WAIFI 2007: Arithmetic of Finite Fields pp 134-146

# A Construction of Differentially 4-Uniform Functions from Commutative Semifields of Characteristic 2

• Nobuo Nakagawa
• Satoshi Yoshiara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4547)

## Abstract

We construct differentially 4-uniform functions over GF(2 n ) through Albert’s finite commutative semifields, if n is even. The key observation there is that for some k with 0 ≤ k ≤ 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 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.

## Keywords

Finite field Almost perfect nonlinear function Differentially δ-uniformity Cubic function of a finite semifield Absolute trace

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