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Classification of (q, q)-biprojective APN functions

Publication at Faculty of Mathematics and Physics |
2022

Abstract

-In this paper, we classify (q, q)-biprojective almost perfect nonlinear (APN) functions over L x L under the natural left and right action of GL(2, L) where L is a finite field of characteristic 2. This shows in particular that the only quadratic

APN functions (up to CCZ-equivalence) over L x L that satisfy the so-called subfield property are the Gold functions and the function κ : F64 RIGHTWARDS ARROW F64 which is the only known APN function that is equivalent to a permutation over L x L up to CCZequivalence as shown in (Browning, Dillon, McQuistan, and

Wolfe, 2010). Deciding whether there exist other quadratic APN functions 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.