Finding periods of Zhegalkin polynomials
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Svetlana N. Selezneva
Abstract
A period of a Boolean function f(x1, …, xn) is a binary n-tuple a = (a1, …, an) that satisfies the identity f(x1 + a1, …, xn + an) = f(x1, …, xn). A Boolean function is periodic if it admits a nonzero period. We propose an algorithm that takes the Zhegalkin polynomial of a Boolean function f(x1, …, xn) as the input and finds a basis of the space of all periods of f(x1, …, xn). The complexity of this algorithm is nO(d), where d is the degree of the function f. As a corollary we show that a basis of the space of all periods of a Boolean function specified by the Zhegalkin polynomial of a bounded degree may be found with complexity which is polynomial in the number of variables.
Originally published in Diskretnaya Matematika (2021) 33, №3, 107–120 (in Russian).
Funding statement: Research was supported by RBRF, project 19-01-00200-a, and by the Ministry of Education and Science of the Russian Federation as part of the program of Moscow Center of Fundamental and Applied Mathematics under the agreement 075-15-2019-1621
Acknowledgment
The author thanks Prof. V. B. Alekseev for discussion of the research and useful remarks.
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The site-perimeter of compositions
- Some cardinality estimates for the set of correlation-immune Boolean functions
- Continuality of classes of functions in multivalued logic with minimal logarithmic growth rate
- Completeness criterion with respect to the enumeration closure operator in the three-valued logic
- Some families of closed classes in Pk defined by additive formulas
- Finding periods of Zhegalkin polynomials
- Admissible and Bayes decisions with fuzzy-valued losses
Articles in the same Issue
- Frontmatter
- The site-perimeter of compositions
- Some cardinality estimates for the set of correlation-immune Boolean functions
- Continuality of classes of functions in multivalued logic with minimal logarithmic growth rate
- Completeness criterion with respect to the enumeration closure operator in the three-valued logic
- Some families of closed classes in Pk defined by additive formulas
- Finding periods of Zhegalkin polynomials
- Admissible and Bayes decisions with fuzzy-valued losses
