Generalized Rotation Symmetric and Dihedral Symmetric Boolean Functions − 9 Variable Boolean Functions with Nonlinearity 242

Kavut, Selçuk; Yucel, M.D.

doi:10.1007/978-3-540-77224-8_37

Cited by 24 publications

(22 citation statements)

References 18 publications

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“…The class of rotation symmetric Boolean functions has turned out to be an especially fertile source of functions which are useful in coding theory and cryptography. Some recent papers giving coding theory applications are [7,8]. Some recent papers giving cryptography applications are [5,[10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

“…(e) = |D n | = (n 2 − 3n + 2)/6(8) from Lemma 1.2, Fix(σ n−1 ) = q − 1 by Lemmas 2.2 and 2.3 (since we proved above that the only classes fixed by σ n−1 contain (1, 2, 3),(1,3,5)) and Fix(σ k ) = Fix(σ k 2 Mod q ) = (q − 1)/3 by Lemma 2.4. Plugging our data into Lemma 1.6 gives Theorem 2.1, and the other assertions in the theorem then follow by computation using(8).…”

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confidence: 99%

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Affine equivalence for cubic rotation symmetric Boolean functions withn=pqvariables

Cusick

Cheon

2014

Discrete Mathematics

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a b s t r a c tRotation symmetric Boolean functions have been extensively studied in the last fifteen years or so because of their importance in cryptography and coding theory. Until recently, very little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in a 2009 paper of Kim, Park and Hahn. The much more complicated analogous problem for cubic functions was solved for permutations using a new concept of patterns in a 2011 paper of Cusick, and it is conjectured that, as in the quadratic case, this solution actually applies for all affine transformations. The patterns method enables a detailed analysis of the affine equivalence classes for various special classes of cubic rotation symmetric functions in n variables. Here the case of functions generated by a single monomial and having pq variables, where p and q are primes, is examined in detail, and in particular, a formula for the number of classes is proved. This is significant because it is the first time that a complete enumeration of the number of classes has been found when the number of variables is divisible by two distinct primes.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Affine equivalence for cubic rotation symmetric Boolean functions withn=pqvariables

Cusick

Cheon

2014

Discrete Mathematics

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“…The relationship between nonlinearity and explicit attack on symmetric ciphers was discovered by Matsui [23]. For results on constructions of Boolean functions with high nonlinearity we refer to [1,4,5,18,19,[26][27][28][29]. The Walsh transform of f ∈ B n at λ ∈ F 2 n is defined by…”

Section: Introductionmentioning

confidence: 99%

“…The lower bounds of the third-order nonlinearities of f (x) = Tr n 1 (x 57 ) for all n ∈ {7,8,10,11,13,14,16,17,19, 20} are listed below.n a Dimension of kernel of L a (x) nl(D 1 D a f ) Lower bound of nl 3 ( f )Remark 1 When gcd(2 n − 1, 57) = 1, then, Tr n 1 (μx 57 ) is affine equivalent to Tr n 1 (x 57 ), for all μ ∈ F * 2 n . Thus for n= 7, 11, 13, 17, 19, the lower bounds of higher order nonlinearities of f and f μ are same.…”

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confidence: 99%

Third-order nonlinearities of a subclass of Kasami functions

Gode

Gangopadhyay

2010

Cryptogr. Commun.

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The rth-order nonlinearity, where r ≥ 1, of an n-variable Boolean function f , denoted by nl r ( f ), is defined as the minimum Hamming distance of f from all n-variable Boolean functions of degrees at most r. In this paper we obtain a lower bound of the third-order nonlinearities of Kasami functions of the form Tr n 1 (μx 57 ). It is demonstrated that for large values of n the lower bound of the third-order nonlinearities of the functions of this form is larger than the general lower bound obtained by Carlet (IEEE Trans Inf Theory 54 (3):1262-1272, 2008) for Kasami functions. Further we show that our result along with the computational results obtained by Fourquet and Tavernier (Designs Codes Cryptogr 49:323-340, 2008) provide us an estimate of the nonlinearity profiles of these functions for n = 7, 8, 10.

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“…It has been experimentally demonstrated that there are functions in this class which are good in terms of balancedness, nonlinearity, correlation immunity, algebraic degree and algebraic immunity (resistance against algebraic attack) [16]. It is interesting to note that the famous Patterson-Wiedemann functions [33] that achieve nonlinearity 16,276 (strictly greater than nonlinearity 2 15 [25][26][27] proved that there exist rotation symmetric functions in 9 variables having nonlinearity 241 and 242 (which is also strictly greater than the bent concatenation nonlinearity 2 9−1 − 2 (9−1)/2 ), which was rather surprising and gives further motivation for the investigation of rotation symmetric Boolean functions.…”

Section: Introductionmentioning

confidence: 99%

Circulant matrices and affine equivalence of monomial rotation symmetric Boolean functions

Canright

Chung

Stănică

2015

Discrete Mathematics

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a b s t r a c tThe goal of this paper is two-fold. We first focus on the problem of deciding whether two monomial rotation symmetric (MRS) Boolean functions are affine equivalent via a permutation. Using a correspondence between such functions and circulant matrices, we give a simple necessary and sufficient condition. We connect this problem with the well known Ádám's conjecture from graph theory. As applications, we reprove easily several main results of Cusick et al. on the number of equivalence classes under permutations for MRS in prime power dimensions, as well as give a count for the number of classes in pq number of variables, where p, q are prime numbers with p < q < p 2 . Also, we find a connection between the generalized inverse of a circulant matrix and the invertibility of its generating polynomial over F 2 , modulo a product of cyclotomic polynomials, thus generalizing a known result on nonsingular circulant matrices.Published by Elsevier B.V.

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Generalized Rotation Symmetric and Dihedral Symmetric Boolean Functions − 9 Variable Boolean Functions with Nonlinearity 242

Cited by 24 publications

References 18 publications

Affine equivalence for cubic rotation symmetric Boolean functions withn=pqvariables

Affine equivalence for cubic rotation symmetric Boolean functions withn=pqvariables

Third-order nonlinearities of a subclass of Kasami functions

Circulant matrices and affine equivalence of monomial rotation symmetric Boolean functions

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