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":null,"pages":null},"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}
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":null,"pages":null},"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 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":null,"pages":null},"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}
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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
Abstract We present a class of irregular languages defined by means of the change of number systems.
摘要我们提出了一类由数制变化定义的不规则语言。
{"title":"On a class of irregular languages","authors":"Kirill I. Groshev","doi":"10.1515/dma-2022-0031","DOIUrl":"https://doi.org/10.1515/dma-2022-0031","url":null,"abstract":"Abstract We present a class of irregular languages defined by means of the change of number systems.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42106751","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 Estimates for the cardinality of the set of correlation-immune n-ary Boolean functions with fixed weight are obtained.
得到了一组具有固定权值的相关免疫n元布尔函数集的基数的抽象估计。
{"title":"Some cardinality estimates for the set of correlation-immune Boolean functions","authors":"E. Karelina","doi":"10.1515/dma-2022-0032","DOIUrl":"https://doi.org/10.1515/dma-2022-0032","url":null,"abstract":"Abstract Estimates for the cardinality of the set of correlation-immune n-ary Boolean functions with fixed weight are obtained.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43018702","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 local probabilities of lower deviations for branching process Zn=Xn,1+⋯+Xn,Zn−1 ${{Z}_{n}}={{X}_{n,1}}+cdots +{{X}_{n,{{Z}_{n-1}}}}$in random environment η. We assume that η is a sequence of independent identically distributed random variables and for fixed environment η the distributions of variables Xi,j are geometric ones.We suppose that the associated random walk Sn=ξ1+⋯+ξn ${{S}_{n}}={{xi }_{1}}+cdots +{{xi }_{n}}$has positive mean μ and satisfies left-hand Cramer’s condition Eexp(hξi)<∞ if h−
{"title":"Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants","authors":"Konstantin Yu. Denisov","doi":"10.1515/dma-2022-0026","DOIUrl":"https://doi.org/10.1515/dma-2022-0026","url":null,"abstract":"Abstract We consider local probabilities of lower deviations for branching process Zn=Xn,1+⋯+Xn,Zn−1 ${{Z}_{n}}={{X}_{n,1}}+cdots +{{X}_{n,{{Z}_{n-1}}}}$in random environment η. We assume that η is a sequence of independent identically distributed random variables and for fixed environment η the distributions of variables Xi,j are geometric ones.We suppose that the associated random walk Sn=ξ1+⋯+ξn ${{S}_{n}}={{xi }_{1}}+cdots +{{xi }_{n}}$has positive mean μ and satisfies left-hand Cramer’s condition Eexp(hξi)<∞ if h−<h<0 $mathbf{E}exp left( h{{xi }_{i}} right)<infty text{ if }{{h}^{-}}<h<0$for some h−<−1. ${{h}^{-}}<-1.$Under these assumptions, we find the asymptotic representation of local probabilities P(Zn=⌊ exp(θn) ⌋) for θ∈[ θ1,θ2 ]⊂(μ−;μ) $mathbf{P}left( {{Z}_{n}}=leftlfloor exp (theta n) rightrfloor right)text{ for }theta in left[ {{theta }_{1}},{{theta }_{2}} right]subset left( {{mu }^{-}};mu right)$for some non-negative μ−.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46693708","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 In 1988, Friese et al. put forward lower estimates for the lengths of shortest nonzero vectors for “almost all” lattices of some families in the dimension 3. In 2004, the author of the present paper obtained a similar result for the dimension 4. Here by means of results obtained in part of the paper we show that these estimates also hold in the dimension 5.
{"title":"Estimates of lengths of shortest nonzero vectors in some lattices, II","authors":"A. S. Rybakov","doi":"10.1515/dma-2022-0028","DOIUrl":"https://doi.org/10.1515/dma-2022-0028","url":null,"abstract":"Abstract In 1988, Friese et al. put forward lower estimates for the lengths of shortest nonzero vectors for “almost all” lattices of some families in the dimension 3. In 2004, the author of the present paper obtained a similar result for the dimension 4. Here by means of results obtained in part of the paper we show that these estimates also hold in the dimension 5.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46127729","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 Reed–Muller codes we consider subcodes of codimension 1. A classification of Hadamard products of such subcodes is obtained. With the use of this classification it has been shown that in most cases the problem of recovery of the secret key of a code-based cryptosystem employing such subcodes is equivalent to the problem of recovery of the secret key of the same cryptosystem based on Reed–Muller codes, which is known to be tractable.
{"title":"Classification of Hadamard products of one-codimensional subcodes of Reed–Muller codes","authors":"I. Chizhov, M. Borodin","doi":"10.1515/dma-2022-0025","DOIUrl":"https://doi.org/10.1515/dma-2022-0025","url":null,"abstract":"Abstract For Reed–Muller codes we consider subcodes of codimension 1. A classification of Hadamard products of such subcodes is obtained. With the use of this classification it has been shown that in most cases the problem of recovery of the secret key of a code-based cryptosystem employing such subcodes is equivalent to the problem of recovery of the secret key of the same cryptosystem based on Reed–Muller codes, which is known to be tractable.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46591335","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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