On Inversion in Z_{2^n-1}

In this paper we determined explicitly the multiplicative inverses of the Dobbertin and Welch APN exponents in Z_{2^n-1}, and we described the binary weights of the inverses of the Gold and Kasami exponents. We studied the function \de(n), which for a fixed positive integer d maps integers n\geq 1 to the least positive residue of the inverse of d modulo 2^n-1, if it exists. In particular, we showed that the function \de is completely determined by its values for 1 \leq n \leq \ordb, where \ordb is the order of 2 modulo the largest odd divisor of d.Comment: The first part of this work is an extended version of the results presented in ISIT1

Crooked maps in F2n

AbstractA map f:F2n→F2n is called crooked if the set {f(x+a)+f(x):x∈F2n} is an affine hyperplane for every fixed a∈F2n∗ (where F2n is considered as a vector space over F2). We prove that the only crooked power maps are the quadratic maps x2i+2j with gcd(n,i−j)=1. This is a consequence of the following result of independent interest: for any prime p and almost all exponents 0⩽d⩽pn−2 the set {xd+γ(x+a)d:x∈Fpn} contains n linearly independent elements, where γ and a≠0 are arbitrary elements from Fpn

Constructing irreducible polynomials recursively with a reverse composition method

We suggest a construction of the minimal polynomial $m_{\beta^k}$ of $\beta^k\in \mathbb F_{q^n}$ over $\mathbb F_q$ from the minimal polynomial $f= m_\beta$ for all positive integers $k$ whose prime factors divide $q-1$. The computations of our construction are carried out in $\mathbb F_q$. The key observation leading to our construction is that for $k \mid q-1$ holds $$m_{\beta^k}(X^k) = \prod_{j=1}^{\frac kt} \zeta_k^{-jn} f (\zeta_k^j X),$$ where $t= \max \{m\mid \gcd(n,k): f (X) = g (X^m), g \in \mathbb F_q[X]\}$ and $\zeta_{k}$ is a primitive $k$-th root of unity in $\mathbb F_q$. The construction allows to construct a large number of irreducible polynomials over $\mathbb F_q$ of the same degree. Since different applications require different properties, this large number allows the selection of the candidates with the desired properties

On sets determining the differential spectrum of mappings

Special issue on the honor of Gerard CohenInternational audienceThe differential uniformity of a mapping $F : F 2 n → F 2 n$ is defined as the maximum number of solutions $x$ for equations $F (x+a)+F (x) = b$ when a ̸ = 0 and $b$ run over $F 2 n$. In this paper we study the question whether it is possible to determine the differential uniformity of a mapping by considering not all elements a ̸ = 0, but only those from a special proper subset of $F 2 n \ {0}$. We show that the answer is " yes " , when $F$ has differential uniformity 2, that is if $F$ is APN. In this case it is enough to take a ̸ = 0 on a hyperplane in $F 2 n$. Further we show that also for a large family of mappings F of a special shape, it is enough to consider a from a suitable multiplicative subgroup of $F 2 n$

A New Family of Perfect Nonlinear Binomials

We prove that the binomials $x^{p^s+1}-\alpha x^{p^k+p^{2k+s}}$ define perfect nonlinear mappings in $GF(p^{3k})$ for an appropriate choice of the integer $s$ and $\alpha \in GF(p^{3k})$. We show that these binomials are inequivalent to known perfect nonlinear monomials. As a consequence we obtain new commutative semifields for $p\geq 5$ and odd $k$