New Construction of Differentially 4-Uniform Bijections

  • Claude Carlet
  • Deng Tang
  • Xiaohu Tang
  • Qunying Liao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8567)


Block ciphers use Substitution boxes (S-boxes) to create confusion into the cryptosystems. For resisting the known attacks on these cryptosystems, the following criteria for functions are mandatory: low differential uniformity, high nonlinearity and not low algebraic degree. Bijectivity is also necessary if the cipher is a Substitution-Permutation Network, and balancedness makes a Feistel cipher lighter. It is well-known that almost perfect nonlinear (APN) functions have the lowest differential uniformity 2 (the values of differential uniformity being always even) and the existence of APN bijections over \(\mathbb {F}_{2^n}\) for even \(n\ge 8\) is a big open problem. In real practical applications, differentially 4-uniform bijections can be used as S-boxes when the dimension is even. For example, the AES uses a differentially 4-uniform bijection over \(\mathbb {F}_{2^8}\). In this paper, we first propose a method for constructing a large family of differentially 4-uniform bijections in even dimensions. This method can generate at least \(\big (2^{n-3}-\lfloor 2^{(n-1)/2-1}\rfloor -1\big )\cdot 2^{2^{n-1}}\) such bijections having maximum algebraic degree \(n-1\). Furthermore, we exhibit a subclass of functions having high nonlinearity and being CCZ-inequivalent to all known differentially 4-uniform power bijections and to quadratic functions.


Block cipher Substitution box Differential uniformity CCZ-equivalence Nonlinearity 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Claude Carlet
    • 1
  • Deng Tang
    • 1
    • 2
  • Xiaohu Tang
    • 2
  • Qunying Liao
    • 3
  1. 1.LAGA, Department of MathematicsUniversity of Paris 8, CNRS, UMR 7539Saint-Denis Cedex 02France
  2. 2.Provincial Key Lab of Information Coding and TransmissionInstitute of Mobile Communications, Southwest Jiaotong UniversityChengduChina
  3. 3.Institute of Mathematics and Software ScienceSichuan Normal UniversityChengduChina

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