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

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


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.


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


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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Nobuo Nakagawa
    • 1
  • Satoshi Yoshiara
    • 2
  1. 1.Department of Mathematics, Faculty of Science and Technology, Kinki University, 3-4-1 Kowakae, Higashi Osaka, Osaka 577-8502Japan
  2. 2.Department of Mathematics, Tokyo Woman’s Christian University, Suginami-ku, Tokyo 167-8585Japan

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