2022/06/02 by Faruk Göloğlu, Göloğlu, Faruk
Computer Science · Engineering · Social Sciences · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Islamic Finance and Communication #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2206.00958
openalex publication_date 2022/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we classify (q,q)-biprojective almost perfect nonlinear (APN) functions over \mathbbLL × \mathbbLL under the natural left and right action of GL(2,\mathbbLL) where \mathbbLL is a finite field of characteristic 2. This shows in particular that the only quadratic APN functions (up to CCZ-equivalence) over \mathbbLL × \mathbbLL that satisfy the so-called subfield property are the Gold functions and the function κ: \mathbbF64 → \mathbbF64 which is the only known APN function that is equivalent to a permutation over \mathbbLL × \mathbbLL up to CCZ-equivalence. The κ-function was introduced in (Browning, Dillon, McQuistan, and Wolfe, 2010). Deciding whether there exist other quadratic APN functions (possibly CCZ-equivalent to permutations) that satisfy subfield property or equivalently, generalizing κ to higher dimensions was an open problem listed for instance in (Carlet, 2015) as one of the interesting open problems on cryptographic functions.