On equivalence between known families of quadratic APN functions

Budaghyan, Lilya; Calderini, Marco; Villa, Irene

doi:10.1016/j.ffa.2020.101704

Cited by 23 publications

(14 citation statements)

References 26 publications

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“…In [29], the third author of the present paper showed that the hexanomials introduced in [19] can be seen as a case of the APN functions in Theorem 4.1. The same was proved for trinomials introduced in the same paper [19] and for multinomials introduced in [7], but in [17], it has been shown that these two classes are contained in the class of the hexanomials. In fact, we will show in the following that the construction of Theorem 4.1 coincides with the hexanomials' class.…”

Section: Equivalence Between the Apn Hexanomials And Carlet's Bivariate Apn Constructionmentioning

confidence: 59%

“…From the results in [17] we have that the family of APN hexanomial introduced in [19] (see Theorem 3.1) can be represented as a pentanomial: Theorem 4.3. Let n and i be any positive integers, n = 2m, gcd(i, m) = 1, and c, d ∈ F 2 n be such that d / ∈ F 2 m .…”

Section: Equivalence Between the Apn Hexanomials And Carlet's Bivariate Apn Constructionmentioning

confidence: 99%

“…The first four cases in Table 1, and, for n odd, all cases in Table 2 are also AB. All families in Tables 1 and 2 are pairwise CCZ-inequivalent for general n [17,34,41].…”

Section: Introductionmentioning

confidence: 99%

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On known constructions of APN and AB functions and their relation to each other

Calderini¹,

Budaghyan²,

Carlet³

2021

Rad Hrvatske akademije znanosti i umjetnosti

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This work is dedicated to APN and AB functions which are optimal against differential and linear cryptanlysis when used as Sboxes in block ciphers. They also have numerous applications in other branches of mathematics and information theory such as coding theory, sequence design, combinatorics, algebra and projective geometry. In this paper we give an overview of known constructions of APN and AB functions, in particular, those leading to infinite classes of these functions. Among them, the bivariate construction method, the idea first introduced in 2011 by the third author of the present paper, turned out to be one of the most fruitful. It has been known since 2011 that one of the families derived from the bivariate construction contains the infinite families derived by Dillon's hexanomial method. Whether the former family is larger than the ones it contains has stayed an open problem which we solve in this paper. Further we consider the general bivariate construction from 2013 by the third author and study its relation to the recently found infinite families of bivariate APN functions.

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Section: Equivalence Between the Apn Hexanomials And Carlet's Bivariate Apn Constructionmentioning

confidence: 59%

Section: Equivalence Between the Apn Hexanomials And Carlet's Bivariate Apn Constructionmentioning

confidence: 99%

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On known constructions of APN and AB functions and their relation to each other

Calderini¹,

Budaghyan²,

Carlet³

2021

Rad Hrvatske akademije znanosti i umjetnosti

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“…In the last years, several families of (quadratic) APN functions (see [9] or [12] for a recent list of inequivalent APN families) were constructed. For some of these families the APN property is connected with the existence of polynomials having specific features; see i.e.…”

Section: Introductionmentioning

confidence: 99%

On the infiniteness of a family of APN functions

Bartoli¹,

Calderini²,

Polverino³

et al. 2021

Preprint

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APN functions play a fundamental role in cryptography against attacks on block ciphers. Several families of quadratic APN functions have been proposed in the recent years, whose construction relies on the existence of specific families of polynomials.A key question connected with such constructions is to determine whether such APN functions exist for infinitely many dimensions or not.In this paper we consider a family of functions recently introduced by Li et al. in 2021 showing that for any dimension m ≥ 3 there exists an APN function belonging to such a family.Our main result is proved by a combination of different techniques arising from both algebraic varieties over finite fields connected with linearized permutation rational functions and partial vector space partitions, together with investigations on the kernels of linearized polynomials.

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“…One of these properties has been captured in the definition of APN functions. Because of their applications, APN functions have been widely investigated; see for instance [1,6,8,9,11] and the survey [12]. In the design of symmetric primitives, APN functions are often required to be permutations.…”

Section: Introductionmentioning

confidence: 99%

On a conjecture on APN permutations

Bartoli¹,

Timpanella²

2021

Preprint

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The single trivariate representation proposed in [C. Beierle, C. Carlet, G. Leander, L. Perrin, A Further Study of Quadratic APN Permutations in Dimension Nine, arXiv:2104.08008] of the two sporadic quadratic APN permutations in dimension 9 found by Beierle and Leander [4] is further investigated. In particular, using tools from algebraic geometry over finite fields, we prove that such a family does not contain any other APN permutation for larger dimensions.

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On equivalence between known families of quadratic APN functions

Cited by 23 publications

References 26 publications

On known constructions of APN and AB functions and their relation to each other

On known constructions of APN and AB functions and their relation to each other

On the infiniteness of a family of APN functions

On a conjecture on APN permutations

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