A construction of differentially 4-uniform functions from commutative semifields of characteristic 2. (English) Zbl 1213.11196

Nakagawa, Nobuo; Yoshiara, Satoshi

Carlet, Claude (ed.) et al., Arithmetic of finite fields. First international workshop, WAIFI 2007, Madrid, Spain, June 21–22, 2007. Proceedings. Berlin: Springer (ISBN 978-3-540-73073-6/pbk). Lecture Notes in Computer Science 4547, 134-146 (2007).

Summary: We construct differentially 4-uniform functions over \(\text{GF}(2^{n }\)) through Albert’s finite commutative semifields, if \(n\) is even. The key observation there is that for some \(k\) with \(0 \leq k \leq n - 1\), the function \(f_{k}(x):=(x^{2^{k+1}}+x)/(x^{2}+x)\) is a two to one map on a certain subset \(D _{k }(n)\) of \(\text{GF}(2^{n }\)). We conjecture that \(f _{k }\) is two to one on \(D _{k }(n)\) if and only if (\(n,k)\) belongs to a certain list. For (\(n,k)\) in this list, \(f _{k }\) is proved to be two to one. We also prove that if \(f _{2}\) is two to one on \(D _{2}(n)\) then \((n,2)\) belongs to the list.
For the entire collection see [Zbl 1121.11002].

MSC:

11T55	Arithmetic theory of polynomial rings over finite fields
11T71	Algebraic coding theory; cryptography (number-theoretic aspects)
94A60	Cryptography

Keywords:

finite field; almost perfect nonlinear function; differentially \(\delta \)-uniformity; cubic function of a finite semifield; absolute trace