Generalized isotopic shift construction for APN functions

Budaghyan, Lilya; Calderini, Marco; Carlet, Claude; Coulter, Robert S.; Villa, Irene

doi:10.1007/s10623-020-00803-1

Cited by 15 publications

(14 citation statements)

References 15 publications

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“…Unfortunately, despite a lot of efforts and some interesting recent advances (see, e.g., [20] and the references therein) on APN functions, studying the existence of APN permutations on 2 2m when m > 3 is a complicated wide-open problem. The existence of APN permutations operating on an even number of bits has been a long-standing open question until Dillon et al, who work for the NSA, provided an example on 6 bits in 2009.…”

Section: On Permutation Quadrinomials With Boomerang Uniformity and The Best-known Nonlinearitymentioning

confidence: 99%

Survey on recent trends towards generalized differential and boomerang uniformities

2021

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Differential cryptanalysis is a general form of cryptanalysis applicable primarily to block and stream ciphers and cryptographic hash functions. The discovery of differential cryptanalysis is generally attributed to Biham and Shamir in the late 1980s, who published several attacks against various block ciphers and hash functions, including a theoretical weakness in the Data Encryption Standard (DES). Boomerang cryptanalysis is a method for the cryptanalysis of block ciphers based on differential cryptanalysis. It was invented by Wagner in (FSE, LNCS 1636, 156-170, 1999 and has allowed new avenues of attack for many ciphers previously deemed safe from differential cryptanalysis. Differential and boomerang uniformities are crucial tools to handle and analyze vectorial functions (designated by substitution boxes, or briefly S-boxes in the context of symmetric cryptography) to resist differential and boomerang attacks, respectively. Ellingsen et al. (IEEE Transactions on Information Theory 66(9), 2020) introduced a new variant of differential uniformity, called c-differential uniformity (where c is a non-zero element of a finite field of characteristic p), of p-ary (n, m)-function for any prime p obtained by extending the well-known derivative of vectorial functions into the (multiplicative) c-derivative. Later, Stănică [Discrete Applied Mathematics, 2021] introduced the notion of c-boomerang uniformity. Both c-differential and c-boomerang uniformities have been extended to the idea of simple differential and boomerang uniformities, respectively, which are recovered when c equals 1.This survey paper combines the known results on this new concept of differential and boomerang uniformities and analyzes their possible cryptographic applications. This survey presents an overview of these significant concepts that might have greater implications for future theoretical research on this subject and applied perspectives in symmetric cryptography and related topics. Along with the paper, we analyze these discoveries and the results provided synthetically. The article intends to help readers explore further avenues in this promising and emerging direction of research. At the end of the article, we present more than nine lines of perspectives and research directions to benefit symmetric cryptography and other related domains such as combinatorial theory (namely, graph theory).

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Section: On Permutation Quadrinomials With Boomerang Uniformity and The Best-known Nonlinearitymentioning

confidence: 99%

Survey on recent trends towards generalized differential and boomerang uniformities

2021

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“…However, it turned out that this construction may be also used for construction of new (up to isotopic equivalence) planar functions [15]. Some generalisations of the isotopic shift constructions are proposed in [14]. In one of them an isotopic shift is applied to Gold-like functions which gives…”

Section: Overview On Known Constructions Of Apn and Ab Functionsmentioning

confidence: 99%

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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“…Many of the known APN functions are quadratic up to CCZ-equivalence, for instance the Gold functions x → x 2 i +1 for gcd(i, n) = 1 (see [30,32]), the two functions from F 2 10 and F 2 12 to itself defined in [27], and the classes defined in [5,6,11,12,13,14,15,16,18,22,40]. Indeed, to the best of our knowledge, at the time of writing only a single APN function is known which is not CCZ-equivalent to either a quadratic or a monomial function (see [28,8]).…”

Section: Quadratic Functionsmentioning

confidence: 99%

“…Note that in the recent works [11,12], the authors have found a new class of quadratic APN permutations which lead to new APN functions in dimension 8 and 9. In [11], it was shown that some of the APN functions from [28] can be classified into an infinite class.…”

Section: Known Apn Functions In Small Dimensionmentioning

confidence: 99%

Linearly Self-Equivalent APN Permutations in Small Dimension

Beierle,

Brinkmann,

Leander

2020

Preprint

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All almost perfect nonlinear (APN) permutations that we know to date admit a special kind of linear self-equivalence, i.e., there exists a permutation G in their CCZ-class and two linear permutations A and B, such that G • A = B • G. After providing a survey on the known APN functions with a focus on the existence of self equivalences, we explicitly search for APN permutations in dimension 6, 7, and 8 that admit such a linear self equivalence. In dimension six, we were able to conduct an exhaustive search and obtain that there is only one such APN permutation up to CCZ-equivalence. In dimensions 7 and 8, we exhaustively searched through parts of the space and conclude that the linear self equivalences of such APN permutations must be of a special form. As one interesting result in dimension 7, we obtain that all APN permutation polynomials with coefficients in F 2 must be (up to CCZequivalence) monomial functions.

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Generalized isotopic shift construction for APN functions

Cited by 15 publications

References 15 publications

Survey on recent trends towards generalized differential and boomerang uniformities

Survey on recent trends towards generalized differential and boomerang uniformities

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

Linearly Self-Equivalent APN Permutations in Small Dimension

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