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16
A class of quadratic APN binomials inequivalent to power functions
, 2006
"... We exhibit an infinite class of almost perfect nonlinear quadratic binomials from F2n to F2n (n ≥ 12, n divisible by 3 but not by 9). We prove that these functions are EAinequivalent to any power function and that they are CCZinequivalent to any Gold function and to any Kasami function. It means t ..."
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Cited by 26 (6 self)
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We exhibit an infinite class of almost perfect nonlinear quadratic binomials from F2n to F2n (n ≥ 12, n divisible by 3 but not by 9). We prove that these functions are EAinequivalent to any power function and that they are CCZinequivalent to any Gold function and to any Kasami function. It means that for n even they are CCZinequivalent to any known APN function, and in particular for n = 12,24, they are therefore CCZinequivalent to any power function. It is also proven that, except in particular cases, the Gold mappings are CCZinequivalent to the Kasami and Welch functions.
New Classes of Almost Bent and Almost Perfect Nonlinear Functions
 IEEE Trans. Inform. Theory
, 2006
"... We construct infinite classes of almost bent and almost perfect nonlinear polynomials, which are affinely inequivalent to any sum of a power function and an affine function. ..."
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Cited by 25 (10 self)
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We construct infinite classes of almost bent and almost perfect nonlinear polynomials, which are affinely inequivalent to any sum of a power function and an affine function.
An infinite class of quadratic APN functions which are Not Equivalent To power mappings
 PROCEEDINGS OF THE IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY 2006
, 2005
"... We exhibit an infinite class of almost perfect nonlinear quadratic polynomials from F 2 n to F 2 n (n 12, n divisible by 3 but not by 9). We prove that these functions are EAinequivalent to any power function. In the forthcoming version of the present paper we will proof that these function ..."
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Cited by 24 (6 self)
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We exhibit an infinite class of almost perfect nonlinear quadratic polynomials from F 2 n to F 2 n (n 12, n divisible by 3 but not by 9). We prove that these functions are EAinequivalent to any power function. In the forthcoming version of the present paper we will proof that these functions are CCZinequivalent to any Gold function and to any Kasami function, in particular for n = 12, they are therefore CCZinequivalent to power functions.
Another class of quadratic APN binomials over $\F_{2^n}$: the case $n$ divisible by 4
 IACR EPRINT
, 2006
"... We exhibit an infinite class of almost perfect nonlinear quadratic binomials from $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{n}}$ with $n=4k$ and $k$ odd. We prove that these functions are CCZinequivalent to known APN power functions when $k\ne 1$. In particular it means that for $n=12,20,28$, they ar ..."
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Cited by 12 (3 self)
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We exhibit an infinite class of almost perfect nonlinear quadratic binomials from $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{n}}$ with $n=4k$ and $k$ odd. We prove that these functions are CCZinequivalent to known APN power functions when $k\ne 1$. In particular it means that for $n=12,20,28$, they are CCZinequivalent to any power function.
Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. Submitted, available at http://eprint.iacr.org/2007/098
"... A method for constructing differentially 4uniform 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 ..."
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Cited by 7 (0 self)
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A method for constructing differentially 4uniform 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 CCZinequivalent to power functions.
On highly nonlinear Sboxes and their inability to thwart DPA attacks
 in "Advances in Cryptology  INDOCRYPT 2005", Lecture Notes in Computer Science, n o 3797
, 2005
"... Prou# has introduced recently, at FSE 2005, the notion of transparency order of Sboxes. This new characteristic is related to the ability of an Sbox, used in a cryptosystem in which the round keys are introduced by addition, to thwart singlebit or multibit DPA attacks on the system. If this ..."
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Cited by 5 (0 self)
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Prou# has introduced recently, at FSE 2005, the notion of transparency order of Sboxes. This new characteristic is related to the ability of an Sbox, used in a cryptosystem in which the round keys are introduced by addition, to thwart singlebit or multibit DPA attacks on the system. If this parameter has su#ciently small value, then the Sbox is able to withstand DPA attacks without that adhoc modifications in the implementation be necessary (these modifications make the encryption about twice slower). We prove lower bounds on the transparency order of highly nonlinear Sboxes. We show that some highly nonlinear functions (in odd or even numbers of variables) have very bad transparency orders: the inverse functions (used as Sbox in the AES), the Gold functions and the Kasami functions (at least under some assumption) .
Determining the Nonlinearity of a New Family of APN Functions
"... Abstract. We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multiterm APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function. ..."
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Cited by 4 (2 self)
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Abstract. We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multiterm APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function.
On Certain 3Weight Cyclic Codes Having Symmetric Weights and a Conjecture of Helleseth
 Sequences and their Applications  Proceedings of SETA'01
, 2001
"... When the binary cyclic code of length 2 m 1 with check polynomial m 1 (x)m t (x) has exactly three nonzero weights, we discuss the symmetry of the two outer weights about the middle weight. This question is connected to a conjecture of Helleseth, that the crosscorrelation function of two msequen ..."
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Cited by 2 (1 self)
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When the binary cyclic code of length 2 m 1 with check polynomial m 1 (x)m t (x) has exactly three nonzero weights, we discuss the symmetry of the two outer weights about the middle weight. This question is connected to a conjecture of Helleseth, that the crosscorrelation function of two msequences of length 2 m 1 must take at least four values when m is a power of 2. We present partial results on this problem, and some equivalent conditions to the symmetry of the weights. 1
On Three Weights in Cyclic Codes with Two Zeros
, 2003
"... We show that the dual code of the binary cyclic code of length 2 1 with two zeros ; cannot have three weights in the case that m is even and d 0 (mod 3). The proof involves the partial calculation of a coset weight distribution. ..."
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Cited by 1 (0 self)
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We show that the dual code of the binary cyclic code of length 2 1 with two zeros ; cannot have three weights in the case that m is even and d 0 (mod 3). The proof involves the partial calculation of a coset weight distribution.
Spectral Methods for CrossCorrelations of Geometric Sequences
 in « IEEE Trans. Inform. Theory
"... Families of sequences with low pairwise shifted crosscorrelations are desirable for applications such as CDMA communications. Often such sequences must have additional properties for speci c applications. Several ad hoc constructions of such families exist in the literature, but there are few s ..."
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Families of sequences with low pairwise shifted crosscorrelations are desirable for applications such as CDMA communications. Often such sequences must have additional properties for speci c 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 crosscorrelations from old families of such sequences. The bounds are obtained in terms the maximum crosscorrelation in the old family and the Walsh transform of certain functions.