Proof of a conjecture about rotation symmetric functions

Zhang, Xiyong; Guo, Hua; Feng, Rongquan; Yi-fa, LI

doi:10.1016/j.disc.2011.03.012

Cited by 14 publications

(9 citation statements)

References 4 publications

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“…Since then, much attention has been given to rotation symmetric Boolean functions. In 2002, Cusick and Stȃnicȃ [2] found a recursion for the weights of the monomial rotation symmetric Boolean functions generated by x 1 x 2 and x 1 x 2 x 3 and gave a conjecture about the nonlinearities of the latter functions which was proved in 2010 by Zhang et al [6]. Furthermore, in 2009, the weights and nonlinearity of quadratic monomial rotation symmetric Boolean functions were completely determined by Kim et al in [4].…”

Section: Introductionmentioning

confidence: 96%

“…Split T .f n r;s / into 2 s equally sized portions and look at each portion individually. It is clear from equation (6) and the earlier work in this section that the j th portion of T .f n r;s / is simply Xor's a sequence onto T .g n s r;s / it does not matter if we do that first and then Xor on the monomials or Xor the monomials onto T .g n s r;s / first and then perform the operation. Then by concatenating each of these portions together we get the first equation in the lemma.…”

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confidence: 95%

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Recursive weights for some Boolean functions

Brown

Cusick

2012

Journal of Mathematical Cryptology

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Section: Introductionmentioning

confidence: 96%

mentioning

confidence: 95%

Recursive weights for some Boolean functions

Brown

Cusick

2012

Journal of Mathematical Cryptology

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“…In [2], Cusick and Stǎnicǎ conjectured that the nonlinearity of the cubic rotation symmetric Boolean function in n variables generated by the monomial x 0 x 1 x 2 is the same as its weight. Ciungu [1] and Zhang, Guo, Feng and Li [7] confirmed this conjecture. It is proved in [7] that the nonlinearity of the cubic rotation symmetric Boolean function in n variables generated by the monomial x 0 x e x 2e with e being a given positive integer equals its weight.…”

Section: Introductionmentioning

confidence: 79%

Nonlinearity of quartic rotation symmetric Boolean functions

Yang,

Wu,

Hong

2012

Preprint

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Nonlinearity of rotation symmetric Boolean functions is an important topic on cryptography algorithm. Let e ≥ 1 be any given integer. In this paper, we investigate the following question: Is the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial x 0 xex 2e x 3e equal to its weight? We introduce some new simple sub-functions and develop new technique to get several recursive formulas. Then we use these recursive formulas to show that the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial x 0 xex 2e x 3e is the same as its weight. So we answer the above question affirmatively. Finally, we conjecture that if l ≥ 4 is an integer, then the nonlinearity of the rotation symmetric Boolean function generated by the monomial x 0 xex 2e ...x le equals its weight.

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“…The simplest special case f = (1,2,3) of this conjecture was already stated in [9, p. 300], and this case was proved by a complicated argument in [20]. This work was extended to the quartic case f = (1,2,3,4) in [18].…”

Section: Conjecturesmentioning

confidence: 92%

“…This has led to many papers which study different aspects of the theory of rotation symmetric functions. Some relevant papers are [12,13,17,20]. In particular, Kavut and Yücel [13] give some applications of rotation symmetric functions in coding theory and the importance of rotation symmetric functions for cryptographic hashing is explained in [17].…”

Section: Introductionmentioning

confidence: 99%

Permutation equivalence of cubic rotation symmetric Boolean functions

Cusick

2014

International Journal of Computer Mathematics

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Rotation symmetric Boolean functions have been extensively studied for about 15 years because of their applications in cryptography and coding theory. Until recently 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 2009. For the much more complicated case of cubic rotation symmetric functions generated by a single monomial, the affine equivalence classes under permutations which preserve rotation symmetry were determined in 2011. It was conjectured then that the cubic equivalence classes are the same if all nonsingular affine transformations, not just permutations, are allowed. This conjecture is probably difficult, but here we take a step towards it by proving that the cubic affine equivalence classes found in 2011 are the same if all permutations, not just those preserving rotation symmetry, are allowed. The needed new idea uses the theory of circulant matrices.

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Proof of a conjecture about rotation symmetric functions

Cited by 14 publications

References 4 publications

Recursive weights for some Boolean functions

Recursive weights for some Boolean functions

Nonlinearity of quartic rotation symmetric Boolean functions

Permutation equivalence of cubic rotation symmetric Boolean functions

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