New Families of Differentially 4-Uniform Permutations over \({\mathbb F}_{2^{2k}}\)

  • Yin Tan
  • Longjiang Qu
  • Chik How Tan
  • Chao Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7280)


Differentially 4-uniform permutations over \({\mathbb F}_{2^{2k}}\), especially those with high nonlinearity and high algebraic degree, are cryptographically significant mappings as they are good choices for the substitution boxes (S-boxes) in many symmetric ciphers. For instance, the currently endorsed Advanced Encryption Standard (AES) uses the inverse function, which is a differentially 4-uniform permutation. However, up to now, there are only five known infinite families of such mappings which attain the known maximal nonlinearity. Most of these five families have small algebraic degrees and only one family can be defined over \({\mathbb F}_{2^{2k}}\) for any positive integer k. In this paper, we apply the powerful switching method on the five known families to construct differentially 4-uniform permutations. New infinite families of such permutations are discovered from the inverse function, and some sporadic examples are found from the others by using a computer. All newly found infinite families can be defined over fields \({\mathbb F}_{2^{2k}}\) for any k and their algebraic degrees are 2k − 1. Furthermore, we obtain a lower bound for the nonlinearity of one infinite family.


Permutation polynomial differentially 4-uniform mapping S-box switching method 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yin Tan
    • 1
  • Longjiang Qu
    • 2
    • 3
  • Chik How Tan
    • 1
  • Chao Li
    • 2
  1. 1.Temasek LaboratoriesNational University of SingaporeSingapore
  2. 2.Department of Mathematics and System Science, Science CollegeNational University of Defense TechnologyChangShaChina
  3. 3.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingapore

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