Abstract V. I. Solodovnikov had employed the shift-composition operation to investigate homomorphisms of shift registers into linear automata; in his papers, conditions for the absence of nontrivial inner homomorphisms of shift registers were derived. An essential role was played by the condition of linearity of the left component of the shift-composition operation in the corresponding polynomial decomposition. In this paper we consider the case where the left component is a function belonging to a wider class, which includes the class of linear functions.
{"title":"Decomposition of polynomials using the shift-composition operation","authors":"V. Nozdrunov","doi":"10.1515/dma-2023-0009","DOIUrl":"https://doi.org/10.1515/dma-2023-0009","url":null,"abstract":"Abstract V. I. Solodovnikov had employed the shift-composition operation to investigate homomorphisms of shift registers into linear automata; in his papers, conditions for the absence of nontrivial inner homomorphisms of shift registers were derived. An essential role was played by the condition of linearity of the left component of the shift-composition operation in the corresponding polynomial decomposition. In this paper we consider the case where the left component is a function belonging to a wider class, which includes the class of linear functions.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"87 - 97"},"PeriodicalIF":0.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44534856","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 construct an explicit finite basis for admissible rules in K-saturated tabular logics that extend the logic Grz.
本文构造了k饱和表逻辑中可容许规则的显式有限基，扩展了逻辑Grz。
{"title":"Explicit basis for admissible rules in K-saturated tabular logics","authors":"V. V. Rimatskiĭ","doi":"10.1515/dma-2023-0011","DOIUrl":"https://doi.org/10.1515/dma-2023-0011","url":null,"abstract":"Abstract We construct an explicit finite basis for admissible rules in K-saturated tabular logics that extend the logic Grz.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"105 - 115"},"PeriodicalIF":0.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48169772","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 Nonlinear invariants of round transformations in XSL-schemes are studied. The emphasis is on invariants which may be found by means of the approach suggested at the conference ASIACRYPT 2016. Some known results on the inertia groups of decomposable functions are used to describe conditions on S-boxes and matrices of XSL-schemes which are necessary for the existence of such invariants. It is shown that for a number of schemes these conditions are not satisfied.
{"title":"On the existence of special nonlinear invariants for round functions of XSL-ciphers","authors":"D. Burov","doi":"10.1515/dma-2023-0007","DOIUrl":"https://doi.org/10.1515/dma-2023-0007","url":null,"abstract":"Abstract Nonlinear invariants of round transformations in XSL-schemes are studied. The emphasis is on invariants which may be found by means of the approach suggested at the conference ASIACRYPT 2016. Some known results on the inertia groups of decomposable functions are used to describe conditions on S-boxes and matrices of XSL-schemes which are necessary for the existence of such invariants. It is shown that for a number of schemes these conditions are not satisfied.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"65 - 75"},"PeriodicalIF":0.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42661238","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 the branching process Zn=Xn,1+⋯+XnZn−1 $ Z_{n} =X_{n, 1} + dotsb +X_{nZ_{n-1}} $, in random environmentsη, where η is a sequence of independent identically distributedvariables, for fixed η the random variables Xi, j areindependent, have the geometric distribution. We suppose that the associated random walk Sn=ξ1+⋯+ξn $ S_n = xi_1 + dotsb + xi_n $ has positive meanμ,0 < h(μ;μ+) and someμ+.
{"title":"Asymptotic local probabilities of large deviations for branching process in random environment with geometric distribution of descendants","authors":"Konstantin Yu. Denisov","doi":"10.1515/dma-2023-0008","DOIUrl":"https://doi.org/10.1515/dma-2023-0008","url":null,"abstract":"Abstract We consider the branching process Zn=Xn,1+⋯+XnZn−1 $ Z_{n} =X_{n, 1} + dotsb +X_{nZ_{n-1}} $, in random environmentsη, where η is a sequence of independent identically distributedvariables, for fixed η the random variables Xi, j areindependent, have the geometric distribution. We suppose that the associated random walk Sn=ξ1+⋯+ξn $ S_n = xi_1 + dotsb + xi_n $ has positive meanμ,0 < h<h+satisfies the right-hand Cramer’s condition Eexp(hξi) < ∞ for, some h+. Under theseassumptions, we find the asymptotic representation for local probabilities P(Zn=⌊exp(θ n)⌋) for θ ∈ [θ1, θ2]⊂</given−names><x> </x><surname>(μ;μ+) and someμ+.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"77 - 86"},"PeriodicalIF":0.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45029747","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 identify the classes of Boolean functions that may be implemented by easily testable circuits in the Zhegalkin basis for constant type-1 faults on outputs of gates. An upper estimate for the length of a complete fault detection test for three-place functions is obtained.
{"title":"Some classes of easily testable circuits in the Zhegalkin basis","authors":"Y. Borodina","doi":"10.1515/dma-2023-0001","DOIUrl":"https://doi.org/10.1515/dma-2023-0001","url":null,"abstract":"Abstract We identify the classes of Boolean functions that may be implemented by easily testable circuits in the Zhegalkin basis for constant type-1 faults on outputs of gates. An upper estimate for the length of a complete fault detection test for three-place functions is obtained.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"1 - 6"},"PeriodicalIF":0.5,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41420779","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 sequences of independent random variables having a Bernoulli distribution with parameter p the limit joint distribution of statistics of four tests of the NIST statistical package (« Monobit Test », « Frequency Test within a Block », « Runs Test » and a generalization of « Non-overlapping Template Matching Test ») is obtained. Conditions of asymptotic uncorrelatedness and/or asymptotic independence of these statistics are given.
摘要对于具有参数为p的伯努利分布的独立随机变量序列，获得了NIST统计包的四个测试（“Monobit Test”、“Frequency Test in a Block”、“Runs Test”和“Non-duclapping Template Matching Test”的推广）的统计的极限联合分布。给出了这些统计量的渐近不迟性和/或渐近独立性的条件。
{"title":"The limit joint distributions of statistics of four tests of the NIST package","authors":"Maksim P. Savelov","doi":"10.1515/dma-2023-0006","DOIUrl":"https://doi.org/10.1515/dma-2023-0006","url":null,"abstract":"Abstract For sequences of independent random variables having a Bernoulli distribution with parameter p the limit joint distribution of statistics of four tests of the NIST statistical package (« Monobit Test », « Frequency Test within a Block », « Runs Test » and a generalization of « Non-overlapping Template Matching Test ») is obtained. Conditions of asymptotic uncorrelatedness and/or asymptotic independence of these statistics are given.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"55 - 64"},"PeriodicalIF":0.5,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44800901","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 A classical scheme of random equiprobable allocations of n particles into N cells is considered. We find an asymptotic formula for the probability that the number of empty cells is equal to k under the condition that n, N → ∞ in such a way that n/(N − k) is bounded and separated from 1 from below.
{"title":"Local limit theorem for the number of empty cells in a scheme of random equiprobable allocations","authors":"O. P. Orlov","doi":"10.1515/dma-2023-0004","DOIUrl":"https://doi.org/10.1515/dma-2023-0004","url":null,"abstract":"Abstract A classical scheme of random equiprobable allocations of n particles into N cells is considered. We find an asymptotic formula for the probability that the number of empty cells is equal to k under the condition that n, N → ∞ in such a way that n/(N − k) is bounded and separated from 1 from below.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"31 - 39"},"PeriodicalIF":0.5,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42357219","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 The asymptotic behavior of general independence numbers of random hypergraphs for the binomial model is studied. We prove that for some types of parameter variations the distribution of independence numbers is concentrated on two neighboring values.
{"title":"On the concentration of the independence numbers of random hypergraphs","authors":"I. O. Denisov, D. A. Shabanov","doi":"10.1515/dma-2023-0002","DOIUrl":"https://doi.org/10.1515/dma-2023-0002","url":null,"abstract":"Abstract The asymptotic behavior of general independence numbers of random hypergraphs for the binomial model is studied. We prove that for some types of parameter variations the distribution of independence numbers is concentrated on two neighboring values.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"7 - 18"},"PeriodicalIF":0.5,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43077547","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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