Differentially low uniform permutations from known 4-uniform functions
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Differentially low uniform permutations from known 4-uniform functions Marco Calderini1 Received: 1 March 2020 / Revised: 9 July 2020 / Accepted: 24 September 2020 © The Author(s) 2020
Abstract Functions with low differential uniformity can be used in a block cipher as S-boxes since they have good resistance to differential attacks. In this paper we consider piecewise constructions for permutations with low differential uniformity. In particular, we give two constructions of differentially 6-uniform functions, modifying the Gold function and the Bracken–Leander function on a subfield. Keywords Low differentially uniform · Boolean functions · Permutations · High nonlinearity Mathematics Subject Classification 94A60 · 11T71 · 06E30
1 Introduction Let n be a positive integer, we will denote by F2n the finite field with 2n elements and its multiplicative group by F2n . Permutation maps defined over F2n are used as the S-boxes of some symmetric cryptosystems. So, it is important to construct permutations with good cryptographic properties in order to design a cipher that can resist known attacks. In particular, among these properties we have a low differential and boomerang uniformity for preventing differential and boomerang attacks [1,36], high nonlinearity for avoiding linear cryptanalysis [25] and also not a too low algebraic degree to resist higher order differential attacks [21]. Over a field of even characteristic, the best differential uniformity of a function F is two. Functions achieving this value are called almost perfect nonlinear (APN). Many works have been done on the construction of APN functions (see for instance [4,8–11]). For odd values of n there are known families of APN permutations; while for n even there exists only one example of APN permutation over F26 [7] and the existence of others remains an open problem. For ease of implementation, usually, the integer n is required to be even in a
Communicated by C. Carlet.
B 1
Marco Calderini [email protected] Department of Informatics, University of Bergen, PB 7803, 5020 Bergen, Norway
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M. Calderini Table 1 Primarily-constructed differentially 4-uniform permutations over F2n (n even) with the best known nonlinearity Name
F(x)
deg
Conditions
In
Gold
i x 2 +1
2
n = 2k, k odd gcd(i, n) = 2
[18]
Kasami
2i i x 2 −2 +1
i+1
n = 2k, k odd gcd(i, n) = 2
[20]
Inverse
n x 2 −2
n−1
n = 2k, k ≥ 1
[27]
Bracken–Leander
2k k x 2 +2 +1
3
n = 4k, k odd
[5]
Bracken–Tan–Tan
i m −m m+i ζ x 2 +1 + ζ 2 x 2 +2
2
n = 3m, m even, m/2 odd, gcd(n, i) = 2, 3|m + i
[6]
and ζ is a primitive element of F2n
cryptosystem. Therefore, finding permutations with good cryptographic properties over F2n with n even is an interesting research topic for providing more choices for the S-boxes. The construction of low differentially uniform permutations with the highest nonlinearity over F2n (with n even) is a difficult task. In Table 1 we give five families of primarily constructed differentially 4-uniform permutations with the best known nonlineari
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