Carl Bracken

New families of quadratic almost perfect nonlinear trinomials and multinomials

We introduce two new infinite families of APN functions, one on fields of order 2^2^k for k not divisible by 2, and the other on fields of order 2^3^k for k not divisible by 3. The polynomials in the first family have between three and k+2 terms, the second familys polynomials have three terms.

A few more quadratic APN functions

We present an infinite family of quadrinomial APN functions on GF(2n) where n is divisible by 3 but not 9. The family contains inequivalent functions, obtained by setting some coefficients equal to 0. We also discuss the inequivalence proof (by computation) which shows that these functions are new.

Determining the nonlinearity of a new family of APN functions

We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multi-term APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function.

On the equivalence of quadratic APN functions

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to the APN Gold function

On the Walsh spectrum of a new APN function

{x^{2^r+1}}

Pseudo quasi-3 designs and their applications to coding theory

if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2n where n ≡ 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function.

New Families of Triple Error Correcting Codes with BCH Parameters

In this paper we compute the Fourier spectra of some recently discovered binomial almost perfect nonlinear (APN) functions. One consequence of this is the determination of the nonlinearity of the functions, which measures their resistance to linear cryptanalysis. Another consequence is that certain error-correcting codes related to these functions have the same weight distribution as the 2-error-correcting Bose-Chaudury-Hocquenghem (BCH) code. Furthermore, for field extensions of

An Infinite Family of Quadratic Quadrinomial APN Functions

\mathbb{F}_2