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On APN and AB power functions

By Lilya Budaghyan

Appears in collection : ALCOCRYPT - ALgebraic and combinatorial methods for COding and CRYPTography

APN and AB functions are S-boxes with optimal resistance to the linear and differential cryptanalysis. In this talk we survey known constructions and classifications of these functions and discuss big open problems for the monomial case. Among these problems are the Dobbertin's conjecture on nonexistence of new APN monomials (open since 2000), the Walsh spectrum of Dobbertin's APN monomials (open since 2000), the existence of APN permutations of the form $x^d+L(x)$ where $x^d$ is some of the known APN monomials and $L$ is a nonzero linear map.

Remark : On page 18 the speaker refers the classification result by Brinkmann [3] for functions from the field $F_{2^4}$ of order 16 to itself.

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Citation data


  • Lilya Budaghyan, Marco Calderini, Claude Carlet, Diana Davidova, Nikolay S. Kaleyski: On Two Fundamental Problems on APN Power Functions. IEEE Trans. Inf. Theory 68(5): 3389-3403 (2022) - https://doi.org/10.1109/TIT.2022.3147060
  • Claude Carlet, Stjepan Picek: ​​​​​​​On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials. IACR Cryptol. ePrint Arch. 2017: 1179 (2017) - https://ia.cr/2017/1179
  • ​​​​​​​Marcus Brinkmann: Extended Affine and CCZ Equivalence up to Dimension 4. IACR Cryptol. ePrint Arch. 2019: 316 (2019) - https://ia.cr/2019/316

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