A new large class of functions not APN infinitely often

In this paper, we show that there is no vectorial Boolean function of degree 4e, with e satisfaying certain conditions, which is APN over infinitely many extensions of its field of definition. It is a new step in the proof of the conjecture of Aubry, McGuire and Rodie

Planar functions over fields of characteristic two

Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over $\mathbb{Z}_4$. We then specialise to planar monomial functions $f(x)=cx^t$ and present constructions and partial results towards their classification. In particular, we show that $t=1$ is the only odd exponent for which $f(x)=cx^t$ is planar (for some nonzero $c$) over infinitely many fields. The proof techniques involve methods from algebraic geometry.Comment: 23 pages, minor corrections and simplifications compared to the first versio

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

A few more functions that are not APN infinitely often

We consider exceptional APN functions on ${\bf F}_{2^m}$, which by definition are functions that are not APN on infinitely many extensions of ${\bf F}_{2^m}$. Our main result is that polynomial functions of odd degree are not exceptional, provided the degree is not a Gold member ($2^k+1$) or a Kasami-Welch number ($4^k-2^k+1$). We also have partial results on functions of even degree, and functions that have degree $2^k+1$

Some More Functions That Are Not APN Infinitely Often. The Case of Kasami exponents

We prove a necessary condition for some polynomials of Kasami degree to be APN over F_{q^n} for large n.Comment: 1