Claude Carlet's publications and their abstracts

Books, Chapters in books and articles in an encyclopedia:

1) Special Issue on Coding and Cryptography. Guest Editor: C. Carlet. Guest co-editors: P. Charpin, M. Girault, G. Kabatiansky and J.H. Van Tilborg.  Discrete Applied Mathematics, vol. 111, nos 1-2, pp. 1-218 (2001).

2) Special Issue on Coding and Cryptography (Proceedings of International Workshop on Coding and Cryptography 2001). Guest Editor: C. Carlet. Guest co-editors: M. Girault, T. Helleseth, T. Hoehold, F. Morain, N. Sendrier.  Discrete Applied Mathematics, vol. 128, no 1, pp. 1-316 (2003).

3) C. Carlet, associate editor of the "Special Section on Sequence Design and its Application in Communications" of the journal IEICE Transactions on Fundamentals of Electronics, Communications and computer sciences, 2006.

4) WAIFI, Proceedings of "First International Workshop on the Arithmetic of Finite Fields 2007", Lecture Notes in Computer Science 4547, 353 pages (C. Carlet and B. Sunar editors).

5) C. Carlet  "Boolean Functions for Cryptography and Error  Correcting Codes", Chapter of the monography  ``Boolean Models and Methods in Mathematics, Computer Science, and Engineering" published by Cambridge University Press, Yves Crama and Peter L. Hammer (eds.), pp. 257-397, 2010. Please find here a preliminary version.

6) C. Carlet, "Vectorial Boolean Functions for Cryptography". Idem, pp. 398-469, 2010. Please find here a preliminary version

7) C. Carlet, Five articles in the Encyclopedia of Cryptography and Security, Springer (Henk van Tilborg, Editor, new version)

8) SETA, Proceedings of the "Sixth Conference on Sequences and their Applications 2010", Lecture Notes in Computer Science 6338, 464 pages (C. Carlet and A. Pott editors)

9) C. Carlet and T. Helleseth, "Sequences, Boolean functions, and Cryptography", chapter of the "Handbook of Codes, Sequences and Their Applications", published by CRC Press, Serdar Boztas (ed.), to appear

10) C. Carlet, "Boolean functions",  Section of the "Handbook of Finite Fields", published by CRC Press, Gary Mullen and Daniel Panario (eds.), pp. 241-252, 2013.

11) S. El Hajji,  A. Nitaj, C. Carlet, El M. Souidi (Eds.); Codes, Cryptology, and Information Security; First International Conference, C2SI 2015 Rabat, Morocco, May 26–28, 2015, In Honor of Thierry Berger, Lecture Notes in Computer Science 9084

12) C. Carlet, M. A. Hasan and V. Saraswat (Eds). Security, Privacy, and Applied Cryptography Engineering - 6th International Conference, SPACE 2016, Hyderabad, India, December 14-18, 2016, Proceedings. Lecture Notes in Computer Science 10076, Springer 2016

Full papers in international journals:

1)"The Automorphism Groups of The Kerdock Codes", C. Carlet, Journal of Information and Optimization Sciences" vol. 12 no. 3, 387-400 (1991)
We prove that the automorphism group of the Kerdock code of length 2^m  (m even at least 6) is the group of permutations on GF(2^m) = GF(2^{m-1}) x GF(2) generated by the automorphism group of the field GF(2^{m-1}), the affine group of GF(2^{m-1}), and the translations on GF(2).

2) "Partially-bent functions", C. Carlet, Designs Codes and Cryptography, 3, 135-145 (1993) and Proceedings of CRYPTO 1992
We study a conjecture about the numbers of non-zeros of, respectively, the auto-correlation function and the Walsh transform of the function (-1)^{f(x)}, where f(x) is any boolean function on {0,1}^n. The result that we obtain leads us to introduce the class of partially-bent functions. We study within these functions the propagation criterion. We characterize those partially-bent functions which are balanced and prove a relation between their number (which is unknown) and the number of non-balanced partially-bent functions on {0,1}^{n-1}. Eventually, we study their correlation immunity .

3) "The automorphism groups of the Delsarte-Goethals codes", C. Carlet, Designs Codes and Cryptography, 3, 237-249 (1993)
We determine the automorphism groups of the Delsarte-Goethals codes DG(m,d). The groups that we obtain are the same as those of the Kerdock codes K(m) of the same length .

4)"A General Case of Formal Duality Between Binary Non-linear Codes", C. Carlet, Discrete Mathematics 111, 77-85 (1993)
We give a general context in which a binary code C admits at least one code whose weight enumerator is dual to that of C. We study two examples of applications of this general result.The first one is the formal duality between the Kerdock code and the generalized Preparata codes of same length and the second one gives a new example of dual weight enumerators.

5) "The divisors of x^{2^m} + x of constant derivatives and degree 2^{m-2}", C. Carlet, SIAM Journal on Discrete Math. vol 7, no. 2, 238-244 (1994)
We completely characterize those polynomials of degree 2^m- 2 over the Galois field GF(2^m) which are fully reducible and admit no multiple factor (i.e. which divide x^{2^m} + x), and whose derivatives are constants. We show that this problem is connected with a problem in algebraic coding theory.

6)"Generalized Partial Spreads", C. Carlet, IEEE Transactions on Information Theory vol. 41 no 5 (correspondence) pages 1482-1487 (1995)
We exhibit a simple condition under which the sum (modulo 2) of characteristic functions of n/2 - dimensional vector-subspaces of (GF(2))^n (n even) is a bent function. The "Fourier" transform of such a bent function is the sum of the characteristic functions of the duals of these spaces. The class of bent functions that we obtain contains the whole Partial Spreads class. Any element of Maiorana-McFarland's class or of class D is equivalent to one of its elements. Thus, this new class gives a unified insight of both general classes of bent functions studied by J.F. Dillon in his thesis. We deduce a way to construct new classes of bent functions and exhibit an example.

7) "On Z_4-duality", C. Carlet, IEEE Transactions on Information Theory vol 41 no. 5 (correspondence) pages 1487-1495 (1995)
Recently was introduced the new notion on binary codes called Z_4-linearity. This notion explains why Kerdock codes and Delsarte-Goethals codes admit formal duals in spite of their nonlinearity. The "Z_4-duals" of these codes (called 'Preparata' and 'Goethals' codes) are new nonlinear codes which admit simpler decoding algorithms than the previously known formal duals (the generalized Preparata and Goethals codes). We prove, by using the notion of exact weight enumerator, that the relationship between any Z_4-linear code and its Z_4-dual is stronger than the standard formal duality and we deduce the weight enumerators of related generalized codes.

8) "A characterization of binary bent functions", C. Carlet and Philippe Guillot, Journal of Combinatorial Theory, Series A, Vol. 76, No. 2, 328-335 (1996)
A recent paper by Carlet introduces a general class of binary bent functions on (GF(2))^n (n even) whose elements are expressed by means of characteristic functions (indicators) of {n/2}-dimensional vector-subspaces of (GF(2))^n. An extended version of this class is introduced in the same paper; it is conjectured that this version is coinciding with the whole class of bent functions. In the present paper, we prove that this conjecture is true.

9) "An alternate characterization of the bentness of binary functions, with uniqueness", C. Carlet and Philippe Guillot, Designs Codes and Cryptography, Vol 14, no 2, p. 133-140 (may 1998)
In a previous paper, we have obtained a characterization of the binary bent functions on (GF(2))^n (n even) as linear combinations modulo 2^{n/2}, with integral coefficients, of characteristic functions (indicators) of n/2-dimensional vector-subspaces of (GF(2))^n. There is no uniqueness of the representation of a given bent function related to this characterization. We obtain now a new characterization for which there is uniqueness of the representation.

10) "Z_{2^k}-linear codes", C. Carlet, IEEE Transactions on Information Theory, Vol. 44, no 4 (correspondence), pages 1543-1547 (1998)
We introduce a generalization to Z_{2^k} of the Gray map and generalized versions of Kerdock and Delsarte-Goethals codes.

11) "Codes, bent functions and permutations suitable for DES-like cryptosystems", C. Carlet, P. Charpin and V. Zinoviev, Designs Codes and Cryptography, 15, p. 125-156 (1998)
Almost bent functions oppose an optimum resistance to linear and differential cryptanalysis. We present basic properties of almost bent functions; particularly we give an upper bound on the degree. We develop the "coding theory" point of view for studying the existence of almost bent functions, showing explicitly the links with cyclic codes. We also give new characterizations of almost bent functions by means of associated Boolean functions.

12) "On Cryptographic Propagation Criteria for Boolean Functions", C. Carlet, Special Issue on Cryptology of "Information and Computation", in Honor of Professor Arto Salomaa on Occasion of His 65th Birthday (article invité) (1999)
We determine the functions on GF(2)^n which satisfy the propagation criterion of degree (n-2), PC(n-2). We study subsequently the propagation criterion of degree l and order k and its extended version EPC. We determine those Boolean functions on GF(2)^n which satisfy PC(l) of order greater than or equal to (n-l- 2). We show that none of them satisfies EPC(l) of same order. We finally give a general construction of nonquadratic functions satisfying EPC(l) of order k. This construction uses the existence of nonlinear, systematic codes with good minimum distances and dual distances (e.g. Kerdock codes and Preparata codes).

13) ``On cryptographic properties of the cosets of R(1,m)", A. Canteaut, C. Carlet, P. Charpin and C. Fontaine, IEEE Transactions on  Information Theory (regular paper) Vol. 47, no 4, pp. 1494-1513 (2001)
We introduce a new approach for the study of weight distributions of cosets of the Reed-Muller code of order 1. Our approach is based on the method introduced by Kasami, using Pless identities. By interpreting some equations, we obtain a necessary condition for a coset to have a ``high'' minimum weight. We next examine the impact of our results when some cryptographic criteria of Boolean functions are considered.

14) ``Spectral Domain Analysis of Correlation Immune and Resilient Boolean Functions", C. Carlet and P. Sarkar, Finite fields and Applications (revue)  8, pp. 120-130 (2002).
We use a general property of Fourier transform to obtain direct proofs of recent divisibility results on the Walsh Transform of correlation immune and resilient functions. Improved upper bounds on the nonlinearity of these functions are obtained from the divisibility results. We deduce further information on correlation immune and resilient functions. In particular, we obtain a necessary condition on the algebraic normal form of correlation immune functions attaining the maximum possible nonlinearity.

15) ``Covering sequences of Boolean functions and their cryptographic significance", C. Carlet and Yu. Tarannikov,  Designs, Codes and Cryptography, 25, pp. 263-279 (2002).
We introduce the notion of covering sequence of a Boolean function, related to the derivatives of the function. We give complete characterizations of balancedness, correlation immunity and resiliency of Boolean functions by means of their covering sequences. By considering particular covering sequences, we define subclasses of (correlation-immune) resilient functions. We derive upper bounds on their algebraic degrees and on their nonlinearities. We give constructions of resilient functions belonging to these classes. We show that they achieve the best known trade-off between order of resiliency, nonlinearity and algebraic degree.

16) ``Spectral Methods for Cross-Correlations of Geometric Sequences", A. Klapper and C. Carlet. IEEE Transactions on Information Theory, Vol. 50 (correspondence), pp. 229-232, 2004.
Families of sequences with low pairwise shifted cross-correlations are desirable for applications such as CDMA communications. Often such sequences must have additional properties for specific applications. Several ad hoc constructions of such families exist in the literature, but there are few systematic approaches to such sequence design. In this paper we introduce a general method of constructing new families of sequences with bounded pairwise shifted cross-correlations from old families of such sequences. The bounds are obtained in terms the maximum cross-correlation in the old family and the Walsh transform of certain functions.

17)  ``Highly Nonlinear Mappings". C. Carlet and C. Ding. Special Issue ``Complexity Issues in Coding and Cryptography" du Journal of Complexity, Edition spéciale  pour les 60 ans de Harald Niederreiter, vol 20, numbers 2-3, pp. 205-244, 2004.
Functions with high nonlinearity have important applications in cryptography, sequences and coding theory. The purpose of this paper is to give a well-rounded treatment of non-Boolean functions with optimal nonlinearity. We summarize and generalize known results, and prove a number of new results. We also present open problems about functions with high nonlinearity.

18)  ``On the  confusion and diffusion properties  of Maiorana-McFarland's and extended Maiorana-McFarland's functions". C. Carlet. Special Issue ``Complexity Issues in Coding and Cryptography" du Journal of Complexity, Special Issue in honour of Harald Niederreiter, vol 20, numbers 2-3, pp. 182-204, 2004.
A practical problem in symmetric cryptography is finding  constructions of Boolean functions leading to reasonably large sets  of functions satisfying some desired cryptographic  criteria.  The main known construction, called Maiorana-McFarland,  has been recently extended. Some other constructions exist, but lead to smaller classes of functions.  Here, we study more in detail the nonlinearities and the resiliencies of the functions produced by all these constructions.  Further we see how to obtain functions satisfying the propagation  criterion (among which bent functions) with these methods, and we give a new construction  of bent functions based on the extended Maiorana-McFarland's construction.

19) ``Concatenating indicators of flats for designing cryptographic functions.  C. Carlet.  Designs, Codes and Cryptography, volume 36, No 2, pp. 189 - 202,  2005.
Boolean functions with good cryptographic  characteristics are needed for the design of robust pseudo-random  generators for stream ciphers and of S-boxes for block ciphers.  Very few general constructions of such cryptographic Boolean functions are  known. The main ones correspond to concatenating affine or quadratic functions. We introduce a  general construction corresponding to the concatenation of indicators  of flats. We show that the functions it permits to design can present very good cryptographic characteristics.

20) ``On the degree, nonlinearity, algebraic thickness and  non-normality  of Boolean functions, with developments on symmetric functions".  C. Carlet. IEEE Transactions on Information  Theory, vol. 50 (correspondence), pp. 2178-2185, 2004.
The two main criteria evaluating, from cryptographic viewpoint,  the complexity of Boolean functions are the nonlinearity and the algebraic degree.  Two other criteria can also be considered: the algebraic thickness and the non-normality. We give simple proofs that, asymptotically,  almost all Boolean functions have high algebraic thicknesses and  are deeply  non-normal, as well as they have high algebraic degrees and high nonlinearities. We also study in detail the relationship  between non-normality and nonlinearity. We derive simple proofs of known results  on symmetric Boolean functions and we prove several new and more general  results on a class containing all symmetric functions.

21) ``Normal Extensions of Bent Functions", C. Carlet, H. Dobbertin and G. Leander. IEEE Transactions on Information Theory, vol. 50 (correspondence), pp. 2873-2879, 2004.
The notion of a normal extension is introduced for bent functions, i.e. maximally non-linear Boolean functions. We apply this concept to characterize when the direct sum of bent functions is normal, and we prove that the direct sum of a normal and a non-normal bent function is always non-normal.

22) ``Cubic Boolean functions with highest resiliency", C. Carlet, and P. Charpin. IEEE Transactions on Information Theory, vol. 51 (regular paper),  pp. 562-571, 2005.
We classify those cubic m-variable Boolean functions which are (m-4)-resilient. We  prove that there are four types of such functions, depending on the stucture of the support of their Walsh spectra. We are able to determine, for each type, the Walsh spectrum and, then, the nonlinearity of the corresponding functions.  We also give the dimension of their linear space. This dimension equals m-k where k=3 for the first type, k=4 for the second type, k=5 for the third type and 5<= k<= 9 for the fourth type.

23) "Hyper-bent functions and cyclic codes", C. Carlet and P. Gaborit. Journal of Combinatorial Theory, Series A 113, no. 3, pp. 466-482, 2006.
Bent functions are those Boolean functions whose Hamming distance to the Reed-Muller code of order 1  equals 2^{n-1}- 2^{n/2-1} (where the number n of variables is even). These combinatorial objects, with  fascinating properties, are rare. Few constructions are known, and it is difficult to know whether the bent functions they produce are peculiar or not, since no way of generating at random bent functions on 8 variables or more is known.

24) "Linear Codes from Perfect Nonlinear Mappings and  their Secret Sharing Schemes", C. Carlet, C. Ding and J. Yuan. IEEE Transactions on Information  Theory 51 (regular paper), pp. 2089-2103, 2005.
In this paper, error correcting codes from perfect nonlinear  mappings are constructed, and then employed to construct secret sharing schemes. The error correcting codes obtained in this  paper are very good in general, and many of them are optimal  or almost optimal. The secret sharing schemes obtained in this paper have two types of access structures. The first type is democratic in the sense that every participant is involved in the same number of minimal access sets. In the second type of access structures, there are a few dictators who are in every minimal access set, while each of the remaining participants is in the same number of minimal access sets. 

25) "Piecewise Constructions of Bent and Almost Optimal Boolean Functions", C. Carlet and J. L. Yucas. Codes and Cryptography, vol. 37, No 3, pp. 449-464, 2005.
The first aim of this work was to generalize the techniques used in MacWilliams' and Sloane's presentation of the Kerdock code and develop a theory of piecewise quadratic Boolean functions. This generalization led us to construct large families of potentially new bent and almost  optimal functions from quadratic forms in this piecewise fashion. We show how our motivating example, the Kerdock code, fits into this setting. These constructions were further generalized to non-quadratic bent functions. The resulting constructions design n-variable bent  (resp. almost optimal) functions from n-variable bent or almost optimal functions.

26) ``Construction of bent functions via Niho power functions", H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke and P. Gaborit.
Journal of Combinatorial Theory, Series A, Volume 113,  Issue 5, pp. 779-798, 2006.
A Boolean function with an even number $n=2k$ of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions. Boolean functions of the form $f(x) = \tr(\alpha_1 x^{d_1} + \alpha_2 x^{d_2})$, $\alpha_1, \alpha_2, x \in \F_{2^n}$, are considered, where the exponents $d_i$ ($i=1, 2$) are of Niho type, i.e. the restriction of  $x^{d_i}$ on $\F_{2^k}$ is linear. We prove for several pairs of $(d_1,d_2)$ that $f$ is a bent function, when $\alpha_1$ and $\alpha_2$ fulfill certain conditions. To derive these results we develop a new method to prove that certain rational mappings on $\F_{2^n}$ are bijective.
27) Nonlinearities of S-boxes, C. Carlet and C. Ding. Finite Fields and Their Applications, Vol. 13 Issue 1, pp. 121-135, January 2007.
We introduce an indicator of the non-balancedness of functions  defined over Abelian groups,  and deduce a new indicator, denoted by $NB$, of  the nonlinearity of such functions. We prove an inequality relating $NB$ and the classical indicator $NL$, introduced by Nyberg and studied by Chabaud and Vaudenay, of the nonlinearity of S-boxes. This inequality results in an upper 
bound on $NL$ which unifies Sidelnikov-Chabaud-Vaudenay's bound and the 
covering radius bound. We also deduce from bounds on linear codes three new bounds on 
$NL$ that improve upon Sidelnikov-Chabaud-Vaudenay's bound and the covering  radius bound in many cases.
28) The Weight Distribution of a Class of Linear Codes from Perfect Nonlinear Functions, J. Yuan, C. Carlet and C. Ding.  IEEE Transactions on Information Theory, vol. 52, no. 2 (correspondence), pp. 712-717, 2006.
In this correspondence, the weight distribution of a class of linear
codes based on perfect nonlinear functions (also called planar functions) is
determined. The class of linear codes under study are either optimal or among the best
codes known, and have nice applications in cryptography.
29) Authentication Schemes from Highly Nonlinear Functions, C. Carlet, C. Ding and H. Niederreiter. Designs, Codes and Cryptography 40, pp. 71 - 79, 2006.
We construct two families of authentication schemes using highly nonlinear functions on finite fields of characteristic 2. This leads to improvements on an earlier construction by Ding and Niederreiter
if one chooses, for instance, an almost bent function as the highly nonlinear function.
30) New Classes of Almost Bent and Almost Perfect Nonlinear Polynomials. L. Budaghyan, C. Carlet and A. Pott.  IEEE Transactions on Information Theory, vol. 52 (regular paper), pp. 1141-1152, 2006.
New infinite classes of almost bent and almost  perfect nonlinear polynomials are constructed. It is shown that they are affine inequivalent to any sum of a power function and an affine function.
31) Algebraic Immunity for Cryptographically Significant Boolean Functions: Analysis and Construction. C. Carlet, D. Dalai, K. Gupta and S. Maitra. IEEE Transactions on Information Theory 52 (regular paper), pp. 3105-3121, 2006.
Recently, algebraic attacks have received a lot of attention in cryptographic literature. It has been observed that a Boolean function $f$ used as a cryptographic primitive, and interpreted as a  multivariate polynomial over $F_2$, should not have low degree multiples obtained by multiplication with low degree nonzero functions. In this paper, we show that a Boolean function having low nonlinearity is (also) weak against algebraic attacks, and we extend this result to higher order nonlinearities. Next, we present enumeration results on linearly independent annihilators. We also study certain classes of highly nonlinear resilient Boolean functions for their
algebraic immunity. We identify that functions having low degreesubfunctions are weak in terms of algebraic immunity, and we analyse some existing constructions from this viewpoint. Further, we present a
construction method to generate Boolean functions on $n$ variables with highest possible algebraic immunity $\lceil \frac{n}{2} \rceil$ (this construction, first presented at FSE 2005, has been originally the first one producing such functions). These functions are obtained through a doubly indexed recursive relation.  We calculate their Hamming weights and deduce their nonlinearities; we show that they have very high algebraic degrees. We express them as the sums of two functions which can be obtained from simple symmetric functions by a transformation which can be implemented with an algorithm whose complexity is linear in the number of variables. We deduce a very fast way of computing the output to these functions, given their input.
32) Improving the upper bounds on the covering radii of binary  Reed-Muller codes. C. Carlet and S. Mesnager. IEEE Transactions on Information Theory 53 (regular paper), pp. 162-173, 2007.
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 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.

33) On an Improved Correlation Analysis of Stream Ciphers Using Muti-Output Boolean Functions and the Related Generalized Notion of Nonlinearity. C. Carlet, K. Khoo, C.-W. Lim and C.-W. Loe. Advances in Mathematics of Communications, Vol. 2, No. 2, pp. 201-221, May 2008.
We investigate the security of $n$-bit to $m$-bit vectorial Boolean functions in stream ciphers. Such stream ciphers have higher throughput than those using single-bit output Boolean functions. However, as shown by Zhang and Chan at Crypto 2000, linear approximations based on composing the vector output with any Boolean functions have higher bias than those based on the usual correlation attack. In this paper, we introduce a new approach for analyzing vector Boolean functions called generalized correlation analysis. It is based on approximate equations which are linear in the input $x$ but of free degree in the output $z=F(x)$. The complexity for computing the generalized nonlinearity for this new attack is reduced from $2^{2^m \times n+n}$ to $2^{2n}$. Based on experimental results, we show that the new generalized correlation attack gives linear approximation with much higher bias than the Zhang-Chan and usual correlation attack. We confirm this with a theoretical upper bound for generalized nonlinearity, which is much lower than for the unrestricted nonlinearity (for Zhang-Chan's attack) and {\em a fortiori} for usual nonlinearity. We also prove a lower bound for generalized nonlinearity which allows us to construct vector Boolean functions with high generalized nonlinearity from bent and almost bent functions. We derive the generalized nonlinearity of some known secondary constructions for secure vector Boolean functions. Finally, we prove that if a vector Boolean function has high nonlinearity or even a high unrestricted nonlinearity, it cannot ensure that it will have high generalized nonlinearity.

34) Recursive lower bounds on the nonlinearity profile of Boolean functions and their applications. C. Carlet. IEEE Transactions on Information Theory (regular paper). Volume 54, Issue 3, pp. 1262 - 1272, 2008.
The nonlinearity profile of a Boolean function (i.e. the sequence of its minimum Hamming distances $nl_r(f)$ to all functions of degrees at most $r$, for $r\geq 1$)  is a cryptographic criterion whose role against attacks on stream and block ciphers has been illustrated by many papers. It plays also a role in coding theory, since it is related to the covering radii of Reed-Muller codes. We introduce a method for lower bounding its values and we deduce bounds on the second order nonlinearity for several classes of cryptographic Boolean functions, including the Welch and the multiplicative inverse functions (used in the S-boxes of the AES). In the case of this last infinite class of functions, we are able to  bound the whole profile, and we do it in an efficient way when the number of variables is not too small. This allows showing the good behavior of this function with respect to this criterion as well.

35) Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. L. Budaghyan and C. Carlet.  IEEE Transactions on Information Theory (regular paper) 54, Issue 5, pp. 2354-2357, 2008.
A method for constructing differentially 4-uniform quadratic hexanomials has been recently introduced by J.~Dillon. We give various generalizations of this method and we deduce the constructions of new infinite classes of almost  perfect nonlinear quadratic trinomials and hexanomials from $F_{2^{2m}}$ to $F_{2^{2m}}$. We check for $m=3$ that some of these functions are CCZ-inequivalent to power functions.

36) Two classes of quadratic APN binomials inequivalent to power functions. L. Budaghyan, C. Carlet and G. Leander.  IEEE Transactions on Information Theory, vol. 54 (regular paper), pp. 4218-4229, 2008.
This paper introduces the first found infinite classes of almost perfect nonlinear (APN) polynomials which are CCZ-inequivalent to power functions. These are  two classes of APN binomials from $F_{2^n}$ to $F_{2^n}$ (for $n$ divisible by 3, resp. 4). We prove that these functions are EA-inequivalent to any power
function and that they are CCZ-inequivalent to the Gold, Kasami, inverse and Dobbertin functions when $n\ge12$. This means that for $n$ even they are CCZ-inequivalent to any known APN function. In particular for $n=12,20,24$, they are therefore CCZ-inequivalent to any power function.

37) Constructing new APN functions from known ones. L. Budaghyan, C. Carlet and G. Leander. Finite Fields and Applications 15(2), Pages 150-159, 2009.
We present a method for constructing new quadratic APN functions from known ones.
Applying this method to the Gold power functions we construct an APN function $x^3+\tr(x^9)$ over $\F_{2^n}$.
 It is proven that in general this function is CCZ-inequivalent to the Gold functions,  and in the case $n=7$ it is CCZ-inequivalent to any power  mapping (and, therefore, to any APN function from the previously known families of APN mappings).

38)  Further properties of several classes of Boolean functions with optimum algebraic immunity. C. Carlet, X. Zeng, C. Li and L. Hu. Designs, Codes and Cryptography, Volume 52 ,  Issue 3, pp. 303 - 338, 2009.
Thanks to a method proposed by Carlet, several classes of balanced Boolean functions with optimum algebraic immunity are obtained. By choosing suitable parameters, for even $n\geq 8$,  the balanced $n$-variable functions can have nonlinearity $2^{n-1}-{n-1\choose\frac{n}{2}-1}+2{n-2\choose\frac{n}{2}-2}/(n-2)$, and for odd $n$, the functions can have nonlinearity $2^{n-1}-{n-1\choose\frac{n-1}{2}}+\Delta(n)$, where $\Delta(x)$ is a polynomial described in Theorem \ref{thm8}. The algebraic  of some constructed functions is also discussed.

39) On the construction of bent vectorial functions. C. Carlet. and S. Mesnager. Special Issue of the International Journal of Information and Coding Theory (IJICoT), Vol. 1, no 2,  dedicated to Vera Pless, pp. 133-148, 2010.
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)  Self-dual bent functions. C. Carlet, L. E. Danielsen, M. Parker and P. Solé.  Special Issue of the International Journal of Information and Coding Theory (IJICoT) dedicated to Vera Pless,  Volume 1, No. 4, pp. 384-399, 2010 (a preliminary version appeared in the proceedings of the BFCA'08 conference).
A bent function is called self-dual  if it is equal to  its dual. It is called anti-self-dual if it is equal to the complement of its dual. A spectral characterization in terms of the Rayleigh quotient of the Sylvester Hadamard matrix is derived. Bounds on the Rayleigh quotient are given for Boolean functions in an odd number of variables. An efficient search algorithm based on the spectrum of the Sylvester matrix is derived. Primary and secondary constructions are given. All self-dual bent Boolean functions in $\le 6$ variables  and all quadratic such functions in $8$ variables are given, up to a restricted form of affine equivalence.

41) Relating three nonlinearity parameters of vectorial functions and building APN functions from bent. Designs, Codes and Cryptography. C. Carlet. Vol. 59, No. 1, Page 89--109, 2011.
We survey the properties of two parameters introduced by C. Ding and the author for quantifying the balancedness of vectorial functions and of their derivatives. We give new results on the distribution of the values of the first parameter when applied to $F+L$, where $F$ is a fixed function and $L$ ranges over the set of linear functions: we show an upper bound on the nonlinearity of $F$ by means of these values, we determine the mean of the values and we show that their maximum is a nonlinearity parameter as well, we prove that the variance of these values is directly related to the second parameter. \\
We briefly recall the known constructions of bent vectorial functions and introduce two new classes obtained with Gregor Leander. We show that bent functions can be used to build APN functions  by concatenating the outputs of a bent $(n,n/2)$-function and of some other $(n,n/2)$-function.  We obtain this way a general infinite class of quadratic APN functions. We show that this class contains the APN trinomials and hexanomials introduced in 2008 by L. Budaghyan and the author, and a class of APN functions introduced, in 2008 also, by Bracken et al.; this gives an explanation of the APNness of these functions and allows generalizing them. We also obtain this way the recently found Edel-Pott cubic function. We exhibit a large number of other sub-classes of APN functions. We eventually design with this same method classes of quadratic and non-quadratic differentially 4-uniform functions.

42) CCZ-equivalence of Bent Vectorial Functions and Related Constructions. L. Budaghyan and C. Carlet. Designs, Codes and Cryptography, Vol. 59, No. 1-3 , pp. 69-87, 2011.
We observe that the CCZ-equivalence of bent vectorial functions over  F_2^n (n even) reduces to their EA-equivalence. Then we show that in spite of this fact,  CCZ-equivalence can be used for constructing bent functions which are new up to EA-equivalence and therefore to CCZ-equivalence:  applying CCZ-equivalence to a non-bent vectorial function $F$ which has some bent components, we get a function $F'$ which also has some bent components and whose bent components are CCZ-inequivalent to the components of the original function $F$. Using this approach we construct classes of nonquadratic bent Boolean and bent vectorial functions.

43) Comment on ``Constructions of Cryptographically Significant Boolean Functions Using Primitive Polynomials". C. Carlet. IEEE Transactions on Information Theory Vol. 57, no. 7, pp. 4852 - 4853 , 2011
We show that the first of the two constructions by Q. Wang, J. Peng, H. Kan and X. Xue in IEEE Trans. on Inf. Th., vol 56, no 6, 2010, of Boolean functions provably satisfying the main criteria for filter functions in stream ciphers, is the same as the construction studied by the author and K. Feng at Asiacrypt 2008. We observe that the bounds shown on the nonlinearities of the functions in this IEEE paper are similar to those shown in the Asiacrypt paper. We point out that all these functions can be implemented in a more efficient way than usually believed.

44)  X. Zeng, C. Carlet, L. Hu and J. Shan. More Balanced Boolean Functions with Optimal Algebraic Immunity, and Good Nonlinearity and Resistance to Fast Algebraic Attacks. IEEE Transactions on Information Theory, Vol. 57, pp. 6310-6320, 2011.
In this paper, three constructions of balanced Boolean functions with optimal algebraic immunity are proposed. It is checked that, at least for small numbers of input variables, these functions have good behavior against fast algebraic attacks as well. Other cryptographic properties such as algebraic degree and nonlinearity of the constructed functions are also analyzed. Lower bounds on the nonlinearity are proved, which are similar to the best bounds obtained for known Boolean functions resisting algebraic attacks and fast algebraic attacks. Moreover, it is checked that for the number $n$ of variables with $5\leq n\leq 19$, the proposed $n$-variable Boolean functions have in fact very good nonlinearity.
45) C. Carlet and S. Mesnager. On Dillon's class H of bent functions, Niho bent functions and o-polynomials. Journal of Combinatorial Theory, Series A 118, pp. 2392-2410, 2011.
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, 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 $u\GF {n/2}$, $u\in \GF n^\star$, are linear.  We also characterize the bent functions whose restrictions to the $u\GF {n/2}$'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 (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).
 46) C. Carlet. More vectorial Boolean functions with unbounded nonlinearity profile. Special Issue on Cryptography of International Journal of Foundations of Computer Science 22 (6), pp. 1259-1269, 2011.
The nonlinearity profile of Boolean functions is a generalization of the most important cryptographic criterion, called the (first order) nonlinearity. It is defined as the sequence of the minimum Hamming distances $nl_r(f)$ between a given Boolean function $f$ and all Boolean functions in the same number of variables and of degrees at most $r$, for $r\geq 1$. This parameter, which has a close relationship with the Gowers norm, quantifies the resistance to cryptanalyses by low degree approximations of stream ciphers using the Boolean function $f$ as combiner or as filter.  The nonlinearity profile can also be defined for vectorial functions: it  is the sequence of the minimum Hamming distances between the component functions of the vectorial function and all Boolean functions of degrees at most $r$, for $r\geq 1$. The nonlinearity profile of the multiplicative inverse functions has been lower bounded in a previous paper by the same author. No other example of an infinite class of functions with unbounded nonlinearity profile has been exhibited since then. In this paper, we lower bound the whole nonlinearity profile of the (simplest) Dillon bent function $(x,y)\mapsto xy^{2^{n/2}-2}$, $x,y\in F_{2^{n/2}}$ and we exhibit another class of functions for which bounding the whole profile of each of them comes down to bounding the first order nonlinearities of all functions.

47) C. Carlet, J.C. Ku-Cauich and H. Tapia-Recillas. Bent Functions on a Galois Ring  and  Systematic Authentication Codes. Advances in Mathematics of Communications, Volume 6, Number 2, pp. 249–258, 2012.
A class of bent functions on a Galois ring is introduced and based on these functions systematic authentication codes are presented.

48) C. Carlet and S. Mesnager. On Semi-bent  Boolean Functions. IEEE Transactions on Information Theory, Vol. 58, no. 5, pp. 3287-3292, 2012.
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.

49) C. Carlet, J.-C. Faugère,  C. Goyet and G. Renault. Analysis of the Algebraic Side Channel Attack.  Journal of Cryptographic Engineering (JCEN) Vol. 2, no. 1, pp. 45-62, 2012.
At CHES 2009, Renauld, Standaert and Veyrat-Charvillon introduced a new kind of attack called Algebraic Side-Channel Attacks (ASCA). They showed that side-channel information leads to effective algebraic attacks. These results are mostly experiments strongly based on a the use of a SAT-solver. This article presents a theoretical study in order to explain and to characterize the algebraic phase of these attacks. We study more general algebraic attacks based on Gröbner methods. We show that the complexity of the Gröbner basis computations in these attacks depends on a new notion of algebraic immunity defined in this paper, and on the distribution of the leakage information of the cryptosystem. We also study two examples of common leakage models: the Hamming weight and the Hamming distance models. For instance the study in the case of the Hamming weight model gives that the probability of obtaining at least 64 (resp. 130) linear relations is about 50% for the substitution layer of PRESENT (resp. AES). Moreover if the S-boxes are replaced by functions maximizing the new algebraic immunity criterion then the algebraic attacks (Gröbner and SAT) are intractable. From this theoretical study, we also deduce an invariant which can be easily computed from a given S-Box and provides a sucient condition of weakness under an ASCA. This new invariant does not require any sophisticated algebraic techniques to be defined and computed. Thus, for cryptographic engineers without an advanced knowledge in algebra (e.g. Gröbner basis techniques), this invariant may represent an interesting tool for rejecting weak S-boxes.

50)  C. Carlet, F. Zhang and Y. Hu. Secondary constructions of bent functions  and their enforcement. Advances in Mathematics of Communications (AMC) Volume 6,Number 3, pp. 305 - 314, 2012.
Thirty years ago, Rothaus introduced the notion of bent function and presented a secondary construction (building new bent functions from already defined ones), which is now called the Rothaus' construction. This construction has a strict requirement for its initial functions. In this paper, we first concentrate on the design of the initial functions in the Rothaus construction. We show how to
construct Maiorana-McFarland's (M-M) bent functions,  which can then be used as initial functions, from Boolean permutations and orthomorphic permutations. We deduce that at least $(2^n!\times 2^{2^n})(2^{2^n}\times2^{2^{n-1}})^2$ bent functions in $2n+2$ variables can be constructed by using Rothaus' construction. In the second part of the note, we present a new secondary construction of bent functions which generalizes the Rothaus construction. This construction requires initial functions with stronger conditions; we give examples of functions satisfying them. Further, we generalize the new secondary construction of bent functions and illustrate it with examples.

51) G. Gao,   X. Zhang,  W. Liu and C. Carlet. Constructions of Quadratic and Cubic Rotation Symmetric Bent Functions. IEEE Transactions on Information Theory, Vol. 58, no. 7, pp. 4908-4913, 2012.
In this paper, we consider constructions of quadratic and cubic rotation symmetric bent functions, which are of the forms f_{c}(x)=\sum_{i=1}^{m-1}c_{i}(\sum_{j=0}^{n-1}x_jx_{i+j})+c_m(\sum_{j=0}^{m-1}x_jx_{m+j})  and f_t(x)=\sum_{i=0}^{n-1}(x_ix_{t+i}x_{m+i}+x_{i}x_{t+i})+\sum_{i=0}^{m-1}x_{i}x_{m+i},  where n=2m, c=(c_1, c_2,..., c_m) with  c_i in {0,1} for
1\leq i \leq m  and 0<t<m (the subscript u of x_u in the expressions of f_c(x) and f_t(x) is taken as u modulo n).
For each case, a necessary and sufficient condition is obtained.  To the best of our knowledge, this class of cubic rotation symmetric bent functions is the first example of an infinite class of nonquadratic rotation symmetric bent functions.

52) C. Carlet, P. Gaborit, J.-L. Kim and P. Solé. A new class of codes for Boolean masking of cryptographic computations. IEEE Transactions on Information Theory, Vol. 58 No. 9, pp.  6000-6011, 2012.
We introduce a new class of rate one-half binary codes: complementary information set codes. A binary linear code of length 2n and dimension n is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune Boolean functions of use in the security of hardware implementations of  cryptographic primitives.
Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks.
In this paper we investigate this new class of codes: we give optimal or best known CIS codes of length <132. We  derive general constructions based on cyclic codes and on double circulant codes. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths \le 12 by the building up construction. Some nonlinear permutations are constructed by using Z_4-codes, based on the notion of dual distance of a possibly nonlinear code.

53) D. Tang, C. Carlet and X. Tang. On the Second-Order Nonlinearities of Some Bent Functions. Information Sciences (Elsevier) 223; pp. 322-330, 2013.
The rth-order nonlinearity of Boolean functions plays a central role against several known attacks on stream and block ciphers. It plays also an important role in coding theory, since its maximum equals the covering radius of the rth-order Reed-Muller code. But it is difficult to calculate and even to bound. In this paper, we show lower bounds on the second-order nonlinearity of two subclasses of well-known bent functions. We first improve a known lower bound on the second-order nonlinearity of the simplest partial spread bent functions, whose nonlinearity profile has
been bounded from below by the second author. This improvement allows improving the bound for the whole profile.
We subsequently give a lower bound on the second-order nonlinearity of some infinite class of Maiorana-McFarland (M-M) bent functions, which generalizes a result by Gangopadhyay et. al.

54) L. Budaghyan, C. Carlet, T. Helleseth, A. Kholosha and S. Mesnager. Further Results on Niho Bent Functions. IEEE Transactions on Information Theory 58 no. 11, pp. 6979-6985, 2012.
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, the aforecited and two other infinite classes of Niho bent functions are analyzed for their relation to the completed Maiorana-McFarland class. In particular, it
is proven that two families of Niho bent functions do not belong to the completed Maiorana-McFarland class. The latter result gives a positive answer to an open problem whether one of the classes of bent functions introduced by Dillon in his thesis of 1974, differs from the completed Maiorana-McFarland class.

55)  D. Tang, C. Carlet and X. Tang. Highly Nonlinear Boolean Functions with Optimal Algebraic Immunity and  Good Behavior Against Fast Algebraic Attacks. IEEE Transactions on Information Theory 59 (1), pp. 653-664, 2013.
Inspired by the previous work of Tu and Deng, we propose two infinite classes of Boolean functions of 2k variables where k\ge 2. The first class contains unbalanced functions having high algebraic degree and nonlinearity. The functions in the second one are balanced and have maximal algebraic degree and high nonlinearity (as shown by a lower bound that we prove; as a by-product we also prove a better lower bound on the nonlinearity of the Carlet-Feng function). Thanks to a combinatorial fact, first conjectured by the authors and later proved by Cohen and Flori, we are able to show that they both possess optimal algebraic immunity. It is also checked that, at least for numbers of variables n\leq 16, functions in both classes have a good behavior against fast algebraic attacks. Compared with the known Boolean functions resisting algebraic attacks and fast algebraic attacks, both of them possess the highest lower bounds on nonlinearity. These bounds are however not enough for ensuring a sufficient nonlinearity for allowing resistance to fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for bounded numbers of variables ($n\leq 38$). Moreover, these values are very good. The infinite class of functions we propose in Construction 2 presents, among all currently known constructions, the best provable trade-off between all the important cryptographic criteria.

56) C. Carlet. More constructions of APN and differentially 4-uniform functions by concatenation. Science China Mathematics, Vol. 56  Issue (7), pp. 1373-1384, 2013.
We study further the method of concatenating the outputs of two functions for designing an APN or  a differentially 4-uniform $(n,n)$-function for every even $n$. We deduce several specific constructions of APN or differentially 4-uniform $(n,n)$-functions from APN and differentially 4-uniform $(n/2,n/2)$-functions. We also give a construction of quadratic APN functions which includes as particular cases a previous construction by the author and a more recent construction by Pott and Zhou.

57) C. Carlet and B. Merabet. Asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions. Advances in Mathematics of Communications, Vol. 7, Issue 2, pp. 197 - 217, May 2013.
This paper extends the work of F. Didier (IEEE Transactions on Information Theory, Vol. 52(10): 4496–4503, October 2006) on the algebraic immunity of random balanced Boolean functions, into an asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions.

58)  C. Carlet, J.-L. Danger S. Guilley, H. Maghrebi and E. Prouff. Achieving side-channel high-order correlation immunity with Leakage Squeezing. Journal of Cryptographic Engineering  (JCEN) 4(2), pp. 107-121, 2014.
This article deeply analyses high-order (HO) Boolean masking countermeasures against side-channel attacks in contexts where the shares are ma partated anh ptograrely fohe infationship witween the co kage modracteristic 2.nd the Relack calicient wcs linfocd as,ads t the oe initiduced n of the leaion of duaHO-CPAmunity are smethod bencalcacterize theaakage modction. We Ourin critribute t is obt oeearke minimum Hamack caler (HO)led a cHO-CPAmunity ar) the sumng uof its rmation leads k It iresting to the afoHO-CPAmunity are be conh loweer clan the Zhmber of varres arethe secking- sidemes uhis impraximde sible algthe autkage modsezing. Jouis alsoptimal tion of the sharct
58)&9) Q.Wa C. Carlet, L. Sol&ce tiod Cubihikweveng. .yptographic Enouffties of two afoHiddtheWht moBooliunctions. idieibeeclaAied tothematics of 17pp. 3841- 2011.
This igheddthewht moBoo anoction for(HWBF),troduced by JohR. Bryainfor E Trans. on pare. 40d Cubioussi by usiD. Knor it . 52(4 Tu aAlgeof th puter Sciuffphing wes sembsp; sliction fort sidee in be def simplest pa areh Leaonents t Boolry  Re Dficn appDiaphin (BDD)mplzuhis imopertiesy introdsting toom a givptographic criwpoint. Fnce itsBDDsed on acks and received tooe recention in cre comptographic computity. It But, be defuse to stream ciphers, is tfunctions in mualgebstudsfying the imper infn criteria fon this paper, we shoestigate the se ptographic primifties of two afoHWBFd providehat if as basedced mu, h optimum algebraic immree and higdsfyin the resict reqavdced se ptrion th. calculate the maxexaconlinearity for hige a lower bound on the maxebraic immunity. It eover, theshoestigate the maxm, qal and Res maxistance to inst sidt algebraic attacks. ComTafoHWBFdshowlest ,n theimplemented in icient wa the a strh nonBDDss theaCubiar infd behptographic primifties of, theshoen ato an acptermat if assmber of variables. Ae be conh lo eer clan the odd r infctions with the same feplementations olicient wcsomTafore, to aafoHWBFdshoood beh bedid form teim up c in thxiscodeers. It pnplys,atribu infthe comp of themetric benctions, which are wingch funcaimproventations ol of o obtarent the comacks. sub- in&cific cossibly ty o the to Dobir relmetric it cas sctures. not toosuIpiveto builelated to itsh funicated to acks.
In > In 60. Carlet and B. Khollearern the SecAhms acal Wo oh paicient waf varlean Functions. IEsp; Re Dfns, Codes and Cryptography, Vosp; Volume 1, 73ssue 2, pp. 197299-318011.
ThiWeneralize the he sumnhms acal Wo oh nsformati (AWT)b- in&ults arich fouwerrevious woknown con the whoWo oh amard matnsformati ooolean functions ane first noteralize the knossifie stuPoissstreumion leamatite e oe iniAWTbsp; We dedt andined ogeneralizatinonlin of theistalce Ch h respect to thiall r anu infnceticatedl sures a Boolean functions of.> ThiWenlyinge iniPoissstreumion leamatite thiain thiandition is ivalent toe oeistalce Che odd nesh funnceticatedl sures absp; We Laimheshsp; Issw that the comAWT a sprge numss of fusp; Boolean Functions can be impressionin the s of the RayAWT a sprlean function $h$fusp; Booebraic immree and ost $r$3 a preeer claber of variables. We> In > In 61) ShaLi. Carlet, L. Zeng, Cd C. CarLi.o clastructions of APNanced Booolean functions with optimum algebraic immunity ar,gh nonlinearity (as Resod behavior of inst sidt algebraic attacks. ComDfns, C des and Cr ptography of 76 , pp. 107279-302012.<5r> Abstract:
In this paper, threclastructions of APNlean functions with optimum alggebraic immunity of proposed. It T boteralize thevious cons), pect to a give a y PotRizoies in nd Cryg, Cdal. innd show nonlewm ctions with optigni witmifties of obtained. By functions in nstructed funihis paper we be conanced and have matimum algebraic immmree anurther, we ew claer bound on the alglinearity of the Carposed $n$nctions in taxesle. ishas,a proa subcial Isse, a niives a pos claer bou nd on the alglinearity of the Carlet-Feng function). shich is now ghtly larter lown the Zhmt provaous woknown cons. L. F$n=12q 19$, the pr aberon cresults, wbiouethis at lg all complructed functions is ahis pap er, threeeall lw a cstencow nonctions with optlinearity of her throuthe besal to the sat the Carlet-Feng function). shese bouctions are also ana cked thatove good behavior of inst sidt algebraic attacks. C least forr small numbers of input variables, te> In > In 62. Carlet, J.-C. Danger S., Guilley, Hd H. Taphrebi an.akage Sq ueezing. J:timal AlgIoventations ol S. Mrity of Eues on. Scirnal of Cryhematics oof ptogralogy 8(3pp. 107249-295011.
ThiHware imvoteic Ae be conpostiveto inst side-channel attac- ks basey troduced of $nesdom balking s. deenive anriables, he complextions a thrh fo an inbalar.&nfunihe simpes are(kingvia able Boo S. kingre int ceedionin cateoietawa theahisdisjoig trequigence It N$neless, as s pap setupe be ig be claacks. sunihe simp-channel infosdsezd wi, clcaushis ne en pron codeaushs infintect ce). In > In 63. Carlet, J.-Gao,&ne claLiu an specndary construction of a mor nsformation of tntation symmetric benction). shi their enfacn of tntt fund show-bent functions inournal of Combinatorial Theory, Volies A 11ol 56,127s se 2, pp. 697161 - 175011.
Thistudy fure in details andtinfationship witween theation symmetric ben (RS)nctions and illuigmentialinto ivariate for blocriate for resent anan). shi theuce a larstruction of bent funRSnctions from ben w-bent funRSnctions fre deduce that irst notinite classes of nd funthe uigmentiald RelaSnt functions on afgebraic immree andmothe uthe3e ded roduce a newnsformation of m a ginylaSnlean function $f$ andr if $\GF(2)^n$to an initigmentialdlean function $f$ a'(z)=f(z,z^2,\ldoega ,z^{n-1}})^2$ver $\F_{GF(n})(2^$,ads t th thial r infaSnlean funnction.
38)64) Zhang an. Carlet, L. Yu and J. Tarleoecondary constructions of be hly Nonlinearitysp; Boolean Functions casp; and 0&ljoint inf ctral csp; andped taufunctions isnformation Sciences (El28pp. 30594-106011.
Thithis paper, we shoelsifygeneralizatinona lctly msof struction of thipstruct Maictions with opth nonlinearity (arsp; TheBy ug bng the new elsifiin catuction of,gh nononlinear mulctions in $2n+(n+m)ariables are be conained. Bom known onet functions in $2n+n$riables and ofsp; The hly Nonlinearitynctions in $2n+m$riables anrsp; Theis provible algthipain thi cla+(n+15)ariable functions canh optlinearity of nd n-1}+15-{n-n-1(}+15-{)/2}+20mes2^{ n/2}}$ nd all cla+12ariable fun $2aristalce o ctions canh opsp; Thelinearity of n2000nd all Braic immmree an$ vahich are evingatimum algebraic immunity. It eover, the new elsifiin catuction ofn also be deed as iniinfiature anrstruction of bena antitiu of an joint infctral csp; andped taufunctions isnform adion isbsp;  X. present a nficient condition isAe bo aantitiu of an joint infctral csp; andped taufunctions is  itse nonlielinzeror inritynsctures. fe> In > In 65) Carlet, F. ZhaFreit bo, Guilley, H. Mim aetioi theL. Kim and P. Sollé.&nbghly ea-er (HO-codes, aEEE Transactions on Information Scieory, S 60(9pp. 107528p-5295011.
Thiintroduce a nplementary information set code (Cf be hly ea-er (H spe ary linear code of length 2n $tk$d dimension n $k$ called a com plementary information set code (CIour r (HO$t$ ($t$- code for short) if it has two$t$ pairwisesjoint information sets. This uals of them as es permit to imprce that irt of masking cryptographic al orithms aga inst side-channel attacks inr> As sma comp of codes con everrroorrelaton of,ge a y autkah 2n fas sum ension n a spr$t$- code foe lowerokn the whohest lowvible algimum Hamstances $n this paper, we s new class of codes: w introtigate thBy fun stence in codd behg CIS codes, aIour r (HO$3$ di&#ved. Poa lowplerm cry argut. AdvGral algstructions based on cyclic cod fasntisi-lic codes and on the dislding up construction of given. Alsp; itsAamatite ilar to e oeaskinsamatite gives an spessified can). $2^3- codes of length < $ 12 by$ gives an slinear Booes of ter lown theear Booes of imvoted. Po Cohtak bentry linm agof len$\Z_4$-es. A bineral algebrthm based on the Edms to'sis tecps i algebrthm basm knorix oidr seory di&#vvelopwith thee mosfowing respertiesy:ge a y anary linear code of lenid t $1/t$ to eir infvides a s$t$ joint information sets. T difmed bhat the proe of ne e too$t$- cosing this apprbrthm ba,d numimum algbest known CIS$[tk, k]$ es perre $F$ t=pp.4, \doegun2 nnd allr1e 12 ke 12 \lferorn2 n/t \rferor$ e invtn thatuilel$t$- cod bo aself as $k$d dim$t$,hibcepor the t=p$ h opsp; The$k=44$d dim$t=4$dh the$k=37$e> In > In 66. Tang, C. Carlet and X. Tang. On lass of ben1-Rstalce o lean funnctions with Optimal Algebraic Immunity and  Good Behavior Ag ainst Fast Algebraic Attacks. IEsp; Theirnational Journal of Com ndations of Computer Science 22 (IJFCS)ol. 52(25o. 1-36p. 107763-780, 1.
ThiRntly fo,ng, C. Cet and X. g, Cesented.  46)bers of variables ($ouOuriputations in w that the firstions hav Mats class of e maty good nonlinearity. In > In 6C. Carlet and B. Dang. On Enhed Boolean functions wittable par bo sum ter.&nnel givihepseudo-dom baleral at ThiDfns, Codes and Cry ptography ofsp; The76 ,3pp. 107571-5 2011.<5r> >Abstract:
Thiis igter.&nnel givihepseudo-dom baleral at Tn paream ciphers, is tak rently kno sami if $nes which boe known, dlinite classes of almlean funnction.
49)68. Tang, C. Carlet and X. Tang. On Derentially 4-uniUorm $(nBiting onTo CohPutati ben ZhmIrse functions. idifns, Codes and Cryptography, Vsp; The77, pp. 653117-141011.<5r> >Abstract:
ThiiBk ciphers. IlushiStitution layes are(oxes. <)ose resaim obt oecre tum plnfus in bithe comptosystem. WL. Ftion.
49)69. Carlet and B. Yn ll Salamin laNewnstruction 2 palmDerentially 4-u niform $(n,n)$--1)$-ctions. IEEances in Mahematics of Communications, Vol. 7, 9o. 7, pp. 384541 - 562012.<5r> Abstract:
In this paper, thros cla tha oestruct Maiferentially 4-uniform $(n ,n)$--1)$-ctions in taxsented.
49)70. Carlet. Morlean fun alltorial funped taufunctions isnd on fun ctions doEEE Transactions on Information Sciory, S . 7, 61. 11, ppsp; andp653-272-6289012.<5r> Abstract:
In lean permed taufunctions is& alltorial functions casp; andh themed taufu ponent fulays aaubcing d carole in coditography, Voyeence ofre ens cunications, Vol Cubioed to pleatorial .nd theegnign$ouOuriwledge in a them dir toothe firdsa g the Syit ortant ccfe alsroduce a nlewm cacterize tons of Commed taufunlean functions. AdvWive opt cacterize tons of Comtorial functions caose responent fula all lin med taufun(h themibly nonferential le otiod, this we canwlestyas num med taufu, mean fuf the Mamues cstribution ofs the Syit ans (wholcacterize theilar tokno sofunctions havse responent fula all tial splynt futhithe Syit autorelation attctions is& allthe Carpr bou mom waf vae Syit Wo oh nsformatiy fapprws imour oeive a Veral spsp; Iss cacterize tons of Com functions whiahxt meameterk ofn poine a b ball th e impt criults arewn, dl predratic APN functions whiends t oemed taufunnctions, whiaseng rest addy twovae Syit fu-s undbuilelwlestifiine dedsw that theif,iadion isy de,l complnent fuuctions are also lbalanced fu,ge mosdy in os ig be wlest r:e cla fu-s undthem as ctions do nnds on a 4-u the alionues cstribution ofy fapprws imoperg thefinstance the sany Boo med taufusp; The$n)$-functions f,e n$ n $n,ving higilar to ues cs tribution of asd funer functions, ws sem and difh hee feaends tfunWo oh ctral Ham he fir andGolunctions wIEsp; The> As product wes,gwedain thianfewm r insp; Thelewiults aror instance th,ggeBoomed taufunctions i even dimension, eich is nowCCZ-ivalent to oeap GoluninsKae fiN functions w dir esary aesy EA-ivalent to oeitr>
49)71) KhoDeansaco, N.sKaitiv,hR. Kai we Xuillo. Carlet and X. Guilley, H. EainoinscrySl numkage Sqn Mahesto a cTur preSnd-Order Nonacks. to an ai Ft exder Nonacks. he comence tc cosWorlunrnal ofoliept012.<5r Hindawi Pu. ish CCZ-ortanon. SciDOI 10.1155/2.<5/743618< 49) r>
Thii Ming cryptermeasures as,gc in the ce i ee-channel attac- ks ba, e mat n exhtn thatuilelvuln cy so fasking-mpct In > In 72. Carlet, J.-Gao,oand Y. Hu.g. .ydratic andZerofDerentiaitsBnced BoooFtions, ws funFtions is& allSnger sy Regr carGhy, sidifns, Codes and Cryptography, Vl. 7, I8 ,3pp. 107629-652013.<6r> >>Abstract:
Thii Lec rF$ batunction in m the$\gf_{p/n}$r oeitf-dud dim$\delta$positive ansfinte. Fu rF$ a gled a czerofderentiaits$\delta$-anced funihe simuivaons a $F(x+a)-F=\su0$fh heexacosy $\delta$psolon ofs bo asellinzeror $a\in\gf_{p/n}$brAsdacticular care, a sp; Booebnown attdratic APNmedntynnction.
54)73. Carlet, J.-Dournyn we Sol&ce tiod CubDang. On ptographic Eng pifties of twomoionoBoolean functions of. Shahematics oof ptogralogy . 7, 10), pp. 6531-12013.<6r> Abstract:
Thiintne a biabl conrlts on NihmoionoBoolean functions of. particular, it< woine a bete ctured bdposed $n$rntly fo,nncetithe ballre is new tu moionoBoobialdlean function $fsurther, we gene a lon uarerund on the algnlinearity of themoionoBooctions, whiceducibed is sumWo oh--amard mat ctral Ham it istigate thew non r infptographic primifties of twomoionoBo olean functions.

58)74. Carlet and B. E. uff. Ac Polynom sprEues on. S& allS chaCnel att Ayses isol&ial Issume 1, icated to fasDaes Kahnol&ig ra Fu R, C. Prs i Yn l., Naccames,sDaes ,ydrisezd, whiJ fu-Jacqu(Elseds.). 653315-34101 3.<6r> Abstract:
ThiS chaCnel attAysis of (SCA) tax tss of theack. Ns balleainoinswkage mod inputmation of m a gifptographic priroventations oldng a sexecon ofy el fe i e J.-king crytax tsommoonstermeasures ahe comig rci of obt oe dom bacals otiery eveeenive anrirmeasude foriable funocrent crytnl sum plextions a o an eral speces are cla shbler of varres ar,gled a c sum {\ao king crsp; and r No},ays a c shquoof an icurity of ameters iy fun t crise 2, ch of rying CCZking crytoopostive anak ciphers. iplementations o obt oewific lonoicient wayemes ulan ercngen Zhmsxes m plextions asol&ral speking cryemes uh,haied ce par bo r anu infer tos,t h veen exhrntly fosroduced bydveost the Camsfowing ilar to aiero of ginal cofosroduced by the sectipof varlet and\etfunpu. ishasloeaFSE1 3.<2;n Zhmsxes mtoopostive obtwpoias iniiemilynom spr showyid tg of obt rotigate thBich is imum the knoler of varfieldlti-oued cens on ch boe new toosezd ngsy fapptipof aimC leasented.iall curse valk ofav Maub plexrehenivvlg thy funhod ofre alsfescus as,a Syit oerentiaitnd Cry ilar tois of obtlo w ed ine cla shbrems u of $ problem whf obtlencedbr>
54)75. Carlet and B. Mesnager. Fu F knodeca of lenent arcortntt fun ctions doEE&ial IssJub ofesue 2, almDfns, Codes and Cryptography, Vol. 7, I8p. 3845-50013.<6r> Abstract:
Thithis papes vee,lshoioussi a shbR r aupptipof acla shbchaptof varlon in's e imsdi&#vated to fast functions, whioillalsucibed is sumt criults are ained. Bothe alfunctions havdng a s alfunl for40 y arsbsp; We dedo ana ca $\Fmothebriefcalsupof-sses of Boolean functions of,mtorial funt fun ctions dor bloc functions in $2noddmpacteristic 2.r>
58)76) Zhang an. Carlet, L. Yu and J. Whang an.aNewnondary construction 2 e enfnt Functions. IEEAAECC (rnal of)ol. 5 1, 2Issue 2, 5p. 3 413211;450332013.<6r> Abstract:
Thithis paper, we shost exasent a nuflia g cndary construction of bent fun ctions dor(lding up bent function isrom the whiawd beydined odu thes)bsp; We ther, wm, to aafoebraic degree and higebraic immunity of ra e Mamplructed functions are alsoyses htr>
38)77. Budaghyan, C. Kholosha an. Carlet and X. Helleseth, Aninariate for o Bent Functions. Iom theo-Polynom spoEEE Transactions on Information Theory 58 62 (4pp. 1072254-2262012.<6r> Abstract:
Thithis paper, we shotriering ballvariate form of theaho bent function con tax tsof thections is e m the Booc of theahnder an-losha andt fun ctions dken as h themicular careaicient waf m theDn (n thery evemeas. iintwn a ballo bent functions are anaated to itso-polynom spoEETCarpr bo e s of sma comvariate foro bent function consbe imp#ved. Poa lk of wes thianft exancep,otheh casnonom sprthe Mamplrpectitioof $-polynom spn w-amet Mly,m it is icurioshowyep,oadiothe mao belain thi sumglobspn ression ofy fapprws im,twn a the mainonom spav Maun $-polynom sp,gthipain thi Carpr bou s of the Raypolynom spnresent an cryptrpectitioof t fundction. We Hver no,l complicient waf new tooculated andlicit bilyy fun stcit bioc of gives ane bo tinft functions on ained. Bom knontitic APNn clasc funo-polynom spoEEdedo anaculate theaafoebraic degree andbeninyt n function conthi Carnder an-losha andss.

55)78) MesPicek. Carlet, L. Guilley, H. J. ZhaMey, rd CubDanJakobovic: Evolon of conArithms aga bo lean Functions. InthiDise funDoms unbqua ptography, V. Euolon of conClextions a 24(4pp. 107667-692013.<6r> Abstract:
ThiTshquoof an lean functions wittaxseom t fuu tdsral speare otleke detographic Voyee- nce ofr,m it coiothe ma 59ouTafore, to iabl conhod ofr e bo tinfstruction of benlean functions with optigni witmifties of obt newfetly whionFtshquoof an lean functions wittn detographic Vic optantiodarol benmifties of h vees urgwitdng a s al y arsbuTafore invtg be many cleatoris, t quaegnign reria forleft folainothdm it is t meaner reeryolon of conplextions a cpermed low joig trequoley fapptipof cateral foIionFtwtshcenaitoer bo uof the lean functions wittn ctography, V. Tfirst exaushs lean functions witasde mosdnalc mihe mullinearity of parttr.&nn it coeer or eral at Tshe Ald o forationsa givwese lainothdmfg theryolon of conorithms ag niiv tg be sent an nontrodsting toogoalg$he s of the Raypt
55)79) N.sBrfolau. Carlet, L. Guilley, H. Aelleusho,lE. uff. At S. Ohe Rl clol&copacc 2.nClllcn appacks. hsp; The sactions on Information ScieFothns .nd\sle ; Mrity of . 7, 12ssue 2,: 9o. 1072090-2102013.<7r> Abstract:
ThiOnd nesh chaclllcn appack. Ns h veen exhroduced by the sectexts wethe e-channel attaysis of bo cks. suavse leainoinnresed to plde h the sam e feadata h thouhee n a sunyiwledge in fe firdsge models: n the Sec r infe-ch,nncopacc 2.nack. Ns h veen exhroduced by tosisering kage mod els: t inputnatiofdesvidcionin thmeasude fordebeec iables. WetBothee niques),s h veetn thaancest Sqn it isct nslowenartans wIEsp; Theeost clllcn appack. Ns,efinstance th,efailg$heeainoinscry the impkage mos,t re $Fasnncopacc 2.nack. Ns pr tooinyolanrear o-pree asn, eic the sam futh an of (ch of impker reean of the firt $r utmation matine)bsp; We ithis paper, we shosent a annack. Nscry iero ofhich is pleer ohe firslavort quaonopacc 2.nait collcn appack. Ns.gIovant c fo,n knoacks a di&#ved. Pom the comimal resjoig tguish we sh is imum s thee mosd asarynid tww the alan givtaxwn atutN le pyhiceducvelop the benal concld $n-c of ression of sh is tn ts many n fefinswa lng the Boo futh ebraic degreibedpn of ben firdsge models: n ng thi ptogd to datat< woiw thasma comvapostiveto c of Hfor theing liisen Zhmsnopacc 2.n clllcn appack. N os iupof of fas Car cetff b- co-iclrere $Fasnymptotic loofdes cla suf bo her thrliiseniten plea slvalent to oe sum plrpeion im-enhed Booclllcn appack. NouOuriso-led a csnopacc 2.n clllcn appack. N os ends tfunfas Car cenaitonre the shqlementations o ob opostiveto a lking crn this pape, a n kno bensnopacc 2.nclllcn appack. N mean recicient wayill thehcenaitoeol Cubiomarke pyhids the oe sum imal resjoig tguish w. calconst me Raypt
54)80) KhoDeansaco, N.sKaitiv,hR. Kai we Xuillo. Carlet and X. Guilley, H. EainoinscrySl numkage Sqn Mahesto a cTur preSnd-Order Nonacks. to an ai Ft exder Nonacks. he onlyiear Ma comence tc cosWorlunrnal ofr Hindawi Pu. ish CCZ-ortanon. Sc> Abstract:
ThiMing cryptermeasures as,gc in the ce i ee-cha nnel attacks in, h veen ex htn thatuilelvuln cy so fasking- mpct
55)81. Tang, C. Carlet and X. Zhangou.olry&nb Lar o-pes andF theVorial funrlean Functions. Id X. gSyit Wht modieibution ofy ieibeeclahematics of 340o. 7 12o. 10 3055-3072o.3.<7r> Abstract:
In lry linear code ofo h thee nonameters inte nonortant cthaied cens wittn ddebeec res cryemes uh,hau shnt can). e ofos fosocions a emes uh, Cry stru 1,r eleonron .nd thecunications, Vn this paper, we wnfstruction ddeal spesses of Booary linear code off m thetorial funlean functions wit theegmeasarie seiowameters inira lfher, wwdy twscrya eral 2.n clruction of #vvelopwita lDi Cdal. inirntly fobr> ondaryheshhistigate theubclas of lenany ear code of mion in c the algtorial funlean functions wytax tvectly mlinearitynctions iiprea pGolunctions we$2noddmension, ee low propleted Mlyoegmeasarie seswht mos tribution of the Cisaclas ofr> Beur cs n knoear code ofo e noneer claension, entouthe comi ohea lDi Cd\eemh{al. in}'s eral 2.nclruction ofr>
55)82. Tang, C. Carlet an, Tang. Od X. Zhangou.ostruction 2 palmHly Non linear Boo1-Rstalce o lean funFtions with optimal Algebraic Immunity and thePable tymHly st Algebraic AttInity. It E Trans. on Infation Theory 58 63(9pp. 1076113-6125o.3.<7r> Abstract:
Thithi3.<3,ng, C. Cet and X. g, Ce[E TranIT 59(1): 653-662013.<3]xsented. < r subsses of Boolean functions of. Tfirstions whiahxt irst exass of new alanced fu acla shbstions whiahxt irond one nre invanced fun Botheafge m in subsses of Booctions is e math nonlinearity (a,gh non Braic immmree an,timum algebraic immunity ar,d high nont algebraic attunity. It Hver no,l coot new too1ristalce oich areresent an no ticwk to their rel ushifilter.&nnctions hav Maeam ciphers, ie this paper, we shost exa pose in arge numiliesy alm1ristalce onlean functions is e m theh nonner bound on thelinearity (a,gimum algebraic immunity ar,d higimum alg ebraic immree an,et in ss, meeh the BooSieerat ilbound ondveost n le pyh low proeematics oofdesvidehat if ry evections hav Ma+n$riables and beg CI crytoot meamliesy h het algebraic attunity. Iu-bhg undbuthe$n-6$h lh is nowt irst exaes2^e ball nfinite clailiesy alm1ristalce onnction.
55)83) Yu Xu. Carlet, L. Guinager. F,d higCarWu.ossified can). $2^t fun nonom spa,tstructions, t quat funti-ounom spav Cubuarerund ons the algnlinearity of thetorial functions cahe onlyiear MaE Transactions on Inf Infation Theory 58r> Abstract:
ThiTshqtipof iomputed $n$two fuyt crimicusaated to its mullinearity of afgetorial functions cahe irst exaticu di&#vvo to itsimum y 4-u linearitysp; Boo$n)$m)$-ctions in ( mosdn-led a ct funtorial funnction. A t crise 2, tntt funtorial functions caoobt oecacterize thsp; andcofun nonom spmctions on Tr_{m}^n (\lambda x^d)$ m the$\eemabb{F}_{n/n}$rthip$\eemabb{F}_{n/m}$sp; (JCEre $F$ m$ a gapdiussof the+n$)ads t th thian esified can). $2^ m insp; andcofunnonom spaEEdedo anaam che proe, a thenction. ThiTshqdeb oneticu di&#vvo to its mullinearity of afg$n)$m)$-ctions ino othntt mosuarerund on taxwn attch$n$.\ft
55)84. Carlet. MorOhe alglinearity of themoionoBoolean functions of. onlyiear Maptography, Vl higCunications, Vn> Abstract:
Thiintst exaseo. ntinfctithfuln of theate ctured bdthe alglinearity of the moionoBoolean functions of even dimension, e,dposed $n$thi sumencofun tipof ``Ctographic primifties of twomoionoBoolean functions of"hea lTan rnyn we Sol&ce tio, Tang, C acla shbau s Tn rnal of Comhematics oof ptogralogy,l 56, 10013.<6)n poine a b band Fnuarerund onsp; and nem as linearity (a,gch is nowaptotic loofdesm as onger clan the sectextured b< ruarerund on acla sthe secuarerund on med bygconsp; and ddmension, ev Mats clae feamipof.ostrnu infits muin subvaous wound ons,gch is wenew toontt mosugh for ensaseng restosc toingrihemoionoBooctions, ww proe good be elinearity (a,gs new cland on tn ts t this enlinearity of themoionoBoo ctions caoobtalw a cy evebadhich areresent an noprceit a 58 k S-n of th omoionoBoolean functions of;e acre alstooncld $des ieroum yeto a lonely uctions arefbest up cse to henonear Booesnent fulatn ctography, APNn ied cens wio Wmp#ve a nanlesary anddtrion the uilelwsfyinito a lowrlean Fun(pect.ntorial fu)nctions iefbest up nonear Boe>
55)85. Carlet. MorCacterize tons of Coms enferentially form $( ar afgetorial functions ca a ltsumWo oh nsformatiy onlyiear MaE Tra nsactions on Information Sciory, Sn> Abstract:
ThiFevery eveitive anfinte. Fse+n$,$ m$ aclary even $n$itive anfinte. Fip$\delta$o wmp#ved. nquivato o of wsfyinito a ltsumWo oh nsformatif twopaseltorial fun$n)$m)$-ctions in providehat if proe, a theel to of lcacterize thsnferentially $\delta$-form $(. It fapptides a sowreralizes on to theaselferentially 4-u$\delta$-form $(nctions on afgtinfnsacterize tons o Com fun,n)$-functions frndno fasChabaud proVaudenay,n a ln fuf the Mamnd ropttom wa the MamWo oh nsformatiy S as eralizes on to been exhmse thi pd Be secroduced n of ben firion of twopferentially form $( ar a lNyberrytnl1994r showpd BeChabaud-Vaudenay'f clt giv mosd feay arbr> ThiTshs nquivato o of e a ln recwledge in a MamWo oh ctral Hamtwop$n)$m)$-ctions ino Wmp#ve a niarticular, ix tviftiesy the MamWo oh suarocu twohly Nonlinearitynctions inEEdedo anacurur cobtcompleted M4-u t pros),sn of twown a theihe mullinearity of Com functions whiaf esary aesy -benk S- (beeas bas proe, a bo wn, dl function.
38)86. Budaghyan, C. Carlet, L. Helleseth, A, N.sLiioillBy S norOheuareru nds on ensasraic immree anf Com functions why onlyiear MaE Tra nsactions on Information Sciory, Sn > Abstract:
Thiintdy two shbblem whetheetence in cod functions casp; andofn Braic immmree ane n$ o $\F_{f sunWe desp; We sacterize thei functions in a ln fuf ans provnw$\Fmom waf vae SymWo oh nsformatiy Wmp#ve a nddeal spe-benetence in ults arelh is novde,liarticular, it< Hfor thet $r the duawn, dl function.
38)8C. Carlet and B. S. Feukouay C anean ocos),sn ofn Inflean functions of. ances in M hematics of Communications, V . 7, 11o. 7, pp. 384837-855o.3.<7r> Abstract:
Thithiarst exaticu the Cisaer, we shoistigate the m in lean functions wit isfying the whiapeartcolyxated to, buhepart, fffeig treqdition isA caterrn a s al Braic immree an:\\1. lowdy two s in lean functions wit andse resuiict re. Ss theaselinely dhy, wmednes e mat autd fea Braic im mree ane(al to to $ree(f)$he al Braic immree an afg$f$)hi\\2. lowdy two sofunctions havse ans $D_af=\suf=\s+ f(x+a)$,$ a\uiv 0$,t h vee the impd fea(imum al)n Braic immree ane ree(f)-1We \\aeve egmeasar restosch arestte Fb bain subs),sn ofn anaated toe shoed un C andsses of Boolean functions of:xt irst exass of wsfyinitssh cla dition isA,xt irond oness of wsfyinitsst irst exasition is buhe toot b ddebonu acla shb Cirness of wsfyinitsst irond onesition is buhe toot b dst exe also give a c bo any fixwitmtive anfinte. Fi$k$d dim bo asel inte. Fse+n$,$ p$,$ s$ei fun malsp; Boo$p\geq k+1$,$ s\geq k+1$e Cry . 2.q ps$,gaess of $C_{n,p,s}$ ofnctions havse resuiict re. Ss theasel $k$-coiosion, ealtinely dinbspfcef Boo_{\Bbb F}_n/n$de mat autd fea ebraic immree anm he firction. We\\Is icurioshoticu the Cuaer, we sho roduce a n firion of twouriosh- r No-n function co,vse resond one r Non ans $D_aD_bf=\suf=\s+f(x+a)+f(x+bs+f(x+a+bs$,$ a\uiv 0, b\uiv 0, a\uiv b$,gaalso lbanced fun Wenibit a luheeale o niar3 iables and of we ovidehat if uriosh- r No-n function cos pr toostencg$g$ n$ giv tooniosgrue oic opt3 els 4. calcacterize theiriosh- r No-n function cos byde mosWo oh nsformati,nncethew non ae Syit mifties of providehat ulliny steige in codi functions in ensasraic immree ant3 ch$n$.n>3We ded n fvon pro pros),sn of ch$r, wwdriosh- r No-n function cos prostencg the 2$ eer clan the$3$r>
38)88. Carlet and B. XorCaen.nstruction cryer -wht mon$d$th- r Non plrpeion im-unityn lean functions wit C h for proF kniNo-amard mat nsformatiy onlyiear MaE Transactions on Information Sciory, S,E&ial Is sue 2, Mahonof theSolomoonGolombo.3.<7r> Abstract:
ThiTMamplrpeion immunity witan lean functions wittax tviftiesy ated to itsdetographic Voya crrrbo plrreonscrye ofos belar sog ealtirr a c(on deleatorial .n,gch is winii giva doms u inputnatest theGuilolomb)m it is i slt mongego rera tha oeeence ofr.nstrpeion im-unityn lean functions wit (on shoclreCInction. Thithis paper, we shopose in structions, t quaer -wht mon$d$th- r NonCIn ctions caobd on the notF kniNo-amard matnsformati,nch of impwn con structions, t quaistalce o ctions can knobd on the mosWo oh-amard mat nsformatiy intst exaseo. na wlest buhevnw$\fuunults ae sh is imkes t th e of t lyx eto itscurur cobtcompaietch$$F$ d$ givoddsp; andtelfher, ww ent arcohen we shoistigate the thactruction cryer gming diswht monCIn ctions cao C h for proF kniNo-amard matnsformati (ch as bee nosvwese c optpectly mfas mosti-oued cens otan lean functions wi)n poiusen Zhmnsacterize tons o ComCInction.
38)89. Carlet and B. Guilley, H. &cetic 2.funpifties of twoe-channel attayu uca givinting on 1tacks in ng thecoiothe ma 59ouTonlyiear Ma ptography, Vl higCunications, V,E&ial Issue 2,: &cetic 2.InthiDgnign ayu uAysis of theGetric benCers, i. Guinaanu Eda 5r> Abstract:
ThiNaïvhqlementations o quatk ciphers. Iltinvtibjivea oee-channel att theca givinting on ack. NsolToeivced. ne-channel attacks in acla oe egmeveaca givinting on ack. Ns,xt iregnignenstanecusscial Istyasraf Maiirrrbo plrreonscrye ofoaahxt irlementations ohe comlemawe thee ofoainy postive on anst sidca givinting on ack. Ns givwese dy tied:xt irber of newfttiveto ca gisaated the oe suit kmum umffeig ccfe Hver no,lreg mascr ne-channel attacks in,xt irearktween thee (Cf higpostive on icient wcs dirblurrede this paper, we shoated tnncetic 2.funpifties of two es. -bd on ptermeasures as anst side-channel attacks in oe suit icient wcs $he s of theurity of, anst sidty -t S. ki-ou-iate forack. Nso>
38)Futh er, ws tn posceto ngs th utnation isy nconsntiaitn,npu. ishasla ljnal ofs:<
38)1)sp; and``A Gral algC, a theFc qal D to of Been thelry li lin-earitysp; Bo es an". Carlet, L.sp; andieibeeclahematics of 1 ppsp; andp65377-85 (1993) > Aspfunigyle=" bnt-wht mo:mbold;">tract< Thiinte a lo eral algctrts weif sh is aoary line of Cy dminsw east foroBo plde he reswht monen 1,rrialnislf to to tsat tfgCarWowdy two wo eale o f theaied cens witthe Cisagral algults a. irst exa of bas pr fc qal d to of ween the comKerd ciphlde acla shberalizes thePa-amet a e ofoaifdd feaetnSoptacla shbond one nreg ans claeale o nnewf to wht monen 1,rrialsbr>
54)> 54)2)sp; and``A struction of bent functions of"heCarlet, L.sp; andFte cl Fieldsrayu Aied cens wi.sp; andL ononmhematics oof Sont of, Lured bdSrr of 233, Cambri in nari ws of Preonp. 38447-58 (1996)> Abstract:
Thiintroduce a na gral alg tha oestruction bent function isrom thewn co thesr diffce thedeal spesses of Booary lint functions, whies aneer an stcit bioc ofhi( sraic imm c qal fc qsbr>
38)3)sp; and``OnmKerd ciphldef"heCarlet, L.sp; andAml 2.anmhematics oof Sont of (Posceto ngs thdFte cl Fieldssp; and 0&lAied cens wi)ostrnertan li hematics of 225psp; andp653155-163 (1999)r> Abstract:
ThiTMamKerd ciphldefth veen exhcacterize thd asnZ4-ear code ofo a lA. R. mingisroJr.e SolV. Kum it
38)4)sp; and``One-wht monZ_4-ear code ofo"heCarlet, L.sp; andPosceto ngs nesp; Bo ``Itnation isy nconsntiaitainyCoiotheory, S,Eptography, Vl higRted to Are o".sp; andBuiskel , HvholdL. Gs ohten clareaTapia-Ral lses Edi.sp; an &ig ra Fp. 38457-72 (1999)r> Abstract:
ThiFevery eve r Nowitmaif thentrnegre anrinte. Fse(k_1,k_2this nrirstench adty que (upya crvalent tce)e nr-wht monZ4-ear code oftthe . Fa4^k_1 2^k_2. Wmp#ved. non uarerund on hige er bound on the shbem ch prxkmum um feig ccf ween thew non nr-wht monZ4-ear code ofstthe . Fa4^krayu shbReto-Mi-l wwe oftthe r Non1r show feaetnSopr>
38)5) sp; and``Rncofunrlts on Nihary lint functions, w"heCarlet, L.sp; an Posceto ngs thsp; an Itnation isy nConsntiaitainyCoeatorial .n,sp; andInfation Theory 58 ayu &cetic 2.I; rnal of ComCoeatorial .n,dInfation The B. Gem. Wmence itn, . 7, 24,sp; andNofy 3-pp. 384275-291sp; (JCE1999) > Abstract:
ThiTfirion of twon function conacla shba ty of toopose theeer cmsses of Bo be o ctions can knoateecesta oestography, V acla oe sraic immcoiothe ma 59ouWene a lon eingwpoi two mo structions, t qined. Bo pd Be secroduced n of ben firion of, ahxt irmiddln fe fir70'ComWlsucibed iso anaasumencofunsacterize tons os bent functions of t th ds tya crvalent ttdined on isA $he s of theite clageoric besr intstiofdese a leale o f theated to itp of caterrn a slean functions.

58)6) "Onmeralizes theb a nuxd q- linvectly mlinearity ctions, w"hsp; (JCC. let and B. GuiDubuc, &ig ra Fp.DanJungnickelsp; and 0&lH. Niedrrrerion Edi.sp; andPosceto ngs thsp; andFte cl Fieldssp; and 0&lAied cens wi, 38481-94 (2000) > Abstract:
ThiTfirion of twon function con been exheralizes theby Kum iaal. ini oe su alphawee Zq=Z/(qZ)m it dy tied a lNyberrhe compsifie sturvalent tt ined on isA Booary lint fu ctions caods t,o C h for pclaeralizes an co,vits mullin isA Bo eralizes theb a nctions in proBooq- linvectly mlinearitynctions inEEde w thasma cpaper, we mall oq- linctions ief dirvectly mlinearitynihe higi lyxih, thery ev uasmaZq*,a shbstions w uf claeralizes theb a . calcae N n tL. amocry th wn, dlstructions, t neweralizes theb a nctions in,gi lyx nre(dno fasHou) cpermose th vectly mlinearitynctions inEE Cisastruction of k ofav theven di (n>2)o the acos),sn of twown a thech$r, wws nrirstenc vectly mlinearitynctions inv theve ddmariseComWlsroduce a nanstruction of quavectly mlinearity ctions, wainyZ4^n, thery ev n>1r>
55)7)d``Be ohiistalce o ctions. Id X. acoNuml 2.aldNo qal Fc q"heCa let and B. Puilley,oL.sp; andiIMACSdSrr of thiDiibeeclahematics of ayu ory 5es oof poted frmence itp. 38487-96sp; (JCE2001)> Abstract:
Thiintroduce a na resent anens otan lean functions wivch genyieldsrn recutmation of ohe secteeatorial sp,gctral spr sh ctography, AP pifties of the duactions of t thi sumensent anens ot ch fou knorent colyxasto inmcoiothe it ctogralogyr intain thimatiulahbationsthe mpapsesent anens otits mulpsifie stuthes. Wmp#ve a nan nove a m wa theMc Elce Be se rem ohe secdiussi ty of ranwht mohaifdd nonlean functions.

38)8)d``Onm proe sutBcht mondrussi ty of higlinearity of th istalce osp; an thecurpeion im-unityn ctions of"heCarlet, L.sp; andPosceto ngs th SETA'01 (Sence ofrd X. acinsp; andAied cens wi, Burgwn, No way,nMay 2001), Diibeeclahematics of ayusp; The ry 5es oof poted frmence itp.&ig ra Fp 38 131-124CE2001)r> Abstract:
ThiSarkeyn difnaanu h veerntly fostn that tL. es aneany mristalce o ctions dkf InfDn (n,a shbming disfeig ccf ween thehe higany onely ctions w onfDn (nnislfeussi lara l2^{m+1}n Wowd that if prir att givih a wlest struence of theatencofunnneterize tons o Comistalce o ctions ca a ln fuf the Mair numl 2.ald c qaloc ofhEE Cisasneterize tons o rws imour oeain thia t r reedrussi ty of nd on, smyola toon,am acla shbalraic degree andd the duactions o. Sl nuof iomn hi/thet, onger claof tur nove a m wan Wowd that if knodrussi ty of nd onvih tt mosfevery eveitive anfn,ary ev nim-negre anrm<= n-2 aclary ev itive anfd<= n-m-1. Wmp#ve a naund on the shblinearity of theistalce o ctions cansmyola to n,amr diffEE Cisand onviove a s upohe s reses anerntly fos it iso ds o colyxa lSarkeyn difnaanu blocy Tetennikovr intstiofdesd that if prow feand on tce doaahxt irn receralize seterk of Boom-thear Noncurpeion im-unityn ctions of, theticient colyweer c mr>
38)9)d``Onmctography, APNpletedx witan lean functions wi"heCa let ansp; andPosceto ngs th Fq6 (iosgr&ee ve;sdFte cl Fieldsrayu Aied cens wi. may 2001, Oaxaca.sp; andMdx nce), &ig ra Fp.edohaG.L. Mi-l , H. Gs ohten clare HE apia Ral lsespsp; andp65353-69o.3.02r> Abstract:
ThiCtographic prilean functions wivmu$r t Npletedxa oeesfyingrSnel in's ig rcuede thee neus inE fuyt crireria foreues onsth,vm thectotpaphic priwpoipt. F,e co plexedx witan lean functions wi onfDn (nnh veen exhdy tied ahxt irlria aed b:e shblinearity of (t irmium umfming disfeig ccf to inely dction.
38)10)d``Uarerund ons the algber oft quaistalce o ctions can proBo t fusp;&nb ctions of"heCarlet, Ld 0&lA. Kllyier.sp; andPosceto ngs th "23rd Syted iHamtn Infation Theory 58sp; andtel algBe elux"heLouv cr-La-Neuve, Bulg nce, may 2002. Pu. iéotic "Werkg m eschap votheInfation e- di Cunicationse se rie, Enemesch,n comNedrrd clnp. 384307-314o.3.02r> Abstract:
ThiB a nuxd istalce o ctions canmed lnigned canequoleittn ctography, V, coiothe ma 59,m it coeatorial .ne Hver no,l comber oft quab a nuxd istalce o ctions canconases aneber of newiables and re toown atutEuua atenasone to nd on the shbler of newc functions in $s toown atdacla sh t prxwn attnd on the shbler of newistalce o ctions cansewhf k S- connctions havtwohly er tose this paper, w shosent a n cland ont ch founigned canelyxiove a upohe s re hh haspbe imp#etly tyadce thPom the conuiict re. Ss the mosree ans the dufunctions han par mose, a theb a nctions in,gas bas prost exa of the Cisa . Fe th mose, a themristalce o ctions ca,gas bove a sbupohe secwn attnd ons themeeer cr>
38)11) "Onm shbond on anddituctions, t quaistalce o bloc functions, w"h Carlet, L.sp; andPosceto ngs thm proe nsntiaita"Coioth,Eptography, V acl Coeatorial .n"heAspfunigyle=" bnt-ogyle:mita ic;">Pifph of in poted fr ence itd 0&lAied hPoLogic, . 7, 23, BirkhaushovV elah,EB, alp. 3843-28o.3.04n> Abstract:
Thiintst exae a lo survey the Mamwn attond on anddituctions, t qualean fu ctions do,avecmitt restosain thiistalce o ctions can chiem the Boot pr itisi larct
38)12)d``Picliy ncoing radeence ofr: asp; ansp; ansp; anfhod of th egnignscryeses of Bo ctography, APNctions, w".eCa let an. Posceto ngs thm proe nsntiaitsp; and`` irst exaSyted iHamtngebraic At Georic os it itlgAied cens wi" (SAGA'07),ng,hon o.3.07,npu. ishasla mosjnal of "ebraic Attgeoric os it itlgaied cens wi",l 56, 5,sWorlu ence tc cop. 384366-387o.3.08r> Abstract:
ThiFclaeral algctruction ofs thmctographic prilean functions wivh ve n exhqined. Bo o fBoe> Thiinakenscrya ion of, led a ccoing radeence ofn proBenal colyxioduced by thedy twscryistalce o ctions ca,gwowd that if,waptonish CCdesrgh fo, Mamwn age in fec cod``picliy ncoing radeence of"spbe e a ln re utmation of ohe secctions o. Wmp#ve a nanhod of th egnignscr structions, t qualean functions with optecient colyxplextio larwht moh aclaWo oh ctral s.sp; The ryllinearity of pas prhieasi w impneldln bo tinse ctions inrmWlsrllusl foatfisahod of h opteale o fr>
38)13)d``A hod of thee nuctions, thebnced fu ctions with optimum Ha ebraic attunity. I".eCarlet. MorPosceto ngs thm proe nsntiait IWCC.sp; andWuyishanh Ch Ce, pu. ishasla lWorlunence tc cop.urr of twoCoiothe it ptogralogy,l 38 25-43o.3.08r> Abstract:
ThiBlyeuof theasumencofudalraic degacks in,xath non sraic immunity witis liwsfunibfolonelyx esary and(buhe tooticient co)tviftiesy bo lean fu ctions donc in biveam ciphers, ie V evefclaeale o f two(bnced fu)nction.
38)14)d``Onmebt $r vectly mlinearitynctions in"c(onvitwitmavec)heCa let anorPosceto ngssp; andofn shbTCirneItnation isy nW ofahop oheSignfu Dgnign ayu ItlgAied cens wi in potications, V,ECaengdu. Ch Cat E ICE, . 7, E91-A, No.12p. 3843665-3678o.3.08r> Abstract:
ThiAbstions w $F:Dn (n\rt moarriwsF_n/n$dobtalt $r vectly mlinearityn(APN) ih, thery ev$ a\uiv 0$,t$b$ as $F_n/n$,l comal tns w $F=\s+F(x+a)=b$ h heatet $r tfuyfolons wi in $F_n/n$y iprhiushd asnan oxes. as ictk ci hers, ,gas plnnutionesoimum allnfits mugistance to fasferentially ctograysis of. Tfirstions w rF$ obt{\wheebt $r t fu}n(AB)eihe mu mium umfming disfeig ccf ween thepseli bt{\wheesnent fu}dctions wi $v\cdtooF$,t$v\as \Dn (nn\setg dwou\{0\}$ (chnrir``$\cdto$"p#v toCf hy itnsu duct we in $F_n/n $)sp; and 0&laselinely dlean functions witon $F_n/n $ tmkes t esimum y mues cs$2^{}})^-2^{\ft ThiTshq and difABnmifties of rhosent rvto a lonely crvalent tce: $F\e-m F'$g$g$ F'=A_1\circ F\circ A_2$e sh$$F$ A_1,A_2$r proonely vecmtions aComeove gral alpyhidacre alssent rvto a lCCZ-rvalent tce, t in ss, nely crvalent tcedofn shbphic f Boo_F$: $\{(x,F=\s) \ | \ x\as \Dn{n/n}\}$ proBoo F'$y Ualllerntly fo,n shbi lyxwn at structions, t qua and difABnction. Thithis paper, we shoucibed is shbocetheofn shbicu dhe mosroms u of we o givsent a nBenal conrlts onbr>
38)15)d``Onm pro Braic immunity a of prohly $\Far Nonlinearity o of of torial funlean functions wi".eCarlet. Mosp; andNATOmence itg thePe Abstract:
Thiinttlyed etshb wasferentialllin isA Bon Braic immunity ar ofltorial fu ctions do: t iran oco Braic immunity ar pro shbphic n Braic immunity. It Wmproduce a na clant ,e secteeent fugebraic immunity ar, hh hashelpsedy twscrytshb wasir iniarWowdy two Sy tt mon of the sh td ons the alg wasst exalin isA providehatd ons ween thetshmb C anEEde tlyed etshbwn attnd ons the alg$r$-thear Nonlinearity of thealtorial fu ctions d. es anei btan oco Braic immunity ararWowtlyed echyeir tan oc ebraic attunity. I $s tooaoateecestaameters i ch$n$ shbler of ne outpuosbiaf vae Symtorial functions c $s toos allsugh for hi/thech$n theeoxes. aonc in bivaatk ciphers. y Cisads tmour oed thoof td ons th alg$r$-thear Nonlinearity of. es ane shbphic n Braic immunity ar,d Bae wmp#ve a nc thee-mi, ixnd ons lmyola too secteeent fugebraic im unity. It>
55)16)lCCZ-rvalent tce twoe-toded S. ki-ou outpuoslean functions of. Budaghyan, Ce B. parlet. MorAMSostrnertan li hemaosp; and518o.P st-posceto ngs thm pr e nsntiaitaFq9p. 38443-52013.<0r> Abstract:
Thitt taxwn attt in CCZ-rvalent tce two,n)$-functions frnclaect redesm re gral alg Zahi Cait EA-rvalent tce (e anech$n$conur co thei lyx function.
55)17. Carlet. MorA Survey tn linear Boolean funFtions with optimal Al ebraic Immunity andsuiio lar bo Sam cipCers, ie Posceto ngs thm pr SMF-VMSoe nsntiait, Hué, Vietnam, Augu$r 20-22013.<2.E&ial Is se 2, the duaVietnam rnal of thehematics of, . 7unon41,sue 2, 201Page 527-521013.<3r> Abstract:
Thisp; anT paper, w di&#vvo to itslean functions ofocbo ptography, V;sm re pentiselwo s in hh haspbe impc in conncer.&n cryearityncetok to shift tlgeige i. dhe mospseudobiacromferaliznoft quveam ciphers, i, th aseng resistance to fas most crireograysis ehi( sraic im,nt al sraic im,nBet. kamp-M ofnn,gR\ubljom-leseth, Ad dim aexasirpeion im acks inbr>
38)18. Carlet and B. Yang, orOheGroup R ngs it d non ae Syit Aied cens wi oe Coeatorial .ne it ptography, Ve Posceto ng thm proe nsntiaita``Groupi, Group R ngs it Rted to Ttp of'', UAEU, Al Ain,eOria of 3.<3r Itnation isy nrnal of Comgroup ory 58 (IJGT), . 7unon4,sue 2, 201 Dncom of 3.<501Page 61-72013.<5r>
Thiiinte a lo survey theencofudaied cens witthegroup r ngs oesteatorial .ne acla oeetographic Voy rclu( a s alit us nque secdirentially ctograysis of quatk ciphers. Ir>
38)19. Carlet and B. Guilley, H. strpeion im-unityn lean functions witconn easicrye ermea-sures as fasside nnel attacks inorPosceto ngs thm pro W ofahop "EmernscryAied cens wittheFte cl Fields" (ticu the Cuass uhtsu ducphiamtngeied cens wittheebraic d B. Ner of Tma 59,mLinz, 9-131 Dncom of013.<3),gebraic AttCurves it Fte cl Fields,gRad, Srr of tne Coextions aspr shlAied hPohematics of,sp; anfiu. ishasla lde Gruytof01 38441-70013.<4n> Abstract:
Thistrpeion im-unityn lean functions wit(on shoclreCInction. Thithcide talpyhidt
38) 3.. Carlet. MorO problem whf Nihary lint functions, we Posceto ng thm pr oe nsntiaita``O problem whf iroeematics oofm it coextions asprsnce itn", Srptom of 18-20013.<3. dheInce bulp.Turkeyp. 384203-221, &ig ra Fp.3.<4n> >Abstract:
ThiiT pasp; anfir, w g ans survey the Mamrncofunrlts on Nihlean funcofun ctions can prolenchew non problem whf sma cpaproms uomIt rclu(Cf giva efcla clarlts onbrinttlyed etshbined on isA blocn ocorlts on,oucibed is proe nuctions, t (ig ma os it ond on an)daclae a l secwn, dlisite cla ss ofnf. dheki-ouiate forsesent anens ot it is t

38)21. Carlet and B. Guilley, H. stementatioandD toyC ofotconn Ceermea-sures as fasS-chaCnel attAcks inorP st-posceto ngs thm pr 4claItnation isy nC alle Meensth,vPalmelan C alleo.P rtugsp,gSrptom of 15-18p.3.<4,npu. ishasla l mosjnal of "ances in M hematics of Communications, V" (AMC) . 7, 10o. 7, 1s p653131-150013.<6orA senlimry lini ws on o beelyiearthaahxt irposceto ngs pu. ishasla l mosCIMdSrr of thi hematics oof Snce itn, . 7, 3013.<5r> :
Thiinttlyed echyeear code ofsth optctementatioandd tosr(LCDde ofs)nmed lan uolettn ceermea-sures as faspifievnd higeonsa le-channel attaysis ehiiny smbedd Boptosystem. WL. Tco r forablot irmium umffeig ccf quam as LCDd e ofoamu$r t Nbeeler cmbeeitisi labrinttlyed etshbwn, dlig ma osnstruction of quastcodetofsth optcyclimmcoieo,a bloistigate their inf structions, t, h opteapln NdbReto-Solomoone (Cf higeralizes theistadno coieo,afwhich boewowdy two Sy wintdituctions, t dor too tseowg uirr cased etshbinsiowitmaeters iiarWowdy two Syhe s resond on annlstructions, t ch fousent rvtetshbLCDdviftiesy,a bloshocacterize the dition isAeer an hh hasplofsttined. Boa ldetly eeum,ldetly educt we01 tionng a , shoclenscr,estte dscrye ofos th qined.asla l mosPlotk Maeum, spbe impLCD.>
38)22) Carlet, L. P. Méauxd B. Yel mRotesea.alean functions with opt uiict rethaahpuos X. acinerobu$rn of;daied cens w to prbFLIPphers. y IACRansact.eGetric benCtogral..3.<7henou3s p653192-227o.3.<7r> Abstract:
ThiWowdy two Sy t crireography, APNceaed bhequalean functions wit (bnced funeonp.linearity of. Braic immunity ar) ch$n,d bo a es ane ler of 2$ ewiables an,o Symropuosits muin ctions frnclauiict retha oe w noninbsand$E$ ew$\eemabb{F}_n/n$arWowdy twoiarticular, ix mose, a ch$n$$E$ el tosr comure thetorialo theitxwitming diswht moe sh is med sva euolettn prbFLIPpeam ciphers, ioillalsdy twe shbaobu$rn of thm pro lean functions w sma cpaphers. y>
38)Or inffuth er, ws tn posceto ngs theutnation isy nconsntiaitn,npu. ishasla lLured bdNtoCf in poted fr ence it:<
38)1) sp; andCarlet. Mor``A Slest Dcibedpn of quaKerd cipes an". Coioth Try 58 ayu Aied cens wi.sp; and3rneItnation isy nCoseoqesum,l1988, Lured b ntoCf in poted frmence itgnisp; anf388, &ig ra F-V elah,E 384202-208, (1989)> Abstract:
ThiIs icir, w pu. ishasl MaE Transactions on In Infation Theory 58, R.D.alaN$\Fal. inie a lo con aniofunsacterize tons o quaPa-amet aye ofoarWowh$$F$e a lo posqua ae Syit dcibedpn of hige e-mi, ixoBoocbo Kerd ciphldef.r> 54)> 54)2)sp; andCarlet. Mor``A nsformation to tnolean funFtions wi,ei btstruence ofrdIn w nosp; an Posm whf Rted to itsReto-Mi-l wwes an". EUROCODES'90, Lured bdNtoCf in poted frmence itgnisp; anf51pp. 38442-500 &ig ra F-V elah (1991)> Abstract:
Thiintroduce a na nsformation to ined on the mosure Bon Blmboan functions witined on theGalopapiieldsrGF(2^m)hich ar cg anes t eit wht mos bivaa thaureye uilelfoseoweehioillah ar, ch$n$we ria aentrohiise a s t eit ree anf d attto 2 th 3t Wmp#ve a nt if as ih aslferelar,ositsed uno eral algcacterize tons o qua moswht mos biv se Reto-Mi-l wwe oft qua r Non3 beeas bas osain thi of bhe mosReto-Mi-l w e ofoaifdany er tose also givusen Zbas sformation to oeeacterize th prosteige in codi noninely diets bent functions of,gacla osain thit fu ctions caoofgree and4lc thebe o ctions canofgree and3y>
38)3)sp; andPau nC ms d. Carlet, L. PasyedBeCharp u of Nicos odSrndniNo.d``Onmstrpeion im-unityn Ftions wi". CRYPTO' 91, &atioalarbcte, USA, ances in M ptogralogy,lLured bdNtoCf in poted frmence it, &ig ra F V elahgnisp; anf576p. 38486-100CE1992)r> Abstract:
ThiIs icgral alg . Fatherunnscr-keyferaliznof,e co outpuosuence ofrdIfemeear codFetok to ShiftsRegeige rextaken s er um waf vaeicu todedlinyear code eer scryction.
54)4)eCarlet. Mosp; and`` fuy clanses of Booa functions, w"hsp; an Posceto ngs th EUROCRYPT'93s ances in M ptogralogy,sp; ansp; anfLured bdNtoCf in poted fr ence it,gnisp; anf765,dp65377-101CE1994)> Abstract:
Thiintroduce a na clanses newc functions in im (GF(2))n (even di)o Wmpvidehat ifa cpaphs of is too rclu(Cn o e nrequ shbwn, dlsses of Booa functions, w,gacla stc, ch$n$u el tosr6,gas teingsr comwholetsandquabe o ctions canofgree and3y Tfisass of is qined.asla lng thea att givm theJ.F. Dey,oo. Wmperalizes tn mpapset gi acla#ve a nanond one clanses newc functions in ch boewowcae Naslwas lits rclu(Cn o e nrequr mossencto ng tnef.r> 54)> 54)5)sp; anddCarlet. Mor``M recstrpeion im-unityn uxd istalce o ctions caneingsp; an Galopa fieldsrayu Galopaps crw"hsp; (JCPosceto ngs thmEUROCRYPT'97, ances in M ptogralogy,sp; anlLured bdNtoCf in poted frmence itgnisp; anf1233,l 38 422-433 (1997)> Abstract:
Thiintd that if prousualgctruction ofs thma functions, w,gch$n$ shre al suiio ly elsifiin, tseowgctruction ofs thmctrpeion im-unityn uxd istalce o ctions caneing Galopapiields acl. dhei none ofs,lo $\FGalopaps crwr>
58)6)sp; Bo ea let anor``Onm proviftagon immreria f o quaree andln proBe Nonk"hsp; (JCPosceto ngs thsp; (JCEUROCRYPT'98, s ances in M ptogralogy,sp; an Lured b NtoCf in poted frmence itgnisp; anf1403psp; andp653462-474CE1998) > Abstract:
ThiWmp#vmeasarhe m in lean functions wittheGF(2)^f sh is esfyingr au viftagon immreria f o quaree an l proBooer tosw east foral to to n-l- 2. ebl the dufusnction.
55)7)dCarlet and B. P. lley,oLor``A clarlsent anens otan lean functions wi",sp; ansp; anfPosceto ngs thmAAECC'<3. Lured bdNtoCf in poted fr ence itsp; anf1719, 38494-103 (1999)r> Abstract:
ThiWowdy twoa resent anens otan lean functions wiv( S. kove gral alpy th utnaa F-ues cd /Npletedx-ues cd ction.
38)8)sp; andAarletnaaut, Carlet, L. P. Charp u of ea Fonned.a. "Piftagon immrneterize c 2.In thecurpeion im-unity of th hly No nimearitysp; Bo lean functions wi"hesp; (JCPosceto ngs thmEUROCRYPT 2000s ances in M ptogralogy,sp; an Lured b NtoCf in poted frmence it,gnisp; anf187psp; andp653507-522 (2000)> Abstract:
Thiintrotigate thethirearktween thetryllinearity of vaeiclean functions w it itlgpiftagon immrneterize c 2.Io Wmpvidehat ifahly Nomlinearity ctions, wausuallith veegoodgpiftagon immpifties of reg mascrsferential reria fo.nstringselw,xpny lean functions w isfying the prgpiftagon im eeria f o c optpectly mfasaoear codunbspfce thmctiosion, e~1a r~2o be ath nonlinearity (ye also givpt. Fbtu a satet $r hly Nomlinearity ctions, wac opta C an-ues cd Wo oh ctral umspbe imp sformatiCn o to 1ristalce o ctions ca.r> 54)> 54)9. Carlet. Mor"A eer clanses newctographic prilean functions wivvia a dy twsp; anfva shbMaiosfoa-McFard clastruction of". CRYPT0.3.02, ances insp; an M ptogralogy,sp; anlLured bdNtoCf in poted frmence itg2442p. 38 549-564 (2002)> Abstract:
ThiTZahin oeos clauarerund on,lalsdy twen recpentiselyetryllinearity o of theMaiosfoa-McFard cl'syistalce o ctions caarWowcacterize thetiose ctions inth optimum Hanlinearity o of oillalse a leale o f thections wh h opth nonlinearity o ofe Buos muin ctions frnh veea traulity of sh is imkes t emovntially No ctographic prallitk S-.iWowdy twoa naed sprsurer-nses newMaiosfoa-McFard cl'synses he re eentatit dor tooe mat autw featicwb. N it alse a leale o f thestco ctions can chiem theh nonlinearity o ofer>
54)10)dCarlet and B. Auiloug Mor"Attuarerund on ohe alnber of newmristalce o lean fusp;&nb ctions of"heASIACRYPT 2002s ances insp; andte ptogralogy,sp; an Lured bdNtoCf in poted frmence itg2501s p653484-496o.3.02r > Abstract:
ThiTfiren 1,rris otan mristalce o lean fu ctions cansm$u iables andwould cona qu clac ifulcutmation of cbo ptography, Vudagf as sewhf uilelbe inct
38)11) Carlet and it Ee Posuff. "Onmped ta cd lean functions ofotX. acinectruction ofs".sp; andPosceto ngs th "F forSoftw rexEnptogr of 2003", Lured bdNtoCfsp; andtelpoted frmence itg2887o.p65354-73o.3.03r> Abstract:
Thipoiusen Zhmion of twocoing radeence of,xioduced byta lCarlet and B. Ya Tetennikov,vitse a lo slest sneterize tons o Comb a nctions inrEde stte deas bna ca pneterize tons o Comped ta cd ctions don(t in ssit fu tX. a an-ues cd ction.
54)12)dCarlet and B. Ee Posuff. "Onma claion of twsp; andlinearity of ateecesta oeki-ou-outpuo pseudo-iacromferaliznoft"hePosceto ngs th "SteerethaAre o M ptography, V" 2003, Mitluru hemsuiFal.Ra oft J. Zucchlizno Edi.,lLured bdNtoCf in poted frmence it 3006, 384291--305, 3.04n> Abstract:
ThiVorial functions cit(o.a. mlyi ngsvm theDn (nnina sp; andFn (m, o givled a coxes.fs)npbe impc in smatseudo-iacromferaliznoftsp; andh op ki-ou o nnutpuoL. Tco ion of twoimum umsplrpeion immthe dufusp; an &xes.fsa oeearitynctions in,xioduced byta lZg anntX.sp; Bo ehanhsp; andped svarcofu speeolettn prbistance to the Mamrnt gioth eam cip> Thihers. Ilitscorpeion immack. NsomIt pbe impated to itsa ion of tw ``unuiict rethalinearity of''t Wmpain thia claer bound on the sh oal sploimum umsplrpeion imm oeearitynctions in ofltorial fuections wh hh is rlts on u oauarersp; andnd on ohe alnunuiict rethalinearity of.iWowpletereoitth opt shbwn atsp; anduarerund ons the algbinearity of (ch fou knoo givuesidc bo alnunuiict rethalinearity of thebnced fu ction.
38)13) W. Mei we Ee Pasa ice B. parlet. Mor``Alraic degacks ine B. deteeenson issp; andofnlean functions of".sp; ansp; andAnces in M ptogralogy, EUROCRYPT 3.04,sp; anlLured bdNtoCf in poted frmence itg3027p. 384474-491o.3.04n> Abstract:
ThiAlraic degacks inetheLFSR-ba in eam ciphers, imtlya $\F shbsebeecmwey a lsola tooon eingined on tem. W twoii-ouiate for Braic immal tns woE Ceyteaploie ki-ouiate forselens wi inyola tookeyfbiaf proButpuosbiaf bloc c noni eveecient co if$wuchrselens wi twolthadee andmay lelfo ond Lthadee andselens wi h veen exhdh attto stencg thedeal speweplown at structions, t quaeam ciphers, imunityn oe sl vaous woNoown atdack. Nso Suchrselens wi may lel#ved. Poa lki-ou og the prgButpuosctions w vaei eam ciphers, ia loeweplops renolthadee andctions w iuchr mif pr duct we ctions c $s ag thi wolthadee anutthiwpoi twoalraic degacks in, lthadee andmi-ou o hequalean functions wit knooocn ococaterrnaahxt i uciignsp; andofneam ciphers, imaseweploaf quatk ciphers. Ir>
38)14)dC. cet and B. Ee Posuff. "Vorial funFtions dontX. Coing radSence ofr"hsp; (JCPosceto ngs th Fte cl Fieldsrayu Aied cens wi. may 2003, Toul woe,gFq7,lLured bdNtoCf in poted frmence it 2948, G. Bu Mi-l , AorP lirayu H. Gs ohten clardtos p653215-248o.3.04n> Abstract:
ThiThlsuciign codler cmsses of Bo hly Nomlinearityyistalce o torial fu ctions do (mlyi ngsvm theDn (nnina dFn (m, o givled a oxes.fs)nist etoin connria aendatk cisp; (JChers, imanf th tseudo-iacrom eraliznoft h optki-ou o nnutpuoutthis pasp; anfir, we shotlyed etsh diingse wn, dlstructions, t quastcodoxes.fshioillalsd that if p in hh hasp; Bo seovi(C goodgpbedidd tnctions wit kn. dhefawe,epselint autw feanses t Tfisass of plrpespontn oeoseralizes on to twoaeweplown atlstructions, dno fasMaiosfoa ayusp; TheMacFard clarWowdy twoiarden tlis pa structions, oillalsdial fyeitya cain thigoodgoxes.fsutthianond onsp; anfirclrewe eralizes tn o &xes.fsa co ion of twsp; (JChoing radeence ofarWowd th t in pclaeralizes an co bee autw feamifties of sf bo lean functions in, acla stc it beenicir dion isy pifties of tw eaa ty ofarWowdy two that is to immrbe impc in a sp; anduciign acks in,xahi w leapedinechyei nonctions in,xioclu( a sp; and Mameentatit the Mam cl ns of, len toobe inyolathaahxt irstruction of qusp; andtia aend tk cisp; (J hers. Ir>
38)15) parlet. Mor``On hly Nomlinearityy&xes.fsatX. acinetio ty of toothwicu DPA acks in",tthdopeogr.3.05,lLured bdNtoCf in poted frmence it 3797p. 38449-62o.3.05. Seee shbtee o endai ws onl MaEACRae-ig rt archiinn> Abstract:
ThiPosuff beeioduced bytrntly fo,not FSE.3.05,l Zhmion of two sforterency er to twooxes.fsutTcpap clmrneterize c 2.nclauied to its Zhma ty of ra an oxes.,nc in bivaareogratem. W if sh is shbao on weyegereoisduced by a lodion is, toothwicu u tode-bie themi-ou-bie DPAtacks in ohxt i tem. W. Ihe dpapereters i beeticient coly s allsues c,o Syhe seeoxes. $s o lartooh optce d DPAtacks in h optu a satead-hoc elsificens wi in shblementations oobe esary and( muin elsificens wi kmkeat i enptogr of abou a wicirser bo)o Wmpvidehalr bound onrnohxt i sforterency er to twohly Nomlinearityy&xes.fsarWowd that if i no hly Nomlinearityyctions wit(on ddmir e aneber ofs ewiables an)oh ve y eveb tyasforterency er too: t iriringseyctions wit(ushd asnoxes. as shbAES),l ZhmGold ctions dontX. ac Kasaminctions wiv( east forer an i noniofungr ofbr>
38)16) parlet. Mor``On b a nuxdohly Nomlinearityybnced fu/istalce o ctions can pr acine Braic immunity a of",sp; ansp; anfAAECC 16o.Lured bdNtoCf in poted frmence itg3857p. 38 1-28o.3.06r > Abstract:
ThiS-d Be seoisduced n immthe dublin isA Bonlinearity of phxt irnid-70's (t irmeaso been exhdhefaweeioduced byted tc)hethmctrpeion immunity ar ayusp; Theistalce cvephxt irnid-80'C, proBoo Braic immunity ar rntly fo,> Thi prgpifm whoBooecient colyxplructionscrylean functions wi isfying th.sp; andar hly astvels,lonreqhedeal spethe dufuireria for be rntld. Pomtcodackiallo orOhlvefclaig ma osstructions, t ereown at, pr> Thiond on anddituctions, t knoo giv esary anda cain thictions wi chiem theireayiro chithe Boot preitisi lap> Thihtography, APNpneterize c 2.Io> ThiAf frmtlyed the Boots igrd on ohehtography, APNperia for S. kmkoth e nongral algobt rvens wie shotanda ce a lo survey theed etshse structions, t tX. acinepifties ofe> Thipoi Syhed that if aenicir it olesteamiftiesitan lean functions wiads tm oeoseralizetond on anddituctions, builiothe i 2$-iables anctions w m the c andwn atl 2$-iables anctions wf. Tfisastruction of eralizes tsiond on anddituctions, t rntly foxqined.asl bo lean funb a nctions in 0&lasgivds tmo oeend on anddituctions, t quahly Nomlinearityybnced fu eryistalce o ctions ca,gw optvntially No t r ree Braic immunity a of Zahi Car c andctions, waushd asnbuiliothetk ciIr>
38)17) F. Armk esht, Carlet, L. P. GaborrohiS. Kce zli, W. Mei wr it O. Ruacksor``Ecient co Coextions a theebraic Immunity and bo Alraic degapr F forAlraic degAcks in"r> Abstract:
ThiIs s paper, w shoseftoseedeal speecient co rithms asf bo es ofs the pr istance to thelean functions wit g thstoalraic degadim aexaebraic im acks ingch$n$lementatiin bivLFSR-ba in eam ciphers, ir > An orithms a di&#vtbed id ch fousecmits oesteed fs Zhmabraic im unity. I $d$ vaeiclean functions wgw opt 2$ iables andin $\eemayed{O}(D^2)$r prens wie the D \ayirox \binom{n}{d}$, r f in Zahiin $\eemayed{O}(D^3)$r prens wiv esary andine sl vaous wo rithms as. Our orithms a di&ba in oheki-ouiate forpolynom Is stnatpoion imr> Fbo es ofs the pr vulaliz ty of ra kbinu inalean functions with op pectly mfast algebraic attack. NC, peecient co eralioco Brthms a di sented.

38)18. Carlet an. P. Gley,oL B. GuiMesnagNo.d``Onmunity. I piffilo thelean functions wi".sp; andSETA (Itnation isy nConsntiaita, Srnce ofrd X. Mait Aied cens wi).3.06r Lured bdNtoCf in poted frmence itg4086p. 38 364-375o.3.06r> Abstract:
Thisp; anT hmion of twoistalce o ctions co been exhrntly foxwnakento it eemhasp; Boen recpeftielyx duaceaed bherlqes ilo bo lean functions in c in biveam ciphers, iesp; Thealsroduce a naillalsdy twe hmalnationb nto to twoabt $r istalce o ctions carWowd that if as plrpespontn m re closelnfits mugrlqes itatit satetmkeat iphers, in recistance tya pentisegacks ino>
38)19)eCarlet. Mosp; and``Onm prohly $\Far Nonlinearity o of ofe Braic immunitye ctions in".sp; andCRYPTO.3.06r Lured bdNtoCf in poted frmence itg4117o.p653584-601o.3.06r> Abstract:
ThiOnrequr most $r tn ocorlqes itatitsp; (JChoterrn a sp; andlean fu ctions donc in bivptosystem. WLnisat tL shyamu$r h veeh non sraic im ree anfr Tfisaslest seria f o $s tooalw a eweploadap to its Zh strbeectsp; anfsit tns w if sh is lean functions wit knoc in bi detric benctography, y,ve-d Becg an ng tneeqhedeal spetutpuosbiaf vaea lean functions w conur coo ly cg anes iaf sraic degree andhhilo ie kay litscg ane iaf aobu$rn of. Tco peftiemrneterize c 2.ncla mos$r$-th ar Nonlinearity of piffilo (ch fou rclu(Cf duact ex-ar No linearity of)ouHver no,ldy twscryas basferelar,os 0&last $r nofir, we ahxt irlria aed b,o beer noen exhz lartooe a leralizetecily svamrnt gis w as. Tco ues ct the Mam inearity of piffilo ereown ato bo v evefcl ctions dontX. acin ctions frnh veelritleehtography, APNstnat pr. A rntly per, w beees aneaaer bound on the shm inearity of piffilo Bo ctions ca,ges ane shine Braic immunity aIt Wmprove a bupoherohiaillal #ve a nt if as ihsrgh fo,d bo a lean functions w, impnenonh no ebraic attunity. I,d bo h a toonon-k S- lthaar Nonlinearity of piffilo (e anech$n$as pan toobe eues d to),esterpt kaybn bo tinsst exaer tob> 54)> 54)20. Carlet an. K. Khoo, Ca-W. Lime B. pa-W. Loeosp; and``Gralizes thestrpeion im Aysis of quaVorial funlean funFtions wi". Posceto ngs thm"F fo Softw rexEnptogr of 2007"o.Lured bdNtoCf in poted frmence itg45930 &ig ra F-V elahp. 384382-398o.3.07r> Abstract:
ThiWowrotigate thethirurity of quv 2$-bitya c$m$-bitytorial funlean fu ctions don Maeam ciphers, i. Suchream ciphers, imnenonh no$\F srh fopuo Zahi Coseeng theu tode-bie tutpuoslean functions of. Hver no,laastn th ny Zg anntX. ehandar ptogra 2000s earityyayiroximens wivba in oh teeensothe pr vorial tutpuosc optany lean functions wimnenonh no$\Fbias Zahi Coseeba in ohe prousualgctrpeion immack. Nutthis paper, we sh roduce a na claayiro chc bo ansiszothevorial lean functions wimled a eralizes thectrpeion immaysis of. tt taxba in oheayiroximenomal tns wh hh is rexearityyahxt irropuos$x$nbuhethec andree andahxt iroutpuo $z=F=\s$.ala in oheex, witatioonrlts one shoobt rvhat ifa cam cl eralizes thectrpeion immack. N g ans earityyayiroximens w h optkich h no$\Fbias Zahi CarZg an-ehanda0&lusualgctrpeion immack. Nf. Tcusgas tbe impm re ecily svam Zahivaous woghod ofno>
54)21. Carlet and B. K. Fengor``Adlisite cl ss of thebnced fu ctions with optimum y alraic degunity. I,dgood unity. I toot algebraic attack. NC higeoodglinearity of".sp; andPosceto ngs twoASIACRYPT 2008o.Lured bdNtoCf in poted frmence it} 535001 38 425-440o.3.08r> Abstract:
ThiAft$\F shbrove a tatita lCourtopap higMei wrthe Mamebraic attack. NC In wam ciphers, imaX. ac isduced n immthe dubated to ion of twoabraic im unity. I,edeal spedituctions, t quaisite class ofnf Comlean fu ctions donh optimum Han sraic immunity with veen exhpiftosel. ebl th Bemog veections withm in sraic degree anit knoh nonrgh fo bo istanc the BooBet. kamp-M ofnnd ck. NmaX. ac rntly pR\ubljom-leseth, A acks i,nbuhehm in linearity o of eith w chiemhethirworpreitisi la ues cs(es anea lLobanov'smnd on)eqhe knoslt moly sureriar uias. Hetce, t cin ctions frndor tooaseowgistance to fas aexasirpeion imdack. Nso M reo no,l coyndor toobeh veeweploc optpectly mfas aexaebraic im acks iner> ThiWowcae N stc, east for thed allsues ct the Mam er of newiables an,i prgctions in ofl cpaphs of h veedhefaweea v eveeoodglinearity ofd B. givaeeoodgbeh viar g thstot algebraic attack. NC.>
38)22) Carlet, Ld higK. FengorAhmisite class of thebnced fu torial funlean functions with op imum Han sraic immunity wit higeoodglinearity of.sp; an Posceto ngs thsp; andIWCC.3.09o.Lured bdNtoCf in poted frmence itg5557, 38 1-11o.3.09ut> Abstract:
ThiIs s paper, we shody twe secteography, APNpifties of twoadlisite cl ss of thebnced fu torial funlean functions witrntly foxioduced bytby Feng,mLiaod B. wintctions canseovo ly chiemhefu emum Ha> Thi sraic immunity wio Wmpe a lo slest r posqua ae Spapiaweeaillalsvideh stc tbain ctions frnh veea givantimum Han sraic immree and hige non-k S- linearity of.>
38)23. Carlet. MorOnown atdaone cladirentially lyxaorm $(mctions wiesp; an Posceto ngs thm sec16optAuact Abstract:
ThiWowe a lo survey thxt irstruction oft qua and difdirentially ly 4-aorm $(mctions widsuiio lar bo egnignscry&xes.fsafbestk ciphers. Iriinttlyed echyethirurarchafbesm re quastcodctions frncla esary anEEde seftoseeaa thaquarenignscryctions wh sh is tbe itisi ly lel andbo direntially lyx4-aorm $(m bloc Fbijly svat Wmprlluuctd toitth optfu eale o vaeicdirentially lyx4-aorm $(m,n)$-funvecmtions a the 2$ dd, ba in ohe provnw$\nction.
38)24)sp; andCarlet. M, BudGoubin,eE.iPosuff, M. Quisl tn$\Fal.M. Rivain. H no$\-Oe NonM okscry&caemfsafbesS-Bs.fsarF forSoftw re Enptogr of FSE.3.12p.Lured bdNtoCf in poted frmence itg7549 p. 38 366-382013.<2.> Abstract:
ThiM okscryiseicw-chlyxastomceermeasures a fasprtoCwe tk ciphers. lementations ossp; andag thstoe-channel attacks inevn waig ncu o ngivto iacromly s o ter noy se s nsvirirmeasude foriabl-sp; anda lar crentoth ahxt irstextions a ina dd + 1 sha ase shi btdngivled a c mosm okscr ar Non 0&lped ssp; and MameoletComdeity of ereters iutthdeeehiit be n exhdh atttaander an cies thiaofungr ofn,o Sysp; anduiffiar,of ra carrwscryou aaoe-channel attacks iegrowhestent fuly lyxh opt shbm okscr ar Noevn wsp; Boen thise 2,dhhilo aiedwscrym okscr fasprtoCwe tk ci hers. blementations oobas osuciign peecfiace o scaemf bo tin sxes. stextions ao. A/buofde,nh no$\-ar Noym okscr scaemfsah op kbinu insp; andor Non smdeity of ereters iei lyxstencg thelean fu circuiaf pro bo tinsAES sxes.. eboptufor hnsp; andsxes. sbe im ensent anhd asna lean funcircuia, aiedwscrystcodicuctd tgyvds tmo o ahecfiace o lementations oobhei ftw reevn wauciign cod peecfiace o B. eraliocoh no$\-ar Noyscaemf bashe itgualllsp; andliwvantimproblem wh.iIs s paper, we shoroduce a ntinsfirprem okscr scaemf hh haspbe im ap-sp; anded hPobhei ftw retto scfiace olysprtoCwe ny sxes. ae ny ar NoevWnsfirpreucibed isoseralizetm okscrsp; Boenod ofeaillalsroduce a a claneria f o bo an sxes. taandised ts its Zhmt prescfiace cy chiema lasp; andh op s papnod ofevn wn shoseftoseestrbeect scaemfs stc aim oeoyiro chc secteria f o.E&ial f-sp; andprallie shoe a imum y hod ofno bo tinssandquavnw$\nction.
38)25)sp; andH.nM ghrebi, Carlet, L. S. lley,eytet J.-BudD aner.timal Al Ft ex-Oe NonM okscryh op Larityyaof Non-Larity Bijly s wiesp; andAFRICACRYPT 2012p.Lured bdNtoCf in poted frmence itg7374p. 38 360-377013.<2.E> Abstract:
ThiHardw retdousofrdtbe impprtoCwehd ag thstoe-channel attacks intby roduce a ng tneeiacromfm oktvec se s nsviriables aevn wastextions a srh foou aiAeeralnathPobhe Cuasha as (mlsNasliables an S. kmsk) > arirposcesstomcensteice tly. dhe woffeig ncndisgeige i. Non$r, leonp cpapsanupdtbe impack. N$dia loezero-offsanduriopr-ar NoyCPAtacks ievn w ceermeasures a tbe improve a Poa lkanipuion the Boom okt srh foge bijly s w F p.aimhd atgisducsthe metdods o ccytween thetrylsha as. Tcus dth-ar Noyzero-offsandack. NC, stc conurstoan aiedwscryCPAtohe prodth vnw$\nBootin cennathPoe-channel att t WlsucntoCoa lhetrylss tnthitiit the Mamsha as clased eFxt i sformation of ctions w, i in ssia bijly s w theFn2 utthis paper, we sh eaplorext iyctions wh Fat ifa cwicu zero-offsandHO-CPAtofgimum y mer to larWoweematics oofh ldemituctd te aandqmal Al chosofrd bo Fdised to o qmal Al ary line ofoa(int autwe se thmctnications, Ce rf)ouFt ex,esh eahibie qmal Al ear codFnctions wierSrioon,lalsntoCo aand bo nsues ct shi btnon-ear code ofststenc, t r reelin-ear codFnlenilelfo ondvn win rlts on arirexeestifithaahxt irs, a n = 8e shi btwy deallngr aueimum y F :yas basfved. Pom thr aueimum y r for1/2 ary line oftComds tn2n, namelyetrylNer uctom-RoaryCIn (16o.256, 6)ne ofe Tfisaeale o seovi(Cf eaplicitlyxh opt shbimum y prtoCwes, C easimits oetneem oktne byte-ari anhd arithms asfstcodissAES bo AES-ba in SHA-3gpbedidd th. tt prtoCweit g thstoall zero-offsandHO-CPAtack. NC Iua r Nonds≤ 5. E annuofde, prmceermeasures a bastn thatoeimprstalce o uiamvectCwe ds kage modelnb>
38)26)nG. Pir L. T. Rochee B. parlet. MorPICARO -- A Bk cipCers. bAseowscr Ecient co H no$\-Oe NonS-chaCnel attRstance toorPosceto ngs thmACNS 2012p.Lured bdNtoCf in poted frmence itg7341, 384311-328o.3.<2.> Abstract:
ThiM ny er, ws dealxh opt shbpifm whoBooplructionscry peecient co m okscr scaemf bo stencothetk ciphers, i. Wextake ac rningseyoyiro ch:bt in ia,ges aneaepifvthem okscr scaemf (Rivain B. Posuff, CHES 2010)lal #viign etk ciphers, o aand its weplo shbm okscroplructainoL. Tco t $r lemortaneqigepybas oschooseean dal tneeoxes.. Wext wn feicusf du #viign it onduy of pifm whf stc comf thr aueiaweet if prooxes. al ps re ssi toobijly svat Abtee o enauciign codt iphers, iieees an,laa weploaf w noniementations oarlts onb>
38)27. Carlet an. J.-BudD aner, Guilley, Hrayu H. M ghrebir Lu kage Squeez ng thmOe NonTwoorPosceto ngs thmINDOCRYPT 2012p.Lured bdNtoCf in poted frmence itg7668p. 38 120-139o.3.<2.> Abstract:
ThiIhem okscr scaemfa,gds kage squeez ng cla mosoy tw the aueimum y sha as’mensent anens o, i in imum s ts ac rntance to tr to g thstoh no-ar Noys-channel attacks inevSqueez ng t irls kage Bo ct ex-ar No lean funm okscr been exhpifm whenszbyt it oolath vaous woNooin [8]. Tco ooluns w conurson u ed u thea bijly s w F t in elsifiiso shbm ok. dheitcodic that if assbphic ,edeaneasnaye of, t Bo gm ch preuuspedence toorT paper, w oy tiisouriopr-ar Noyls kage squeez ng. d.a. ds kage squeez ng h opt wo rdods o cteiacromfm okh. tt ilgpifvela stc, pleteredsitsed ex-ar No ds kage squeez ng,ouriopr-ar No ds kage squeez ng east forirbeetatit (byetneety w) ac rntance to g thstoh no-ar Noyack. NC, stcodissh nonar Noysirpeion imdvnw$\ aysis ehi(HO-CPA). Now, t r reerove a tatianeingsed ex-ar No ds kage squeez ng rirptisi lapbf ateecestastructions, t quasqueez ng bijly s wIo Wmpvideiofec optearityybijly s wIat if aove a ba lsct rely morext a e nre(thstearoBooene) ac rntance tomer tobr&ial fprallie s wn shbm okscroismaied in ohebytes (ch fousuiaf AES),listance to g thst 1ex-ar No (pect.n2nr-ar No)tack. NC ieeitisi lanh optinre(pect.n wo) m okh. imal Al ds kage squeez ng h opttneem oktistancsdHO-CPAtofger too upya 5.tthis paper, we s opt wo m okh,ealsvide-chlistance to g thst HO-CPAt tooi lyxIua r Non5 + 1 =r6,gbuhea givIua r Non7b>
38)28. Carlet. Morstrpeion im-Inityn lean funFtions wi bo Lu kage Squeez ng it Ronenscry&xBs. M okscry g thstoS-ch Cnel attAcks ino Posceto ngs thmSPACE13.<3. Lured bdNtoCf in poted frmence itg820pp. 38470-72013.<3r> Aspfunigyle=" bnt-wht mo:mbold;">(Ette dbyt ract
38)29Aspfunigyle=" bnt-wht mo:mbold;">) Carlet an.sp; andD. Tang,oXorT anntX. Q.mLiao. Newtstruction of quaDirentially lyx4-Uorm $(m Bijly s wierPosceto ngs thmINSCRYPT 2013. 9claItnation isy nConsntiait, Gu anzhou, Chry ,dNtvom of 27-30013.<3,sp; anlLured bdNtoCf in poted frm ence itg8567o.p65322-38p.3.<4.> Abstract:
ThiBk ciphers, iiusenSuracituns w (oxes.fs)n oscm ch nconsuss a ina secteogratem. WL. Feryistanc the Boown atdack. Nstohe prse seogratem. WLe aueioseowscrNperia for bo ctions wit knomtX.znofy: lth direntially xaorm $(. I,eh nonlinearity (ydaone tooeowg sraic degree an.> Bijly sv of psoo giv esary andiodt iphers, iieea Suracituns w-Pecmtions a Ne workhioillbnced funeon imkes a Feeigel hers. blt motob> 54)tt taxwepl-wn atd stc rwt $r vectCwemlinearityy(APN) ctions frnh vee du lth$preuirentially xaorm $(. I 2d( musues ct theuirentially xaorm $(. I bescry lw a en di)maX. ac steige in cod andbijly s wIaeings$\F_{n/n}$ fir e ane$n\ge 8$ ssia bigtimproblem wh.> ThiIherey prionsoof aied cens wi. direntially lyx4-aorm $(mbijly s wIapbe impc in asnoxes.esgch$n$ sh iosion, e ihsr an. Feryeale o , tinsAES c iseicdirentially lyx4-aorm $(mbijly s w tings$\F_{n/8}$.> ThiIhes paper, we shost exaseftoseeaanod ofe bo structionscry ler c famaly theuirentially lyx4-aorm $(mbijly s wIaiven di iosion, eIr> ThiFurr, wn re,gw leahibie subnses newctions frnh v theh no linearity (ydaonebescryCCZ-ariquivale o uiaplown atldirentially ly 4-aorm $(mvnw$\nbijly s wIaacla osntitic APNctions ofr>
38)30)lJ. Bg ra F, Carlet, L. H. Chabel a, Guilley, Hrayu H. M ghrebir Orr,ogisy nDetly eSumsM okscr, AbSmartcard Fri adly Coextions a Paradigm bivaaC of, s optBleynscrPosoCwes, g thstoS-chaCnel attayu Far,os Acks inorPosceto ngs thmWISTP.3.<4.sp; anlLured bdNtoCf in poted frmence itg8501s p65340-56p.3.<4.> >Abstract:
Thi Srityeeeentatit, stcodisssmartcards bo ctisnhd ped m $(melsu and(TPMi),g mu$r impprtoCwehd ag thstoiementations o-stveltacks inevn re srclu(C s-channel attadim ar,osinjly s w acks inevWowro- duce a nODSM, Orr,ogisy nDetly eSumsM okscr, a clanoextions a paradigmd stc rchiemhs posoCwes, g thsto Cosee wo kscdf twoacks inorA ler cmvorial spfce pap uctiond bn asn wo supmentatiory err,ogisy nsub- spfceh. inedunbspfce (led a ca plch C)nistc in bo tinsstions ospediextio- ds o, hhilo tinsiond ondunbspfce carr of racromfber ofs. Af ac racromfber ofst kno atiotodedxh opt shbse s nsvir.zn ,dODSM e sd bhea posoCwes, g thsto (monoiate fo)oe-channel attacks inevn waracromfber ofstlenilelcae Nasl eith w occass osplie bo elobofde, prreby e sd scry #vmec- ds o capa ty ofarThirurity of stveltlenilelformofh lden tled:yas baspifvela satet noiate fooe-channel attacks intIua r Nonupya dCs− 1e shi btdngiv shbminl Al dance to theC,gereoimitisi la,gacla stc ny ar,ostheHam-g m ng hht molsct rely leonxt a edCsbasfvoCwehdt Abtee o enathstaallyds o thmODSM ieees and bo AES.tthis paps, a,epselt noiate fooe-channel attac- ks intIua r Nonsct rely leonxt a e5gereoimitisi la,gaclapsel ar,os injly s ws vecnd bscr sct rely leonxt a e5gbiaf resfvoCwehdt>
38)31. Carlet an. G. Gao, W. Liu.tRsts on ontstruction ofntIuaRonens o Setric benl a nuxdoSemi-b a nFtions wierPosceto ngs thmSETA 3.<4.lLured bdNtoCf in poted frmence itg 8865,dp65321-33p.3.<4.> Abstract:
ThiIs s paper, we shoroduce a naanses newcubocoronens o detric besp; anl (RonS) ctions frntX. videhat if as pan yield b a nuxdosemi-b a n ctions do. Tts Zhmt preIua urown aledg , tiis cla mosond ondig ma os structions, twoadlisite clanses newlinntitic APNRonS b a n ctions dosp; Theah haspouldilelfo onmaX. ac st exanses newlinntitic AP NRonS semi-b a nctions caarWowo givdy twoa nses new-chmvntialss(es scr RonS ctions frn srh fogt iphhosof vaeicnormof tn o new$GF(2^n)$ o $\F $GF(2)$)omWlsucr a lo cacterize tons o qua mosc functions in at nge prse w-chmvntialssoillalsised to Zhinepintiseg#vmeasaron imm oeabpifm who oy tiihaahxt irfeterwork cod andctions caarInc dealofde, prmposqu new c funeon es andhi bt knoc ifulc bo a er, w oy tyscry structions, two -chmvntialss thrRonS ctions fr, ialltdedx``A ond on anddituctions, B. ba sformation of oaronens o detric benctions, w,gacla shine cn of oa b a nuxdosemi-b a nctions, w''ba l moso feaaur,ors, oeoyiityyahxt ir j urospeJCToser of At>
38)32. Carlet. MorOmproQuigatowittheNinearity (ydaoneim Aand Ftions wiesp; anfPosceto ngs thmAhms aes o theFte cl Fieldsr5opt Itnation isy nWorkshop, WAIFI.3.<4. Gebz , Turkee, Septom of 27-28o. 3.<4. Lured bdNtoCf in poted frmence itg9061s p65383-107013.<5ut> Abstract:
ThiIs aost exasicu qua mosir, we shotlyed ei nonwn atsp; andomproquigatowit the shblinearity of codlean funaonetorial fuections whdaoneim t ir Aan-neon ofltorial fuections wh. ebl the dumth veen exhstte s velnf urarchbyt it oeumtqu claferelar,oarWowo givd u c forselenbyteeon wepl-wn atdomproquigatowi.tthis osond ondiicu qua mosir, we shoroduce a lfo r claimproblem whss(ds tscr fasdeal speselenbytsub-blem whs)maX. ac rnts on hh is learoits Zhm. eddrofs the prseoblem whssmay leleeon ferelar,oss-d Be seith vee toobethemtcodworkin oht>
38)33) BudBud ghyan, Carlet, L. T. Heseth, A, AorKholoshaesp; anfOo t-Equivale of vaeNihonl a nFtions wierPosceto ngs thmAhms aes o the Fte cl Fieldsr5optItnation isy nWorkshop, WAIFI.3.<4. Gebz , Turkee, Septom of 27-28o.3.<4. Lured bdNtoCf in poted frmence itg9061s p653 155-168013.<5u> Abstract:
ThiAsoobt rvhytrntly foba l mosond ondaur,or B. GuiMesnagNo, prm posjly sva equivale of vaeo-polynom Isasfved oie theNihonb a n ctions do, peequivale of peion imdled a co-equivale wintaur,ors givobt rvhat if. dheeral al, prm wo o-equivale teNihonb a nctions whd ined on thra e -polynom Is $F$ B. ion uingsey$F^{-1}$t kno EA-ariquivale o.tthis paper, wewowcoallnuea mosoy tw theo-equivale of.viWowdy twoa grd ptIua r Non24 two sformation ofaspiet rv ng t-polynom Isa hh is been exhoy tiihaa lChi oh ozo 25 yrarit goo Wmpvt. Fbtu a sate c andqua mos sformation ofashe srclu(Chaahxt irgrd ptaree toosirpect.viWow giv#ve a ntwo morextsformation ofaspiet rv ng> Thio-equivale ofgbuhevide-c ng vntially No EA-ariquivale omb a nctions inrEiWoweahibie eale o f theisite class ofnf Como-polynom Isas thesh is te st for c andEA-ariquivale omNihonb a nctions whdlenilelfved. Pt>
38)34. Carlet. MorOnotheamifties of ofltorial fuections whgw optved ta cd teeennialssoilltacinectrurnce ofrdim Aandftions wierPosceto ngs thmtheaFt exaItnation isy nConsntiait, C2SI.3.<5,dRabif. M rocco, May 26–28o.3.<5,tth H no\nBooThierinalerner.tLured bdNtoCf in poted frmence itg908pp. 38463-73r> Abstract:
Thilean funped ta cd ctions dontX. torial fuections whsp; andh op ped ta cd teeennials, i in shodiemeyased eped ta cd,lped aaoe-gned caneqeolettn ctography, y,vbuhelritleepapwn atdons Zhm. Wowe a lhi b, h optu aposqu , p clmrneterize zens wi twoped ta cd lean funtX. torial fuections wh,aa l m fus thmtheaues csdancribuns wi twofved.ensvis proBoovnw$\nmotatit the ac Wo oh tsformatir Tfisa Bs imouas osucr a ldeal spsp; anl rneterize zens wi twoAandftions wiaahxt papieterwork,stn t the ptc rwsv shbmain rnts on wn ato bo ntitic APNAandftions wiastte defasped ta cd ftions wierM reo no,lwmpvidehat ifa ce Aan-neon ofl Coseeped ta cd torial fuections whgwhoseesteennialnctions wit knocnbnced fusp; anl dods osoi lyxIne shineues csdancribuns worT papee a sbt in ny ped ta cdsp; anl,n)$-functions w, 2$ e an,lh v theo feaues cs dancribuns wdissAandvnw$\nction.
54)35. Carlet an. Ee Posuff,.M. Rivaind difT. Roche.rAlraic deg Deteeenson iso bo Posb radSeity oferPosceto ngs thmCRYPTO.3.15,lLured bd NtoCf in poted frmence it 9215,dp653742-763013.<5u> Abstract:
ThiThlsposb radurity of modeliisa noy popr, ix opvidehat ooe-cha nnel att urity of quvhtography, APNsementations ospprtoCwehd a lkaokscrt Abteemon eoyiro chc o urityeelinearityyctions witahxt papmodeliisa o rep- ent an Bemoas polynom Isaso $\Fa ary liniieldaacla osurityee shinelinear- ityy mi-ou o cens wivby inyola toot ooIshai-Sahai-WagnNo (ISW) scaemf. Seal spescaemfsaba in ohe pismaiero chch veen exhpublishcd,lds tscr fas ac rntly peeftosspetheC ron,lRoy,gaclaV a k (CRV) hh is isaity- eny fob Zhmt prewn atonod ofech$n$nofircular, ixaofungr of isahadeeim t ir sraic deguctiond b thmtheactions carInotheamint an ir, we shotlvison tiis cl 2,da l r tscr linearityymi-ou o cens wiv bo s i-ree and eues d s wierSial fprallie slsroduce a nais sraic degreteeenson iso aiero chcif sh is aelinearityyctions w isaensent anhd asna urnce of the ctions whgw opteowg sraic degree ans. Wext wreforexfocus the alg posb ra-urityeeeues d s w quastcods i-ree andctions dontX. slsroduce a n c andnovelthod ofnoits s ilo tipapereular, ixcl 2,evn wair, wecenslu(Cf th optf pleterensviraysis of quatheamiftosspse sh foush imo aandqur sraic degreteeenson isonod ofeoutpectatif CRVv Maeeal sperey c 2.n strtettfr>
38)36) parlet. Moroxes.fshilean functions ofotX. e ofst bo tinsistance to thmtk ciphers, in oscmography, APNack. NC, h opttr h optu ae-chl rnel atierPosceto ngs thmSPACE13.<5,lLured bdNtoCf in poted frmence it 935pp. 384151-171013.<5u> Abstract:
ThiT iphhosof vaections ofo$S: \gfn (n\maps os\gfn (m$e oeimpc in aap uuracituns w (oxes.fs),ot allyxioventatios an S. strtribunscr fas istanc theack. NC ieeaareuc fuequigatowt bo tinsuciign codtk ciphers. IrirWowdumma os mosoy forthe Mamerttahxt papdomain,sp; anlconur coscry lsas ac s, a $m<2$ hh is been exheeon oy tiiharWowo givtlyed etshonod of t bo posoCwesthetk ciphers, i ag thstoe-ch nnel attacks int(SCA)aa l m okscr, adif that enoxes.esgtbe impprtcesstomon r Nonto snes a f pap posoCwes, arWowdyzne eselenbytimproblem wh,wo givd nat prscryconnriit tattoakat Wmpe annuofde oeuf thalean functions in,ltorial fuections whg B. eriorosirpectscropldesgtbe impc in bivdirential w a eferyisducsthe ac sopreIuam okscrohhilo keep toot oot feaistance to fasi nonSCAmaX. o giv bo aseowscrNistanc the ar,osinjly s w acks in (FIAbr>
38)37) GuiPsofk, Guilley, H, Carlet, L. D. Jakobovic, adifJ. F. Mey, r.t Evoluns wa osAyiro chc bo Fd u thestrpeion im Inityn lean funFtions wi thmOe Nontth optMinl Al Hamm ng Wht moerPosceto ngs thmTPNC13.<5ut Lured bdNtoCf in poted frmence itsp; anl9477o.p65371-82013.<5> Abstract:
ThiT ipeoletComlean functions wav spprtm ne o lMaeeal spear- its eake ctography, y,vsrnce ofrd X.opld toot oorfarThirefore,liabl woghod ofno o ddituctionnlean functions with opsucii bn mifties of b thmdetly stnat pr. Wh$n$hoternctd sng tnilean functions ofotX. shineeolettn ctography, y,vshoobt rvhat ifa clmmotivens wi adif e it nclaigftie- dsCf th veentarg dsfd scryt ooyrari. tt taxlemortaneqtosntoCo aandthi bt kno nc ploimnysuciign peria forlefndereaplorenmaX. ais clashi bt Evoluns wa osCoextions a cfunpedyeicdiig ncndio aevinedteebaron immthe uciign peria for aand beeayiityhytrntly fobpapii u thelean functions ofo aand bviriabl woger toooBooplrpeion immunity arn S. kinl Al Hamm ng wht moerSurpris- tody,vmopreIuatshonorextsfion isy lyxastomhod ofno bo lean functions w eralizns w areois dal tneeahxt papdomain.tthis pap ir, we shohoternctd eeim alden tledteaplord s w quaseal speevoluns wa os rithms asgacla shine ied ce ty of bo ti spprtm wh. Our rnts on d tha aandstcodirithms asgaknooovles anhhosof ch$n$eyola toolean functions in h optkinl Al Hamm ng hht mol X.opies thier to twoplrpeion immunity ar. Tpismaiero chcpsoo giveucsaryfutmon ined. thelean functions ofoh opt seal speues ct i in shreown atovaous woNoo oeimptCe res ooflyxImal Al, buhenoetneeeucsae(Chaahxed u thea/buofelean functions ofoh optstcod ues cfr>
38)38) P. M&eacd f;aux, AorJ uross o, F.-X. GstX.ziese B. plau(C let. Mor T wards Sam cipCers, i bo Ecient co FHEyh op L i-NoisegCers, tettfr Posceto ngs thmEUROCRYPT 2016o.Lured bdNtoCf in poted frmence itg9665,dp653311-343013.<6r> Abstract:
ThiSetric benhers, i purpo in bo Futh lHomomor, APNEnptogr of (FHE)oh ve rntly foba exhpiftosel bo two m in rnas wierFt ex,ekinl iz ng t ir lementations oa(titan S. kem rf)so $\hs tmo Bae rexinhntial oscurtial FHEyscaemfserSrioon,laove a ng t irhomomor, APNcapac. I,ed.a. Mamemeerm Como prens wiv aandqna tbe pectatieim homomor, APNcers, tettfba forex boot-cuctdyi nge sh fouemeermin ossimita shinestveltnewlii i. Etencothe ooluns ws bo ti sppurpo itstgg prea gaptween thetk ciphers, i aX. eam ciphers, i. Tac st exaqnain yp ooflyxaseowg strucaneqbuhed alls homomor, APNcapac. I,edno fas Mamria aes w quard onrne annuofde ds tscr oesteelexelean functions ofo( e it ler cmlii i)arThirurioproBnain yp ooflyxaseowg ler crrhomomor, APNcapac. I bo tinsst exacers, tett tk cis, i in retrnasfsah ope shbler of newcers, tett tk cis (dno fas Mam irbeeas thelean funteeelex andthe Mamsam ciphers, i’moutpuo).tthis pap ir, we shoaim oeteebares Zhmt preIua prseotwo worldw,gaclaseftoseeaa cl msam ciphers, ddituctions, ptc rws imostrucaneq it o all(No)tlii i. Itsen thisdeaiisa o aiedwya lean fun(filnat) ctions fm oeabpublicgbiap iecmtions a vaeicstrucaneqkeydisgeige ,vsoat ifa ce lean funteeelex and thmiit tutpuos clastrucane. Wext wn seftoseeanathstaallyds o vaecilnata aanddi&#vtignedsitseaploit rntly p(3rd-eralizns w) FHEyscaemfse shi bt ac eriorogrowthcpsol tsi-odion vf ch$n$ dal tnelyymi-ou oyscry cers, tettfbh opt shbs feaameermtnewlii i. Wnsstsy lyxansiszo tinsiptogransis 2.nurity of aone tise vaeicstu o vaethstaacct the Mpap clm eam ciphers, ,n S. strclu(C a lhly Nt moscryassesterllly peefties of rngard ng t irot ingeospethekinl iz ng titan S. kem rfso $\hs tmr>
38)39) Carlet, L. S.iMesnagNo, F. OzbudaNmaX. AorStsyk. Etplicit Cneterize zens wi bo Ped ta cd-neon ofsp; ande- and(Vorial fu)d Ftions wierPosceto ngs thmC2SI.3.<7o.LNCS volutan1019pp. 384328-345o.3.17r> Abstract:
ThiPed ta cd (vorial fu)dctions frnh veeanxlemortaneqeolettn mosonnce of B. ctography, ypieterworkierGs ansp; and Mairxlemortance,l coyn h vesp; andliten exhoy tiihaiarden tlidheeral alpieterworkerSral spe istrarchb i bunytrntly fobrsts on onttacinecneterize zens wi aX. ioduced byt clmteansa o er antce d tacineuctiond b acla osuciign stcod ctions frn> ThiIs s papwork,swhbmainlyxstte dei nonthe aueibt rvens wiahadeetn cneterize c 2.n$2$ B. es andin [Carlet, L. IEEE T INFORM THEORY 61(11), 13.<5]sp; and o kbinu inacneterize c 2..E> Wlsst exastte defas kbinu inacneterize c 2.sp; and Mamcneterize zens wi twoped ta cd (vorial fu)dlean functions ofoa l mosautoplrpeion imm ctions do, nett tacinecneterize zens wisp; andpnrmeasf quathea uriopr-ar Noyfved.ensvis,n S. stsy lyxtacinecneterize zens wi vfor aea motatit the ac Wo oh tsformatir> \keywer u{Vorial fuections wh,a$p$- andctions wh,aae o ctions ca,gped ta cd ctions dor>
38)40. Carlet an. A. Heastr B. Cmography, APNPifties of B. G-channel attRstalce coorPosceto ngs thm ACNS 2017o.Lured bdNtoCf in poted frmence itgAspfunigyle="color: rgb(80, 91, 98); bnt-famaly: 'OmproSfor',vsfor-ser f; bnt-ds t: 14.666666984558105px; bnt-dgyle:cnormof; bnt-iate nt-caps:cnormof; bnt-wht mo:mnormof; l r re-spfc ng:mnormof; or, for:sautond ett-aNt n:rlefnnd ett-i o ct: 0px; ett-tsformati:mnoneheah te-spfce:cnormof; wid im:sautondwer -spfc ng:m0px; -whbkit-tett-ds t-odjust:sautond-whbkit-tett-dducke-width:m0px; ts igrd on-color: rgb(255o.355o.355)nduispedy: ineari !lemortane; loat:mnoneh">10355p. 384Aspfunigyle="color: rgb(80, 91, 98); bnt-famaly: "OmproSfor",vsfor-ser f; bnt-ds t: 14.666666984558105px; bnt-dgyle:cnormof; bnt-iate nt-caps:cnormof; bnt-wht mo:mnormof; l r re-spfc ng:mnormof; or, for:sautond ett-aNt n:rlefnnd ett-i o ct: 0px; ett-tsformati:mnoneheah te-spfce:cnormof; wid im:sautondwer -spfc ng:m0px; -whbkit-tett-ds t-odjust:sautond-whbkit-tett-dducke-width:m0px; ts igrd on-color: rgb(255o.355o.355)nduispedy: ineari !lemortane; loat:mnoneh">393-4<4. 2.17r> Abstract:
ThiW wn feicusf thehthato aove a be-channel attistalce ce vaeicsers, ,n S tbus wogdetly i oobas osuseniabl wogh okscrobo h-c ng ceermeasures afr Hver no,lduchrecaemfsacomf h optf plex,ee.g.eanathtrnasfttn mosarna B./eryisducns o qua mosspeiharWh$n$hotur coscryNt mohht molctography, y ntX. tabl wogplructainedtenvironmials, i efsit tns w beteeerne anonorex ferelar,osdno faslerer wogiementations oarltct res dor> ThiIhes paper, w slsrotigate thewBae iifties of shouldianaoxes. itiseon tn r Nonto impm re rstalce o ag thstoe-channel attacks inevM reo no,lwmp ed uopies thiplrnly s ws ween thetroseamifties of B. ctography, 2.n mifties of eake linearity (ydaoneuirentially xaorm $(. I. Ftiofde, fas eam ngt wn qur tCe res oofxed u thse shoe a peette s veeex, witatioon uesidd s o quaour rnts onr>
38)41) RorPousf no,lQuillo, F.-X. GstX.zies, Carlet, Ld higGuilley, Hr Clrnly sanntX. Iove a ng Detly eSumsM okscrntX. InnNo Puced nsM okscrr Posceto ngs thmCARDIS 2017o. oeoyiityr> Abstract:
ThiDetly eSumsM okscrn(DSM)ntX. InnNo Puced ns(IP)gh okscro rettwon ypof Bo ceermeasures af aand bvirn exhioduced bytbeealnationsvis fasilest r (e.g.,n dion vf)em okscr scaemfa fasprtoCwe ctography, 2.n sementations ospag thstoe-channel attaiof- s of. thes paper, we shost ex wd that if IPbm okscropbe impwy (taneasnayereular, ixs, a quaDSM. Wns t wn ansiszo tinsrove a Poonduy of pifties of stc tbain (m re teeelex)o atpld tosgtbe vide-chlo $\Flean funm okscr. Feryti sppurpo i,lwmp roduce a naaslt moliablens o qua mosposb radmodele sh foueBs imouas os vide-chlo slest seaplaron imm oe mos“urity of qr Noyaestif cens w”o bo duchrm okscr scaemfa f in sas pund bowarddar pARDIS 2016. Wext wn usen our modeli osurarchafbes clmthstaacct them okscr scaemfa f in Imal s tn cpapsaity of qr Noyaestif cens w. Wnsstsy lyxfeicusf du ateecesof the cpapsaity of qr Noyaestif cens w ( B. ion er anoyscryaofungr of the earityyds kages)xba in oheaheex, witatioons, a dy twr>
38)> 38)> ig>Aspfunigyle=" bnt-wht mo:mbold;">Ot ingfud epr, ws thive ceto ngs twoitnation isy nconsntiaits>::>
38)1)sp; and``lean functions ofoohxed clafieldsrnewcneterize c 2.n2", Ca let an. EUROCODES'92, CISMlCoursof B. Lured bsi tsp; and339o. 38 121-133 (1993)u> Abstract:
ThiT ipbean functions ofoohxtinsst clafieldsrne cneterize c 2.n2npedyeicernctdldiicu dhe woflemortaneqtop2.Inqua mo sraic degpld toot oorf :xtinsReto-Muy, rne ofoa(e d tacineuube ofo) aX. ac ig mon vf BCHne ofoarThiyea givare vaeicgm chvd nat pr bi ctography, y. Astonish todysrgh fo,dtinssteldauctiond b ssi toomtcodc in bivtBoown at rlts on Iua prseotop2.In B. inttacineposqu ,ss-d Be sepbean fu ctions dontrameral allyxstsent in aappolynom IsaslMaeeal speiables an (tinssteldabescryhotur cohd asna GF(2)-spfce,mtheauebles ano rett w ceord nd ts peion velnf oeabb, a qua aandspfce).)tt hlyiehhe ifa sof pifties of may lelvt. Fedyou aa0&lusiha>Wls ined.easnaye roll anda nclaigfof vaeicrntly prlts o tnilCHne ofoa>
38)2)sp; and``Hyporae o ctions ca", Carlet, L.sp; anfPosceto ngs th PRAGOCRYPT'96, Czech Techn oofxUniings of PublishscryHouse,mPraguep. 384145-155 (1996)u> Abstract:
ThiWeg#vmeasareetrosealean functions ofoohx(GF(2))^m (men di)mduchrt if. bo a es ande andinte crrkhioindthe Mamlean functions ofooh (GF(2))^{m-k} qined.aslbiniixscryk ceord nd ts thmtheauebles antaxbe o.>
38)3) "Onothea sraic deg cpckneon aone tn-normof of codlean functions o", Carlet, L.sp; ansp; anfPosceto ngs thm"2003 IEEE Infation TheT oorf Workshop", Pebls, France,l 384147-150o.3.03r> Abstract:
ThiCmography, APNlean functions ofomu$r impsteelexe osuon sfy Snel on's ig ncu o ntwoplnsuss aarThirtwo m in peria foreues d snge th ctography, 2.nviewvt. F,sp; and Mamceeelex andthelean functions ofooh Fn (no rett w linearity (ydaonethea sraic degree an.sp; andTwo ot in peria forh veea givn exhhotur cohd:othea sraic deg cpckneon aonet w tn-normof of. tt taxwn atd stc,laayngros oofly,sp; andrwt $r ofl lean functions wimnenonh noa sraic deg cpckneonnf ayusp; The re deeplysp; andlin-normof,laasweploaf seith veeh noa sraic degree ans aX. h nonlinearity (ins. Wexaove a bupohe pismrlts o aX.,sp; an tlyed scry peion imshsptween the tn-normof of aone tnearity (y,esh pe a ba nclarlts osp; andoo detric benctions, w,ghh is iestinf aeea detly ctrurnce ofvtBoown atbrsts on onttacinelinearity o of ( pis g ans d non clmthst molonttacsp; andrnas wi thmtheinebeh viar)er>
54)4) "Onotheasupmortt the ac Wo oh tsformatisdthelean functions of", Ca let an B. GuiMesnagNoorPosceto ngs thmBFCA (Ft exaWorkshop tnilean fu Ftions wi :xCtography, ypaX. Aied cens wi),lRouan,lMarcha3.05, Publicons ospofoaUniings o&eacd f;spoflRouantet du Havre,l 38465-820 3.05.> Abstract:
ThiIs s paper, we shody tw. dhepeion imshspth optco $\ radurnce ofr,mthe uctiond b thmtho itstbsetn ofsp; andFn (nohh haspbe imsp; and Ma Wo oh supmortt thelean functions wie> Thisp; an> Thi5) "Onotheastructions, twobnced fu lean functions ofoh optaeeoodi sraic immunity wi", Carlet, Ld higP. GabthmsorPosceto ngs thmBFCA (Ft exaWorkshop tnilean fu Ftions wi :xCtography, ypaX. Aied cens wi),lRouan,lMarcha3.05, Publicons ospofoaUniings o&eacd f;spoflRouantet du Havre,l 3841-200 3.05.> Abstract:
ThiIs s paper, w shody twe sec sraic immunity witthelean functions ofo B. hotur co thivreular, ix mosposb whoBooplructionscrylean functions in h optaeeoodn sraic immunity wio Wm ct exwe a lhiue c 2.e rgutatit opvidehat ifa ce sraic immunity witth aaracromflean functions w w nauebles anopsooeast fortinsrnte crrvreu ofsp; andn/2sp; andh op a v eveh nonposbe ty of (hhilo tinsupmin nd on cla mos``cei scr"tth> Thin/2)o Wmpe a lonsupminmnd on,der an adrnas wos an ofungr of, the shb sraic immunity witthelean functions ofoplructionela srh fo Maiarana-MacFet, S. structions, o Wmpe a lo structions, sh fousct rely thtrnasfse shb sraic immunity wittheaflean functions w a lodu thea pies thiler of new clmuebles ano oneuee a ntinsct exwisite clafamaly th ctions whgw optaelin tsiviy prt aneasraic immunity wio Aeas forshoe a eale o f thebnced fu ctions donh optimum speasraic immunity wit higa eoodglinearity ofd B. thebnced fu ctions donh optaeeoodn sraic im unity. I,eaeeoodglinearity ofd B.eaeeoodgplrpeion immunity ar,ghh is tbe impc in bo seography, APNpurpo ioa>
38)6) "Newtss ofnf ComAwt $r l a nuxdoAwt $r PectCwemNinearity Polynom Isa", BudBud ghyan, Carlet, LmaX. AorPotsorPosceto ngs thmWCC, 384306-3<5,llernen,sp; anl3.05.> Abstract:
ThiWeastructioneisite class ofnf Comrwt $r b a nuxdorwt $r vectCwe linearityppolynom Isae sh foue b afed oNooiniquivale o uivnw$\ ctions dor>
38)Aspfunigyle=" bnt-wht mo:mbold;">7)s``TMamceeelex andthelean fu ctions whg thrseography, AP viewvt. F", Carlet, LudD gdy hloSemintyp``Ceeelex andthelean fu Ftions wi"l3.06,.M. Krause,mPorPudlakhiR.tRsischuNmaX. D. vfunMelken ek Eds. http://difts.d gdy> Abstract:
ThiCmography, APNlean functions ofomu$r impsteelexe osuon sfy Snel on's ig ncu o ntwoplnsuss aarBuortinsctography, 2.nviewvt. F ounteeelex an ssi too shbs feaaaslMacircuianteeelex aner>
38)8)s``Anot in pses newntitic APNAandbinom Isaso $\FF_{n/n}:otheas, a n deviso lapbf 4", h optBudBud ghyandaoneGr Lu r an,aWorkshop tniCld to aX. Ctography, y,vp65349-58o.3.07r> Abstract:
ThiW leahibie dlisite clanses newrwt $r vectCwemlinearityyntitic AP binom Isas thrF_{n/{n}}e osF_{n/{n}}eh opsp; andl=4NmaX. k ddEEde sefa stc tbain ctions wim resCCZ-ariquivale o uiwn atbAandvnw$\nction.
38)9)p``Ceructionscrybnced fu ctions donh optimum umi sraic immunity wi. IEEEsp; andItnation isy nSyeensouieim Infation Th T oorf.3.07r> Abstract:
ThiBlyeusonthe auerntly p sraic immack. NC, aeh noa sraic degunity. Itis liwvantaboolunoNoo esary and(buhenondstcient co) piftiesyg thelean fu ctions donc in biveam ciphers, i. Aeuirentiaof vaeonlyx1tween the aci sraic immunity w of ofltwo ctions whdleniimkeeaareuc fueuirentiaof h optpectCwem uiapraic immack. NC. Vaoy fewteale o f the(bnced fu) ctions whgw opth noa sraic degunity. It bvirn exh bunytgiv ardvn win eale o f oeumtto impioolenbytaone tanod ofe bo ined. theduchrction.
38)10)l``Onoiniquivale ofgbeen thewn atovnw$\nAandftions wi"r L. Bud ghyan, Carlet, LmaX. Gr Lu r anorPosceto ngs thmBFCA 3.08p. 38 151-165o.3.09u> Abstract:
ThiT i bt knosixewn atocafnf Comvnw$\nAandftions wievWowrotigate thebe omproquigatow ch$ts, o aese pafnf knoiairwisegCCZ-ariquivale oa>Wlsp; andeefa sp; and Mtc,lsterpt thivreular, ixpafnf,mtheaGold mdyi ngim resCCZ-ariquivale o uitheaKas fimaX. Welchrction.
38)11)l``Ono structions, twontitic APNAandftions wi"r L. Bud ghyan, Carlet, LmaX. Gr Lu r anorPosceto ngs thmIEEE Infation Th T oorf.Workshop 3.09u>Abs> ThiAract
ThiIs aorntly per, we mosaut,ors ioduced bytbsp; Boenod ofefat structionscrynewtntitic APNAandftions wia thrwn atdonns. Aiedwscry pis nod of,l coynqined.asltheactions c $x^3+\tr_n(x^9)$sp; Theah hasisa an eings$\F_{n/n}$ bo anydpos nsvirrnte crr$n$.>
38)12)l``OnotheaDuspethel a nFtions wigw opt$2^r$mNihonExennials''gw optT. Heseth, A, SorKholosha B. GuiMesnagNoorPosceto ngs thmISIT 2011, Saino Pvmeasburgo.3.11.> Abstract:
ThiWeastted fotheauuspections w he aueNihonb a nctions wlh v the$2^r$ eapnnialsspiftosel bf Lu r an B. Kholoshae>
38)13)l``Onol a nFtions wigAssocienbyta ABnFtions wi'', h optBudBud ghyan difT. Heseth, AorPosceto ngs thmIEEE Infation Th T oorf.Workshop 3.11.> Abstract:
ThiIni1998, i efsnd ondaur,or, Charpind difZinovievecneterize zbyt and di ABn,n)$-functions wfoa lm fus thmassocienbyt$2n$-uebles anlean fu ctions doarInivreular, ie coynpifvela stc aos oons c $F$iisa B if di onlyxiodt ipassocienbytlean functions w $\gamma_F$iisabe o.rT pa ibt rvens w leart opvntially No clmc functions in associenbyta aciwn atbABnctions, w,gbo aeast forg ans clmthst molontwn atbc fu ctions dor> ThiIhethirmint an ir, w weg#vmeasaree$\gamma_F$i bo mopreIuatshown at camalinf Com anraX. ABnctions, we>
38)14)rOnoDill w’sanses H ComNihonb a nctions whd di o-polynom Isa,h optparlet. Moroyeensouieim AeulientspeItnallige ofg di Matshion cs (ISAIM.3.<2), Fert Lau(Crdala,gFlthmof, USA, Januerf.3.12er>
54)15) Gralize zbytl a nFtions wige d tacineReion imsp; and o Maiarana-McFet, S. ss of. BudBud ghyan, Carlet, L. T. Heseth, Asp; an LmAorKholoshaerPosceto ngs thmISIT 2012e>
38)16)rOnotheaAhms aes o Wo oh Coecient cot thelean funFtions wierCa let an B. AorKllyieoorPosceto ngs thmtheaItnation isy nWorkshopooh Cld to aX. Ctography, y 2013. April 2013. lernen.> Abstract:
ThiWeagralize zbya tho Ahms aes o Wo oh Tsformati (AWT) d nonrsts on ah hasshreopaous woNoown ati bo tinsWo oh Hadamarddtsformatidthelean fu ctions wh. Wnsstrallyxgralize zbyahlanses oofxPoissonwdumman Th matir, ya tho AWTarWowdnd onlysp; andd that if s o AWTttheafler c nses newlean functions ofopenilelstsent in pnrmeasf quatheaAWTtthea lean functions w newrwraic degree andatet pre3 bivaaler crrler of ne uebles ane>
38)17. Carlet and higGuilley, H,oS-chaCnel attIndiig nguisha ty of, HASP,TattAviv, Israël, Jtyn 3.<3r Publishel bf ACM (Associen Th mat poted to Mrchialiy)> Abstract:
ThiWearoduce a naam okscr sctd egyg thenetdw rett ae iie anns any e-channel attacks int thrrnto $\ raduniquil l mosondreeqkeydthea ctography, 2.ndousof.vthes papm okscr scaemf,rmeasedrhomomor, AP,mthe ue s nsvir.zn ihsrxclus ve-orenmh optaeracromfues cst ae bel wgsa o a es andsetarWowd that if ifes papm okscr set clastrcealaf,l cohe t infation Th abouortinsctography, 2.nkeydds koarIuatshon okscr set cl publicg(thefeiclosel),l cohemnys(h no-ar No)tack. NotlveIsasa grd ptIu equiposbe lo keynevn re rsts on e b aied in a tho casonthe aueAES, shi btue s nsviruebles ano retbytes. Ttsmnysm oktplrpespnndsnaam oked uuracituns w es.o WmpvideCo aandthi btsteigsnaahomomor, APNm okscroh op $16$ m okho( e it ailer of newuuracituns w al tli osf in Iua mo s feaasrthms a h optu am okscr)sf in istancsdt no-iate fooed ex-, uriopr-,gacla sirr-ar Noye-channel attacks inevFurr, wn re,ge andiua mo m okscr set clapublic,ge chcbytenthe auesirpectnkeydii bunytonlyxst aal ogw opt$15$athtirpectno oie mak toot oot-channel attaiofs of instcient coaasqna -- tinsismed. thekeydspfce shad elelstslorenmby ot in m fus ( yp ooflyxexhaust a ldearch)arThuw,gburrhomomor, AP m okscr sctd egygeBs imobothc o thtrnasfttshbler of newt-channel at sures amialssoilltosucm S. sbo a stsy elin negligo lapbrd f-sbocsthe(ne ceeelex and$16^{N_B}=2^{64}$ bo AES, aand bee$N_B=16$ uuracituns w es.fs)arThirnetdw retiementations oathe aueRonenscry&uracituns w M okscrn(RSM)nieeaamiionsoof thstaallyds o vaeburrhomomor, APam okscr ceermeasures ae>
38)18) BudBud ghyan, AorKholosha, Carlet, L. T. Heseth, A.eNihonl a n Ftions wi thrQtitic APNo-Monom IsaerPosceto ngs thmISIT 2014.> Abstract:
ThiIs s paper, we shostte defhlanses newNihonb a nctions wlcoturncothethe $2^r$mmeasf feico $\el bf Lu r an B. KholoshaerThirette s oobas rchiemhdvby insies cryhoecient cot thethirmnw$\nmeasf bivtBoothmgtsy e ctions w. Doscry pis,vshoobtadhepeion im uiaplotBoown atbntitic APN o-monom Isa (upya equivale of).vn wint clmc functions in aree too EA-iquivale o ui ny Iuatshown atass ofnf.vWowo givoofar, fothea sraic degree an newrdndctions w bivtBooette dbytss ofe>
38)19. Carlet an. A. Daif, J.-L. Dara F, Guilley, H,oZ.eNajm, X. T. Ngo. T. xPorteboeuf aX. C. Ta $\nieoorOmal s td LarityyCeeeletatiory C ofoa Iementations oa bo Hetdw retTrojacrPoe anns w. Posceto ngs thmthea22X. EuiftiannConsntiait tniCircuianT oorf.aX. Dciign, ECCTD13.<5,l 3841-4013.<5u> Abstract:
ThiBk ciph ofoa bvirn exhd thntto impamteani osf wicu netdw rettrojacr ,orsfoarThisoy forthe Mamelral onAPNcercuiahgieeatpldela stnksm oeabpair ntwopleeletatiory h ofo:dqna tlch iie anns aNpurpornbytnetdw rettrojacr ,orsfo thrthmgg$\ ra, hhilo tinsot in qna fvoCwes any kscd Comvayload.>
38)20) GuiPsofk, Carlet, L. D. Jakobovic adifJ. Mey, r, BudBenscar Clrpeion im Inity witthelean funFtions wi: An Evoluns wa osArithms asg PerctCwe a . Posceto ngs thmtheaGraltic adifEvoluns wa osCoextions a Consntiait (GECCO.3.15), Jtl 11-<5,lMitiif,lSpain,l 3841095-1102013.<5u> Abstract:
ThiBean functions ofoensent aneaheessially xig mon vf insimnyseam cip hers, i. Tacyt knoc iomon r Nonto addoplnsuss a, indot in wer u,l coyn ensent anet w linearitydiicu qua moseam cipasrthms aarWh$n$c iomon teebarer eraliznors, Bean functions ofonein a bvirstcient colveh non ues cntwoplrpeio- ds o unity. I,eal wgs-chltinsot in pifties oforOnothea ot in smof, plrpeion im-inityn ctions in aseowgisducsthe auesipreIuatsho m okscr siermeasures a fasilchannel attacks intch$n$tseith veed alls Hamm ng hht moarThire been exhailer of newpr, ws ealed. thethea ied ce ty of newevoluns wa osasrt- hms asgch$n$eyola toolean fun ctions whg bo seography, APNusagesarIs i nonthe aoseamr, ws,taur,ors cae Nasl auesir- peion immunity arnpiftiesy,vbuhenr no gaehat oopiftiesy aueh no$$r vrioy oferIs s paper, w shoealed.hat ooeciec- ds aneon ofl c andrirential EAo, n fely,dGraltic Asrthms a,lGraltic pifphymm ng aX. lettesifunGraltic Pifphymm ng ch$n$eyola toobnced fu lean functions ofo Bae rexplrpe- ion immunity . Beilchse shb foremanns wbn mifties of ofl bnced funeon aoneplrpeion immunity ar,ghelcotur coaeeal speot in ateeces pctography, 2.nmifties of hhilo med.ned. thetinsomum spe tsfie-offsaat nge prh. Our rnts on d tha Bae eyola toolean functions ino Bae rexplrpeion immunity (un-obnced fu eryistalce o)gieeante and netdof nbjly sva t a etho t rextsfion isy qna shi bttho eospecla mos m xl s ons oathe auelinearity (ydpiftiesye>
38)21. Carlet. MorOnothealinearity of codmonotonamlean functions oferPosceto ngs thmSETA 3.<6, Chengdu, Chsca, 09-14 Octo of 2.<6r> Abstract:
ThiWlsst exavidehat ootrd hfulneon oflaaroajured bdonothealinearity of cod monotonamlean functions ofmon e andditatss a, piftosel bivtBoorntly p ir, wp``Ctography, 2.nmifties of codmonotonamlean functions of",aa lD. Joyn r, P. GstX ce. D. Tang aX. mosaut,or (J urospecodMatshion cspe Ctogralogr,gyol8410013.<6)o WmpvideCo aohemnsupminmnd onsp; andoo dtcod tnearity (y,esh hasisaaayngros ooflyomtcoddducra F t a etho roajured bd supminmnd on aX. ma etho upminmnd on pifvelaforsp; andodddditatss amon cpaps feair, worT papnd on sh imoasfvepgheakneon oflmonotonamlean fun ctions wh; ciyearestea closelyxaieroxl aehd a lafed o ctions whg bo bescryusas an o linearitydteeennialssin peography, APNaied cens wi. Wns uce a nao esary andceria f o to impson sfi P a la lean fun(pesp. torial fu) ctions fm bo bescrylinearitye>
38)22) BudBud ghyan, Carlet, L. T. Heseth, A aX. N. La. orOnothea (lin-)steigiaof vaeAandftions wianewrwraic degree and 2$ o $\F$Fn (n$r Posceto ngs thmISIT 2016,vp653480-484r>
38)A 38)> ig>Aspfunigyle=" bnt-wht mo:mbold;">Popr, ixmr, ws>::>
38)plau(C let. MorFoions wfoaean&eacd f;ennain Lmtas aaux ert,ogisyux au uriourspofoastrtre-mees af yux acksquigdiicopenyux auxalcairesarGaz r r uc l'InacitunrGalin&eacd f;e, 3.<3r> 38)Aspfunigyle=" bnt-wht mo:mbold;">> 38)Aspfunigyle=" bnt-wht mo:mbold;">>/spfu> < bnt ds t="1"> plau(C let. M(plau(C.let. M@ty> L formodif cens w: < bnt ds t="1">Desar of< bnt ds t="1"> 2017> 38)A/ bnt> )A/body>