Computing the Distance Distribution of Systematic Nonlinear Codes

Guerrini, Eleonora; Orsini, Emmanuela; Sala, Massimiliano

doi:10.1142/s0219498810003884

Cited by 6 publications

(5 citation statements)

References 15 publications

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“…Here we present the main results from [SS07], [Sim09]. The same techniques are also applied in [GOS06] and [Gue05].…”

Section: Preliminary Resultsmentioning

confidence: 99%

Nonlinearity of Boolean functions: an algorithmic approach based on multivariate polynomials

Bellini¹,

Simonetti²,

Sala³

2014

Preprint

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We compute the nonlinearity of Boolean functions with Gröbner basis techniques, providing two algorithms: one over the binary field and the other over the rationals. We also estimate their complexity. Then we show how to improve our rational algorithm, arriving at a worst-case complexity of O(n2 n ) operations over the integers, that is, sums and doublings. This way, with a different approach, we reach the same complexity of established algorithms, such as those based on the fast Walsh transform.

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“…Here we present the main results from [SS07], [Sim09]. The same techniques are also applied in [GOS06] and [Gue05].…”

Section: Preliminary Resultsmentioning

confidence: 99%

Nonlinearity of Boolean functions: an algorithmic approach based on multivariate polynomials

Bellini¹,

Simonetti²,

Sala³

2014

Preprint

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“…Techniques that are similar to those presented in this work, have also been applied in [27,28] to compute the minimum distance and the weight distribution of, respectively, systematic nonlinear codes, and generic binary nonlinear codes.…”

Section: Related Workmentioning

confidence: 99%

Nonlinearity of Boolean Functions: An Algorithmic Approach Based on Multivariate Polynomials

2022

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We review and compare three algebraic methods to compute the nonlinearity of Boolean functions. Two of them are based on Gröbner basis techniques: the first one is defined over the binary field, while the second one over the rationals. The third method improves the second one by avoiding the Gröbner basis computation. We also estimate the complexity of the algorithms, and, in particular, we show that the third method reaches an asymptotic worst-case complexity of O(n2n) operations over the integers, that is, sums and doublings. This way, with a different approach, the same asymptotic complexity of established algorithms, such as those based on the fast Walsh transform, is reached.

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“…(Details will be introduced in Section 2.) The Boolean polynomials and the ring formed from them also play important roles in various fields: e.g., algebraic geometry [4,10,22], Boolean ideal and variety [25,33], circuit theory [35], cording theory [13,29], cryptography [7,18], and Gröbner basis [6,8,34]. Although the context differs depending on the field, solving a system of Boolean polynomial equations is a common problem.…”

Section: Introductionmentioning

confidence: 99%

A formula for systems of Boolean polynomial equations and applications to computational complexity

Machide¹

2019

Preprint

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It is known a method for transforming a system of Boolean polynomial equations to a single Boolean polynomial equation with less variables. In this paper, we improve the method, and show a formula in the Boolean polynomial ring for systems of Boolean polynomial equations. The formula has both operations of conjunction and disjunction recursively, and has a structure of binary tree.As corollaries, we prove computational complexity results for solving a given system. The complexity results include parameters similar to the bandwidth and rank of the matrix, and they are influenced by the order of variables in Boolean polynomials forming the system. Comparisons with existing complexity results of NP-complete problems related to systems of Boolean polynomial equations are also mentioned.

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Computing the Distance Distribution of Systematic Nonlinear Codes

Cited by 6 publications

References 15 publications

Nonlinearity of Boolean functions: an algorithmic approach based on multivariate polynomials

Nonlinearity of Boolean functions: an algorithmic approach based on multivariate polynomials

Nonlinearity of Boolean Functions: An Algorithmic Approach Based on Multivariate Polynomials

A formula for systems of Boolean polynomial equations and applications to computational complexity

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