Abstract
This paper is dedicated to a question whether the currently known families of quadratic APN polynomials are pairwise different up to CCZ-equivalence. We reduce the list of these families to those CCZ-inequivalent to each other. In particular, we prove that the families of APN trinomials (constructed by Budaghyan and Carlet in 2008) and multinomials (constructed by Bracken et al. 2008) are contained in the APN hexanomial family introduced by Budaghyan and Carlet in 2008. We also prove that a generalization of these trinomial and multinomial families given by Duan et al. (2014) is contained in the family of hexanomials as well.
Highlights
Let n and m be two positive integers, an (n, m)-function, or vectorial Boolean function, is a function F from the finite field F2n with 2n elements to the finite field F2m with✩ Some results of this paper were presented at the International Workshop on Coding and Cryptography WCC 2019.2m elements
EA-equivalence is a particular case of CCZ-equivalence, which is the most general known equivalence relation preserving the differential uniformity
Since the algebraic degree is preserved by EA-equivalence, and families in Table 1 have, in general, different algebraic degrees, all these families differ up to CCZ-equivalence
Summary
Let n and m be two positive integers, an (n, m)-function, or vectorial Boolean function, is a function F from the finite field F2n with 2n elements to the finite field F2m with. Boolean functions and vectorial Boolean functions have been intensively studied due to the large number of applications both in mathematics and computer science They have a crucial role in the design of secure cryptographic primitives, such as block ciphers. The differential uniformity, and the APN property, is preserved by some transformations of functions, which define equivalence relations between vectorial Boolean functions. Two of these equivalence notions are the extended affine equivalence (EA-equivalence) and Carlet-Charpin-Zinoviev equivalence (CCZ-equivalence). EA-equivalence is a particular case of CCZ-equivalence, which is the most general known equivalence relation preserving the differential uniformity.
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