Charpin, Pascale Peng, Jie

The associated codes of almost perfect nonlinear (APN) functions have been widely studied. In this paper we consider more generally the codes associated with functions that have differential uniformity at least 4. We emphasize, for such a function F , the role of codewords of weight 3 and 4 and of some cosets of its associated code C F. We give som...

Canteaut, Anne Perrin, Léo Tian, Shizhu

Whether there exist Almost Perfect Non-linear permutations (APN) operating on an even number of bit is the so-called Big APN Problem. It has been solved in the 6-bit case by Dillon et al. in 2009 but, since then, the general case has remained an open problem. In 2016, Perrin et al. discovered the butterfly structure which contains Dillon et al.'s p...

Charpin, Pascale Peng, Jie

In this paper some new links between the nonlinearity and differential uniformity of some large classes of functions are established. Differentially two-valued functions and quadratic functions are mainly treated. A lower bound for the nonlinearity of monomial δ-uniform permutations is obtained, for any δ, as well as an upper bound for differential...

Aubry, Yves Herbaut, Fabien

For any polynomial $f$ of ${\mathbb F}_{2^n}[x]$ we introduce the following characteristic of the distribution of its second order derivative,which extends the differential uniformity notion:$$\delta^2(f):=\max_{\substack{\alpha \in {\mathbb F}_{2^n}^{\ast} ,\alpha' \in {\mathbb F}_{2^n}^{\ast} ,\beta \in {\mathbb F}_{2^n} \\\alpha\not=\alpha'}} \s...

Canteaut, Anne Duval, Sébastien Perrin, Léo

The existence of Almost Perfect Nonlinear (APN) permutations operating on an even number of variables was a long-standing open problem, until an example with six variables was exhibited by Dillon et al. in 2009. However it is still unknown whether this example can be generalised to any even number of inputs. In a recent work, Perrin et al. describe...

Charpin, Pascale Kyureghyan, Gohar M. Suder, Valentin
Published in
Finite Fields and Their Applications

We study the functions Fs,t,γ(x)=xs+γTr(xt) on F2n. We describe the set of such permutations and the explicit expressions of their compositional inverses. Further we consider special classes of such functions, for which we determine the size of their image set, the algebraic degree and the differential uniformity.

Perrin, Léo Paul

S-boxes are key components of many symmetric cryptographic primitives. Among them, some block ciphers and hash functions are vulnerable to attacks based on differential cryptanalysis, a technique introduced by Biham and Shamir in the early 90’s. Resistance against attacks from this family depends on the so-called differential properties of the S-bo...

Budaghyan, L. Carlet, C. Leander, Gregor

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 F2(n). It is proven that for n >= 7 this function is CCZ-inequivalent to the Gold functions, and in the case n = 7 it is CCZ-inequivalent to any power mapping (and, ther...

Budaghyan, L. Carlet, C. Leander, Gregor

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 F2(n). It is proven that for n >= 7 this function is CCZ-inequivalent to the Gold functions, and in the case n = 7 it is CCZ-inequivalent to any power mapping (and, ther...

Budaghyan, L. Carlet, C. Leander, Gregor

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 F2(n). It is proven that for n >= 7 this function is CCZ-inequivalent to the Gold functions, and in the case n = 7 it is CCZ-inequivalent to any power mapping (and, ther...