On the c-differential uniformity of certain maps over finite fields

Hasan, Sartaj Ul; Pal, Mohit; Riera, Constanza; Stănică, Pantelimon

doi:10.1007/s10623-020-00812-0

Cited by 51 publications

(56 citation statements)

References 9 publications

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“…It is well known that the power function x p k +1 is PN over F p n when n gcd(n,k) is odd [13]. Theorem 8 shows the c-differential uniformity of a variant of this PN power function for c = ±1.…”

Section: It Can Be Seen Inmentioning

confidence: 95%

“…According to the definition of the c-differential uniformity, it was shown in [13] that the power functions x d and x dp j ( j ∈ {0, 1,…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, they exhibited the c-differential uniformity of some known APN power functions in odd characteristic finite fields. By using the first kind of Dickson polynomials [15], Hasan et al [13] introduced some classes of power maps with low c-differential uniformity for c = −1. Some PcN power maps over finite fields of odd characteristic were exhibited.…”

Section: Introductionmentioning

confidence: 99%

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Some classes of power functions with low c-differential uniformity over finite fields

Zha

2021

Des. Codes Cryptogr.

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Functions with low c-differential uniformity have optimal resistance to some types of differential cryptanalysis. In this paper, we investigate the c-differential uniformity of power functions over finite fields of odd characteristic. Based on some known almost perfect nonlinear functions, we present several classes of power functions f (x) = x d with c Δ f ≤ 3. Especially, two new classes of perfect c-nonlinear power functions are proposed.

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Section: It Can Be Seen Inmentioning

confidence: 95%

“…According to the definition of the c-differential uniformity, it was shown in [13] that the power functions x d and x dp j ( j ∈ {0, 1,…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some classes of power functions with low c-differential uniformity over finite fields

Zha

2021

Des. Codes Cryptogr.

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“…It is interesting to check the invariance property of c-differential uniformity of vectorial p-ary functions under these equivalences. Hasan et al [55] first addressed the invariant property on c-differential uniformity. Suppose [ (c, F) ∶ c ∈ p n ] is a differential spectrum of F. Then they proved that the c-differential uniformity spectra preserved (restricted to input, i.e., F � = F•A 2 ) under affine equivalence.…”

Section: Definition 16mentioning

confidence: 99%

“…For example the c-differential uniformity spectra of x 3 over 2 4 is [1,2,3], but the EA-equivalence function of x 3 + x 4 is [1, 2, 3, 4] over 2 4 . They proved that [55,Theorem 6.7] if F and F ′ are CCZ-equivalence via an affine transformation A and also via A ′ , then c-differential uniformity of F is the same as the c ′ -differential uniformity of F ′ , where A 11 , A 12 , A 21 and A 22 are n × n matrices with entries in p . Thus, the c-differential uniformity may not preserve under EA-equivalence or CCZ-equivalences.…”

Section: Definition 16mentioning

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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P℘N functions, complete mappings and quasigroup difference sets

Anbar,

Kalaycı,

Meidl

et al. 2023

J of Combinatorial Designs

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We investigate pairs of permutations of such that is a permutation for every . We show that, in that case, necessarily for some complete mapping of , and call the permutation a perfect nonlinear (PN) function. If , then is a PcN function, which have been considered in the literature, lately. With a binary operation on involving , we obtain a quasigroup, and show that the graph of a PN function is a difference set in the respective quasigroup. We further point to variants of symmetric designs obtained from such quasigroup difference sets. Finally, we analyze an equivalence (naturally defined via the automorphism group of the respective quasigroup) for PN functions, respectively, for the difference sets in the corresponding quasigroup.

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On the c-differential uniformity of certain maps over finite fields

Cited by 51 publications

References 9 publications

Some classes of power functions with low c-differential uniformity over finite fields

Some classes of power functions with low c-differential uniformity over finite fields

Survey on recent trends towards generalized differential and boomerang uniformities

P℘N functions, complete mappings and quasigroup difference sets

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