On a conjecture for balanced symmetric Boolean functions

Cusick, Thomas W.; Li, Yuan; Stănică, Pantelimon

doi:10.1515/jmc.2009.017

Cited by 21 publications

(36 citation statements)

References 3 publications

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“…indeed, Tr mid is the identity on GF (2 k ) because the degree m = [GF (2 n ) : GF (2 k )] is odd. In conclusion, the claimed equivalence will follow once we show that an element a ∈ GF (2 n ) satisfying (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15), if it exists, can be chosen in GF (2 k ). To see this, let a ∈ GF (2 n ) be parity-reversing.…”

Section: General Rs Quadraticsmentioning

confidence: 89%

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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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Section: General Rs Quadraticsmentioning

confidence: 89%

“…According to Lemma 3.7 the function Q n is balanced if and only if there is a Q-parity-reversing element a ∈ GF (2 n ), i.e. one satisfying Q n (x + a) = Q n (x) + 1, ∀x ∈ GF (2 n ); (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) The factorization (5)(6)(7)(8)(9)(10)(11) implies that…”

Section: General Rs Quadraticsmentioning

confidence: 99%

Affine equivalence for quadratic rotation symmetric Boolean functions

Chirvăsitu

Cusick

2020

Des. Codes Cryptogr.

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“…ν 2 (j ) for j = 1, 2, 3, · · · . In other words, it looks like the sequence {t j } j ∈N satisfies the recurrence The general solution to (11) is Fig. 4 you can see a graphical representation of ν 2 (S(σ 32m−4,16m−2 )) versus w 2 (m) for m odd.…”

Section: The Casementioning

confidence: 99%

“…In [11], Cusick, Li and Stǎnicǎ proved their conjecture for elementary symmetric functions of odd degree. They also proved the conjecture for many of these functions when the weight of the degree is 1 or 2.…”

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confidence: 99%

A divisibility approach to the open boundary cases of Cusick-Li-Stǎnicǎ’s conjecture

2015

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In this paper we compute the exact 2-divisibility of exponential sums associated to elementary symmetric Boolean functions. Our computation gives an affirmative answer to most of the open boundary cases of Cusick-Li-Stǎnicǎ's conjecture. As a byproduct, we prove that the 2-divisibility of these families satisfies a linear recurrence. In particular, we provide a new elementary method to compute 2-divisibility of symmetric Boolean functions.

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“…For elementary symmetric Boolean functions, Cusick et al proposed a conjecture in [3] about the nonexistence of nonlinear balanced elementary symmetric Boolean functions σ n,d except for n = l · 2 t+1 − 1 and d = 2 t , where t and l are any positive integers. They also obtained many results towards the conjecture in [4]. Later in [6] for sufficient large number of variables, but a certain bound had not been obtained.…”

Section: Introductionmentioning

confidence: 99%

Recent Results on Balanced Symmetric Boolean Functions

Gao

Guo²,

Zhao³

2016

IEEE Trans. Inform. Theory

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In this paper we prove all balanced symmetric Boolean functions of fixed algebraic degree are trivial when the number of variables grows large enough. We also present the nonexistence of trivial balanced elementary symmetric Boolean function except for n = 2 t+1 l − 1 and d = 2 t , where t and l are any nonnegative integers, which shows Cusick's conjecture for balanced elementary symmetric Boolean function is exactly the conjecture that all balanced elementary symmetric Boolean functions are trivial. In additional, we obtain a bound n0, which depends only on the algebraic degree, such that Cusick's conjecture holds for any n > n0.

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On a conjecture for balanced symmetric Boolean functions

Abstract: Abstract. We give some results towards the conjecture that X(2 t , 2 t+1 − 1) are the only nonlinear balanced elementary symmetric polynomials over GF (2), where t and are any positive integers and

Cited by 21 publications

References 3 publications

Affine equivalence for quadratic rotation symmetric Boolean functions

Affine equivalence for quadratic rotation symmetric Boolean functions

A divisibility approach to the open boundary cases of Cusick-Li-Stǎnicǎ’s conjecture

Recent Results on Balanced Symmetric Boolean Functions

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