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