Abstract An n-place function over a field Fq $ mathbf {F}_q $ with q elements is called maximally nonlinear if it has the largest nonlinearity among all q-valued n-place functions. We show that, for even n=2, a function is maximally nonlinear if and only if its nonlinearity is qn−1(q−1)−qn2−1 $ q^{n-1}(q - 1) - q^{frac n2-1} $ ; for n=1, the corresponding criterion for maximal nonlinearity is q − 2. For q>2 $ q gt 2 $ and even n=2, we describe the set of all maximally nonlinear quadratic functions and find its cardinality. In this case, all maximally nonlinear quadratic functions are quadratic bent functions and their number is smaller than the halved number of the bent functions.
{"title":"Maximally nonlinear functions over finite fields","authors":"V. G. Ryabov","doi":"10.1515/dma-2023-0005","DOIUrl":"https://doi.org/10.1515/dma-2023-0005","url":null,"abstract":"Abstract An n-place function over a field Fq $ mathbf {F}_q $ with q elements is called maximally nonlinear if it has the largest nonlinearity among all q-valued n-place functions. We show that, for even n=2, a function is maximally nonlinear if and only if its nonlinearity is qn−1(q−1)−qn2−1 $ q^{n-1}(q - 1) - q^{frac n2-1} $ ; for n=1, the corresponding criterion for maximal nonlinearity is q − 2. For q>2 $ q gt 2 $ and even n=2, we describe the set of all maximally nonlinear quadratic functions and find its cardinality. In this case, all maximally nonlinear quadratic functions are quadratic bent functions and their number is smaller than the halved number of the bent functions.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"41 - 53"},"PeriodicalIF":0.5,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49075006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We show that the system of elementary conjunctions Ωn,2k=K0,…,K2k−1 $ Omega_{n,2^k} = {K_0,ldots,K_{2^{k} -1}} $ such that each conjunction depends essentially on n variables and corresponds to some codeword of a linear (n, k)-code can be implemented by a separating contact circuit of complexity at most 2k+1 +4k(n − k) − 2. We also show that if a contact (1, 2k)-terminal network is separating and implements the system of elementary conjunctions Ωn,2k $ Omega_{n,2^k} $ , then the number of contacts in it is at least 2k+1 − 2.
{"title":"On implementation of some systems of elementary conjunctions in the class of separating contact circuits","authors":"Elena G. Krasulina","doi":"10.1515/dma-2023-0003","DOIUrl":"https://doi.org/10.1515/dma-2023-0003","url":null,"abstract":"Abstract We show that the system of elementary conjunctions Ωn,2k=K0,…,K2k−1 $ Omega_{n,2^k} = {K_0,ldots,K_{2^{k} -1}} $ such that each conjunction depends essentially on n variables and corresponds to some codeword of a linear (n, k)-code can be implemented by a separating contact circuit of complexity at most 2k+1 +4k(n − k) − 2. We also show that if a contact (1, 2k)-terminal network is separating and implements the system of elementary conjunctions Ωn,2k $ Omega_{n,2^k} $ , then the number of contacts in it is at least 2k+1 − 2.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"19 - 29"},"PeriodicalIF":0.5,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49396502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract For each Boolean function, we find a minimal possible value of the uniform width of a contact circuit implementing this function. We also show that, for almost all n-place Boolean functions, this value is equal to 3.
{"title":"On implementation of Boolean functions by contact circuits of minimal uniform width","authors":"K. A. Popkov","doi":"10.1515/dma-2022-0035","DOIUrl":"https://doi.org/10.1515/dma-2022-0035","url":null,"abstract":"Abstract For each Boolean function, we find a minimal possible value of the uniform width of a contact circuit implementing this function. We also show that, for almost all n-place Boolean functions, this value is equal to 3.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"403 - 415"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43672303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Invertible matrices over some quotient rings: identification, generation, and analysis","authors":"V. Vysotskaya, L. Vysotsky","doi":"10.1515/dma-2022-9999","DOIUrl":"https://doi.org/10.1515/dma-2022-9999","url":null,"abstract":"","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"419 - 419"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42164389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider automata in which transitions are labelled with arbitrary permutations. The language of such an automaton consists of compositions of permutations for all possible admissible computation paths. The membership problem for finite automata over symmetric groups is the following decision problem: does a given permutation belong to the language of a given automaton? We show that this problem is NP-complete. We also propose an efficient algorithm for the case of strongly connected automata.
{"title":"On the membership problem for finite automata over symmetric groups","authors":"Arthur A. Khashaev","doi":"10.1515/dma-2022-0033","DOIUrl":"https://doi.org/10.1515/dma-2022-0033","url":null,"abstract":"Abstract We consider automata in which transitions are labelled with arbitrary permutations. The language of such an automaton consists of compositions of permutations for all possible admissible computation paths. The membership problem for finite automata over symmetric groups is the following decision problem: does a given permutation belong to the language of a given automaton? We show that this problem is NP-complete. We also propose an efficient algorithm for the case of strongly connected automata.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"383 - 389"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47647247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Synthesis of reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs","authors":"D. Zakablukov","doi":"10.1515/dma-2022-9998","DOIUrl":"https://doi.org/10.1515/dma-2022-9998","url":null,"abstract":"","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"418 - 418"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43255591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs are considered. For such circuits implementing maps f : Z2n $begin{array}{} displaystyle mathbb Z_2^n end{array}$ → Z2n $begin{array}{} displaystyle mathbb Z_2^n end{array}$, we study the Shannon complexity function L(n, q) under the condition that the number of additional inputs is q = O(n2). For this range of q, it is shown that L(n, q) ≍ n2n / log2 n. The growth order L(n, q) ≍ n2n / log2 (n + q) for all q ≲ n2n−⌈n/ϕ(n)⌉, where ϕ(n) → ∞ and n / ϕ(n) − log2 n → ∞ as n → ∞, is evaluated.
{"title":"Synthesis of reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs","authors":"D. Zakablukov","doi":"10.1515/dma-2022-0037","DOIUrl":"https://doi.org/10.1515/dma-2022-0037","url":null,"abstract":"Abstract Reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs are considered. For such circuits implementing maps f : Z2n $begin{array}{} displaystyle mathbb Z_2^n end{array}$ → Z2n $begin{array}{} displaystyle mathbb Z_2^n end{array}$, we study the Shannon complexity function L(n, q) under the condition that the number of additional inputs is q = O(n2). For this range of q, it is shown that L(n, q) ≍ n2n / log2 n. The growth order L(n, q) ≍ n2n / log2 (n + q) for all q ≲ n2n−⌈n/ϕ(n)⌉, where ϕ(n) → ∞ and n / ϕ(n) − log2 n → ∞ as n → ∞, is evaluated.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"439 - 444"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42261658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study conditions of the asymptotic normality of the number of repetitions (pairs of equal values) in a segment of strict sense stationary random sequence of values from {1, 2, …, N} satisfying the strong uniform mixing condition. It is shown that under natural conditions for the number of repetitions to be asymptotically normal as the length of the segment tends to infinity it is necessary for the stationary distribution to be different from the equiprobable one. Under additional conditions the accuracy of the normal approximation in the uniform metrics is estimated.
{"title":"On the asymptotic normality conditions for the number of repetitions in a stationary random sequence","authors":"V. Mikhailov, N. Mezhennaya, A. Volgin","doi":"10.1515/dma-2022-0034","DOIUrl":"https://doi.org/10.1515/dma-2022-0034","url":null,"abstract":"Abstract We study conditions of the asymptotic normality of the number of repetitions (pairs of equal values) in a segment of strict sense stationary random sequence of values from {1, 2, …, N} satisfying the strong uniform mixing condition. It is shown that under natural conditions for the number of repetitions to be asymptotically normal as the length of the segment tends to infinity it is necessary for the stationary distribution to be different from the equiprobable one. Under additional conditions the accuracy of the normal approximation in the uniform metrics is estimated.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"391 - 401"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47939711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study matrices over quotient rings modulo univariate polynomials over a two-element field. Lower bounds for the fraction of the invertible matrices among all such matrices of a given size are obtained. An efficient algorithm for calculating the determinant of matrices over these quotient rings and an algorithm for generating random invertible matrices (with uniform distribution on the set of all invertible matrices) are proposed and analyzed. A special version of the latter algorithm for quotient rings modulo polynomials of form xr − 1 is considered and analyzed. These methods may find practical applications for generating keys of cryptographic schemes based on quasi-cyclic codes such as LEDAcrypt.
{"title":"Invertible matrices over some quotient rings: identification, generation, and analysis","authors":"V. Vysotskaya, L. Vysotsky","doi":"10.1515/dma-2022-0036","DOIUrl":"https://doi.org/10.1515/dma-2022-0036","url":null,"abstract":"Abstract We study matrices over quotient rings modulo univariate polynomials over a two-element field. Lower bounds for the fraction of the invertible matrices among all such matrices of a given size are obtained. An efficient algorithm for calculating the determinant of matrices over these quotient rings and an algorithm for generating random invertible matrices (with uniform distribution on the set of all invertible matrices) are proposed and analyzed. A special version of the latter algorithm for quotient rings modulo polynomials of form xr − 1 is considered and analyzed. These methods may find practical applications for generating keys of cryptographic schemes based on quasi-cyclic codes such as LEDAcrypt.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"423 - 438"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46597060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some cardinality estimates for the set of correlation-immune Boolean functions","authors":"E. Karelina","doi":"10.1515/dma-2022-9997","DOIUrl":"https://doi.org/10.1515/dma-2022-9997","url":null,"abstract":"","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"417 - 417"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45802600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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