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":null,"pages":null},"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}
Pub Date : 2023-02-01DOI: 10.1515/dma-2023-frontmatter1
{"title":"Frontmatter","authors":"","doi":"10.1515/dma-2023-frontmatter1","DOIUrl":"https://doi.org/10.1515/dma-2023-frontmatter1","url":null,"abstract":"","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134941007","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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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":null,"pages":null},"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":null,"pages":null},"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":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":"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":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":"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}
{"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":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":"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}
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