaire Codes, Cryptographie et Algorithmique à l'ENSTA, Paris, France, Octobre 2005.
76. Séminaire de combinatoire algébrique de l'université de Paris 13, France, Avril 2005.
77. Séminaire de Cryptographie de l'université de Rennes, Rennes, France, Avril 2005.
78. Séminaire pour la sécurité de l'information de l'université de Paris VIII, France, Juin 2003.
79. Séminaire de géométrie algébrique de l'université de Rennes I, Rennes, France, Avril 2002.
80. Forum des jeunes mathématiciennes et informaticiennes à l'Institut Henri Poincaré, Paris, France, Mars 2002.
1. Invitation en Octobre 2017 par Professeurs Shri Kant, Shanta Laishram et Subhamoy Maitra, New Delhi, Inde.
2. 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 (Hong-Kong university of science and technology, Hong-Kong).
3. Invitation en Fevrier 2017 par Professeurs Marcus Greferath et Camilla Hollanti dans le departement de mathematiques de l'Université d'Aalto, Finlande.
4. Invitation en Septembre 2016 par Professeurs Dongdai Lin, Keqin Feng et Baofeng Wu à l'Acadamie des Sciences, Chine.
5. Invitation en Septembre 2016 par Professeurs Francoise Soulier, Fangwei Fu et Jian Liu à l'univerité de Tianjin et université de Nankai, Chine.
6. Invitation en Juillet 2016 par professeur Zhengchun Zhou, département de mathématiques, l'université de Southwest Jiaotong, Chungdu, Chine.
7. Invitation en Juin 2015 par professeur Cunsheng Ding, Hong-Kong université science et technologie, Hong-Kong, Chine.
8. Invitation en octobre 2014 par professeur Kanat Abdukhalikov, departement of mathematics, El Ain, UAE.
9. Invitation en Septembre 2014 par professeur Ferruh Özbudak, Middle East Technical University, Ankara, Turquie.
10. Invitation en octobre 2013 par professeur Janos Korner, Université de Rome, Italie.
11. Invitation en novembre 2010 par professeur Simon Litsyn, Université de Tel Aviv, Israel.
12. Invitation en septembre 2010 par professeur Marcus Greferath., College Dublin, Irlande.