A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions

Carlet, Claude; Gao, Guangpu; Liu, Wenfen

doi:10.1016/j.jcta.2014.05.008

Cited by 63 publications

(55 citation statements)

References 17 publications

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“…For functions 3-(1, , ) 9 ( = 3), the 6 de ning monomials with exactly one of or equivalent to 1 mod 3 are [1,2,4], [1,2,7], [1,3,4], [1,3,7], [1,4,8], [1,4,9]. = 8, 9; and [1, 8, 9].…”

Section: De Nition 23 (Repeated and Unique Variables)mentioning

confidence: 99%

“…Given a mixed form function 3-( , , ) 3 where is unique and , are repeated, let (3 , ) = 3 (where is as de ned above) and 3 = 3 . We de ne the -th string of 3-( , , ) to be the set of monomials S such that [7,8,13], [13,14,4], [4,5,10], [10,11,1] .…”

Section: [ + 3 + 3 + 3 ] and [ + 3 + 3 + 3 ]mentioning

confidence: 99%

“…More recently, [15] extends the de nition of -rotation symmetric functions to the multi-output case and uses these functions to get new results in the design of cryptographic S-boxes. Also, the 2-rotation symmetric functions are used to construct bent functions in [4].…”

Section: Introductionmentioning

confidence: 99%

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Theory of 3-rotation symmetric cubic Boolean functions

Cusick

Cheon

2014

Journal of Mathematical Cryptology

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Rotation symmetric Boolean functions have been extensively studied in the last 15 years or so because of their importance in cryptography and coding theory. Until recently, very little was known about such basic questions as when two such functions are a ne equivalent. This question in important in applications, because almost all important properties of Boolean functions (such as Hamming weight, nonlinearity, etc.) are a ne invariants, so when searching a set for functions with useful properties, it su ces to consider just one function in each equivalence class. This can greatly reduce computation time. Even for quadratic functions, the analysis of a ne equivalence was only completed in 2009. The much more complicated case of cubic functions was completed in the special case of a ne equivalence under permutations for monomial rotation symmetric functions in two papers from 2011 and 2014. There has also been recent progress for some special cases for functions of degree > 3. In 2007 it was found that functions satisfying a new notion of -rotation symmetry for > 1 (where the case = 1 is ordinary rotation symmetry) were of substantial interest in cryptography and coding theory. Since then several researchers have used these functions for = 2 and 3 to study such topics as construction of bent functions, nonlinearity and covering radii of various codes. In this paper we develop a detailed theory for the monomial 3-rotation symmetric cubic functions, extending earlier work for the case = 2 of these functions.

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“…For functions 3-(1, , ) 9 ( = 3), the 6 de ning monomials with exactly one of or equivalent to 1 mod 3 are [1,2,4], [1,2,7], [1,3,4], [1,3,7], [1,4,8], [1,4,9]. = 8, 9; and [1, 8, 9].…”

Section: De Nition 23 (Repeated and Unique Variables)mentioning

confidence: 99%

Section: [ + 3 + 3 + 3 ] and [ + 3 + 3 + 3 ]mentioning

confidence: 99%

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Theory of 3-rotation symmetric cubic Boolean functions

Cusick

Cheon

2014

Journal of Mathematical Cryptology

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“…The transformation Q → Q ′ is less well behaved than one might hope: it does not, in general, preserve the nonlinearity or the weight (pathological behavior is noted throughout [4, subsections 5.2 and 5.3]). Nevertheless, by [4,Theorem 5.1] nonlinearity is preserved in the quadratic case we are concerned with here.…”

Section: Introductionmentioning

confidence: 99%

“…On ker C| W the restriction of the quadratic form Q is Frobenius-semi-linear, in the sense that Q satisfies the conditions in (2)(3)(4)(5) except that x, y range over W , c ranges over GF (2 n ), and the very last equality no longer holds.…”

Section: Introductionmentioning

confidence: 99%

Affine equivalence for quadratic rotation symmetric Boolean functions

Chirvăsitu

Cusick

2020

Des. Codes Cryptogr.

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Let f n (x 0 , x 1 , . . . , x n−1 ) denote the algebraic normal form (polynomial form) of a rotation symmetric (RS) Boolean function of degree d in n ≥ d variables and let wt(f n ) denote the Hamming weight of this function. Let (0, a 1 , . . . , a d−1 ) n denote the function f n of degree d in n variables generated by the monomial x 0 x a1 · · · x a d−1 . Such a function f n is called monomial rotation symmetric (MRS). It was proved in a 2012 paper that for any MRS f n with d = 3, the sequence of weights {w k = wt(f k ) : k = 3, 4, . . .} satisfies a homogeneous linear recursion with integer coefficients. This result was gradually generalized in the following years, culminating around 2016 with the proof that such recursions exist for any rotation symmetric function f n . Recursions for quadratic RS functions were not explicitly considered, since a 2009 paper had already shown that the quadratic weights themselves could be given by an explicit formula. However, this formula is not easy to compute for a typical quadratic function. This paper shows that the weight recursions for the quadratic RS functions have an interesting special form which can be exploited to solve various problems about these functions, for example, deciding exactly which quadratic RS functions are balanced.

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Open Problems on Binary Bent Functions

Carlet

2014

Open Problems in Mathematics and Computational Science

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A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions

Cited by 63 publications

References 17 publications

Theory of 3-rotation symmetric cubic Boolean functions

Theory of 3-rotation symmetric cubic Boolean functions

Affine equivalence for quadratic rotation symmetric Boolean functions

Open Problems on Binary Bent Functions

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