Юрий Семенович Харин, Yurii Semenovich Kharin, В. А. Волошко, V. A. Voloshko
Мы рассматриваем две малопараметрические модели двоичных цепей Маркова высокого порядка и исследуем их способность аппроксимировать произвольные цепи Маркова высокого порядка. Введены два типа глобальных мер точности аппроксимации, для них и для рассматриваемых малопараметрических моделей получены теоретические и экспериментальные результаты. Новая состоятельная статистическая оценка параметров построена для малопараметрической модели на основе двухслойной искусственной нейронной сети.
{"title":"Об аппроксимации двоичных цепей Маркова высокого порядка малопараметрическими моделями","authors":"Юрий Семенович Харин, Yurii Semenovich Kharin, В. А. Волошко, V. A. Voloshko","doi":"10.4213/dm1710","DOIUrl":"https://doi.org/10.4213/dm1710","url":null,"abstract":"Мы рассматриваем две малопараметрические модели двоичных цепей Маркова высокого порядка и исследуем их способность аппроксимировать произвольные цепи Маркова высокого порядка. Введены два типа глобальных мер точности аппроксимации, для них и для рассматриваемых малопараметрических моделей получены теоретические и экспериментальные результаты. Новая состоятельная статистическая оценка параметров построена для малопараметрической модели на основе двухслойной искусственной нейронной сети.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"9 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74671858","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}
В работе исследуется сложность реализации систем мономов схемами композиции. Под сложностью в этой модели понимается минимальное количество операций, необходимое для вычисления системы мономов по переменным, при этом допускается многократное использование результатов промежуточных вычислений. Основной результат данной работы - для произвольной системы из трeх мономов от двух переменных без нулевых степеней установлена формула, выражающая сложность их совместной реализации схемами композиции с точностью до единицы.
{"title":"О сложности реализации системы из трeх мономов от двух переменных схемами композиции","authors":"Сергей Александрович Корнеев, S. A. Korneev","doi":"10.4213/dm1708","DOIUrl":"https://doi.org/10.4213/dm1708","url":null,"abstract":"В работе исследуется сложность реализации систем мономов схемами композиции. Под сложностью в этой модели понимается минимальное количество операций, необходимое для вычисления системы мономов по переменным, при этом допускается многократное использование результатов промежуточных вычислений. Основной результат данной работы - для произвольной системы из трeх мономов от двух переменных без нулевых степеней установлена формула, выражающая сложность их совместной реализации схемами композиции с точностью до единицы.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"41 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81739951","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}
Серафим Павлович Максаков, S. P. Maksakov, Марина Михайловна Сорокина, M. M. Sorokina
Для непустого множества $omega$ простых чисел В.А. Ведерниковым с помощью функциональных методов были построены $omega$-веерные формации групп. В работе изучаются решеточные свойства $omega$-веерных формаций конечных групп с направлением $delta$, удовлетворяющим условию $delta_{_{0}} leq delta$. Доказана алгебраичность решетки $omegadelta F_{theta}$ всех $omega$-веерных формаций с направлением $delta$ и $theta$-значным $omega$-спутником при условии, что решетка формаций $theta$ является алгебраической. В качестве следствий установлена алгебраичность решеток $omegadelta F$, $omegadelta F_{tau}$, $tauomegadelta F$, $omegadelta^{n} F$ $omega$-веерных формаций групп.
对于大量的omega / omega质数，v.a. vedernikov使用功能方法制造了1。它研究了以美元为基础的有限组的格栅性能，以满足delta的条件。证明了格栅的代数意义，即格栅是代数的。作为调查алгебраичн安装栅栏美元/ delta F $, $,欧米茄omega delta F_ { tau} $, $ tau omega,达美F $, $ omega / delta F ^ {n} $ $ / omega -轮流地层组美元。
{"title":"Об алгебраичности решеток $omega$-веерных формаций конечных групп","authors":"Серафим Павлович Максаков, S. P. Maksakov, Марина Михайловна Сорокина, M. M. Sorokina","doi":"10.4213/dm1659","DOIUrl":"https://doi.org/10.4213/dm1659","url":null,"abstract":"Для непустого множества $omega$ простых чисел В.А. Ведерниковым с помощью функциональных методов были построены $omega$-веерные формации групп. В работе изучаются решеточные свойства $omega$-веерных формаций конечных групп с направлением $delta$, удовлетворяющим условию $delta_{_{0}} leq delta$. Доказана алгебраичность решетки $omegadelta F_{theta}$ всех $omega$-веерных формаций с направлением $delta$ и $theta$-значным $omega$-спутником при условии, что решетка формаций $theta$ является алгебраической. В качестве следствий установлена алгебраичность решеток $omegadelta F$, $omegadelta F_{tau}$, $tauomegadelta F$, $omegadelta^{n} F$ $omega$-веерных формаций групп.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"203 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80289590","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}
We prove that each monotone (antimonotone) Boolean function in n variables can be modeled by a logic circuit with one additional input in the basis “conjunction, disjunction, negation” allowing a complete diagnostic test with length no more than n + 2 (no more than n + 1, respectively) relative to constant faults of type 1 at outputs of logic gates.
{"title":"Short complete diagnostic tests for circuits with one additional input in the standard basis","authors":"K. A. Popkov","doi":"10.17223/20710410/56/6","DOIUrl":"https://doi.org/10.17223/20710410/56/6","url":null,"abstract":"We prove that each monotone (antimonotone) Boolean function in n variables can be modeled by a logic circuit with one additional input in the basis “conjunction, disjunction, negation” allowing a complete diagnostic test with length no more than n + 2 (no more than n + 1, respectively) relative to constant faults of type 1 at outputs of logic gates.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67582800","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}
We suggest an elliptic curve modification of the undeniable signature introduced by D. Chaum and H. van-Antwerpen. The signature generation algorithm is supplemented with a preliminary stage of randomization. For signature verification and disavowal protocols, two options are offered. Theorems showing that these protocols meet their purpose have been proven. A method for converting an undeniable signature into a regular digital signature is described, illustrated by the Schnorr electronic signature scheme as an example.
{"title":"A randomized analog of Chaum - van Antwerpen undeniable signature","authors":"Pavel A. Polyschuk, Alexandr V. Cheremushkin","doi":"10.17223/20710410/57/3","DOIUrl":"https://doi.org/10.17223/20710410/57/3","url":null,"abstract":"We suggest an elliptic curve modification of the undeniable signature introduced by D. Chaum and H. van-Antwerpen. The signature generation algorithm is supplemented with a preliminary stage of randomization. For signature verification and disavowal protocols, two options are offered. Theorems showing that these protocols meet their purpose have been proven. A method for converting an undeniable signature into a regular digital signature is described, illustrated by the Schnorr electronic signature scheme as an example.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67583012","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}
In this paper, we address the problem of intrusion detection for modern web applications and mobile applications with the cloud-based server side, using malicious content detection in JSON data, which is currently one of the most popular data serialization and exchange formats between client and server parts of an application. We propose a method for building a JSON model for the given set of JSON objects capable of detection of structure and type anomalies. The model is based on the models for basic data types inside JSON collection objects and schema model that generalizes objects’ structure in the collection. We performed experiments using modifications of objects’ structures and insertions of code injection attack vectors such as SQL injections, OS command injections, and JavaScript/HTML injections. The analysis showed statistical significance between the model’s predictions and the presence of anomalies in the data gathered from the real web applications’ traffic. The quality of the model’s predictions was measured using the Matthews correlation coefficient (MCC). The MCC values computed on the data were close to one which indicates the model’s high efficiency in solving the problem of anomaly detection in JSON objects.
{"title":"Anomaly detection in JSON structured data","authors":"E. A. Shliakhtina, D. Gamayunov","doi":"10.17223/20710410/56/5","DOIUrl":"https://doi.org/10.17223/20710410/56/5","url":null,"abstract":"In this paper, we address the problem of intrusion detection for modern web applications and mobile applications with the cloud-based server side, using malicious content detection in JSON data, which is currently one of the most popular data serialization and exchange formats between client and server parts of an application. We propose a method for building a JSON model for the given set of JSON objects capable of detection of structure and type anomalies. The model is based on the models for basic data types inside JSON collection objects and schema model that generalizes objects’ structure in the collection. We performed experiments using modifications of objects’ structures and insertions of code injection attack vectors such as SQL injections, OS command injections, and JavaScript/HTML injections. The analysis showed statistical significance between the model’s predictions and the presence of anomalies in the data gathered from the real web applications’ traffic. The quality of the model’s predictions was measured using the Matthews correlation coefficient (MCC). The MCC values computed on the data were close to one which indicates the model’s high efficiency in solving the problem of anomaly detection in JSON objects.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67582747","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}
A toolkit and a method for reducing sequences of integers belonging to the class of factorial-generating recursions to a closed form are presented. The signs and properties of the modified factorial-generating recursion of one and two variables are determined. The best-known factorial-generating recursion of two variables is the sequence of Stirling numbers of the first kind. Modified hyperharmonic numbers are used to synthesize an analytical recursion model. The advantages of these numbers for constructing closed forms of factorial-generating recursions are revealed. An incomplete closed form of the sequence of Stirling numbers of the first kind is synthesized.
{"title":"APPLICATION OF MULTIHARMONIC NUMBERS FOR THE SYNTHESIS OF CLOSED FORMS OF PARAMETRICALLY MODIFIED FACTORIAL GENERATING SEQUENCES","authors":"I. V. Statsenko","doi":"10.17223/20710410/55/1","DOIUrl":"https://doi.org/10.17223/20710410/55/1","url":null,"abstract":"A toolkit and a method for reducing sequences of integers belonging to the class of factorial-generating recursions to a closed form are presented. The signs and properties of the modified factorial-generating recursion of one and two variables are determined. The best-known factorial-generating recursion of two variables is the sequence of Stirling numbers of the first kind. Modified hyperharmonic numbers are used to synthesize an analytical recursion model. The advantages of these numbers for constructing closed forms of factorial-generating recursions are revealed. An incomplete closed form of the sequence of Stirling numbers of the first kind is synthesized.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67582974","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}
The main types of algorithmic bookmarks are considered. A method for constructing asymmetric kleptographic bookmarks in the RSA key generator is presented, which allows the owner of the bookmark key (the developer or an authorized intelligence agency) to access a user key generated by an infected algorithm. Theorems illustrating the performance of the described algorithms are formulated, and the computational complexity of these algorithms is estimated. The resistance of the built tabs to some classes of attacks is demonstrated even if the adversary knows the methods used and has access to the source code of the key generator.
{"title":"Kleptographic (algorithmic) backdoors in the RSA key generator","authors":"A. V. Markelova","doi":"10.17223/20710410/55/2","DOIUrl":"https://doi.org/10.17223/20710410/55/2","url":null,"abstract":"The main types of algorithmic bookmarks are considered. A method for constructing asymmetric kleptographic bookmarks in the RSA key generator is presented, which allows the owner of the bookmark key (the developer or an authorized intelligence agency) to access a user key generated by an infected algorithm. Theorems illustrating the performance of the described algorithms are formulated, and the computational complexity of these algorithms is estimated. The resistance of the built tabs to some classes of attacks is demonstrated even if the adversary knows the methods used and has access to the source code of the key generator.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67582987","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}
The membership problem for finitely generated subgroups (subsemigroups) of groups (semigroups) is a classical algorithmic problem, actively studied for many decades. Already for sufficiently simple groups and semigroups, this problem becomes undecidable. For example, K. A. Mikhailova in 1966 proved the undecidability of the membership problem for finitely generated subgroups (hence and for subsemigroups) of a direct product F2×F2 of two free groups of rank 2. Since, by the well-known Sanov theorem, the group F2 has an exact representation by integer matrices of order 2, the group F2×F2 is a subgroup of the group GL4(ℤ) of integer matrices of order 4. It easily implies the undecidability of this problem for the group GLk(ℤ) for k ≥ 4. Undecidability of the membership problem for finitely generated subsemigroups of semigroups of integer matrices of order ≥ 3 follows from Paterson’s result proved in 1970. In this paper, we propose a strongly generic algorithm deciding the membership problem for semigroups of integer matrices of arbitrary order for inputs from a subset whose sequence of frequencies exponentially fast converges to 1 with increasing size.
群(半群)的有限生成子群(子半群)的隶属性问题是一个经典的算法问题，已被积极研究了几十年。对于足够简单的群和半群，这个问题已经变得无法确定。例如，K. a . Mikhailova(1966)证明了两个秩2的自由群的直接积F2×F2的有限生成子群(因此和子半群)的隶属性问题的不可判定性。由于根据著名的Sanov定理，群F2可以用2阶整数矩阵精确表示，因此群F2×F2是4阶整数矩阵群GL4(0)的子群。对于k≥4的群GLk(0)，可以很容易地推导出这个问题的不可判定性。从1970年Paterson证明的≥3阶整数矩阵半群的有限生成子半群的隶属性问题的不可判定性出发。本文提出了一种确定任意阶整数矩阵半群的隶属性问题的强通用算法，该半群的输入来自一个频率序列随大小的增加而指数快速收敛于1的子集。
{"title":"Generic complexity of the membership problem for semigroups of integer matrices","authors":"A. Rybalov","doi":"10.17223/20710410/55/7","DOIUrl":"https://doi.org/10.17223/20710410/55/7","url":null,"abstract":"The membership problem for finitely generated subgroups (subsemigroups) of groups (semigroups) is a classical algorithmic problem, actively studied for many decades. Already for sufficiently simple groups and semigroups, this problem becomes undecidable. For example, K. A. Mikhailova in 1966 proved the undecidability of the membership problem for finitely generated subgroups (hence and for subsemigroups) of a direct product F2×F2 of two free groups of rank 2. Since, by the well-known Sanov theorem, the group F2 has an exact representation by integer matrices of order 2, the group F2×F2 is a subgroup of the group GL4(ℤ) of integer matrices of order 4. It easily implies the undecidability of this problem for the group GLk(ℤ) for k ≥ 4. Undecidability of the membership problem for finitely generated subsemigroups of semigroups of integer matrices of order ≥ 3 follows from Paterson’s result proved in 1970. In this paper, we propose a strongly generic algorithm deciding the membership problem for semigroups of integer matrices of arbitrary order for inputs from a subset whose sequence of frequencies exponentially fast converges to 1 with increasing size.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67583122","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}
We consider the problem of constructing a median for an odd set of linear order relations defined on a finite set A = {a1,a2,..., an}, which is also sought in the class of linear order relations. We arrive at a similar problem when considering some group choice problems. The distance between binary relations is the Hamming distance between their adjacency matrices. In the case under consideration, the binary relation ρ, which has the minimum total distance to the given set of binary relations, is the median for these relations and, moreover, is unique. However, this median is not always transitive (and in this case is neither linear order, nor even a quasi-order), and therefore cannot be taken as a solution to a given problem. However, the median ρ necessarily belongs to the set LA[n] (of linear asymmetric binary relations on A), to which, in particular, all linear orders on A also belong. Some properties of binary relations from LA[n] are investigated. The concepts of “almost optimal” and Δ-optimal relations are introduced, which are linear orders and, at the same time, exact solutions of the stated problem. Algorithms for finding them are given, based on the obtained statements about binary relations from LA[n] and having polynomial computational complexity. An equivalence relation on the set LA[n] is considered, which allows one to divide this set into equivalence classes, the number of which Kn is much less than the number of elements in LA[n]. For example, |LA[5] | = 1024, K5 = 12. Thus, each binary relation from LA[n] is equivalent to exactly one of the Kn representatives of the equivalence classes and, therefore, has its main properties. But then the study of a wide class of problems can be reduced to considering a relatively small set of them. The process of finding the specified set of equivalence class representatives is illustrated for n = 2,3,4, 5. A method for solving the problem posed is also given, using the representation of binary relations in the form of graphs (the method of selecting the minimum sets of contour representatives in the median ρ), which has exponential computational complexity.
{"title":"Median for an odd number of linear order relations and its use in group choice problems","authors":"Victor N. Nefedov","doi":"10.17223/20710410/57/7","DOIUrl":"https://doi.org/10.17223/20710410/57/7","url":null,"abstract":"We consider the problem of constructing a median for an odd set of linear order relations defined on a finite set A = {a1,a2,..., an}, which is also sought in the class of linear order relations. We arrive at a similar problem when considering some group choice problems. The distance between binary relations is the Hamming distance between their adjacency matrices. In the case under consideration, the binary relation ρ, which has the minimum total distance to the given set of binary relations, is the median for these relations and, moreover, is unique. However, this median is not always transitive (and in this case is neither linear order, nor even a quasi-order), and therefore cannot be taken as a solution to a given problem. However, the median ρ necessarily belongs to the set LA[n] (of linear asymmetric binary relations on A), to which, in particular, all linear orders on A also belong. Some properties of binary relations from LA[n] are investigated. The concepts of “almost optimal” and Δ-optimal relations are introduced, which are linear orders and, at the same time, exact solutions of the stated problem. Algorithms for finding them are given, based on the obtained statements about binary relations from LA[n] and having polynomial computational complexity. An equivalence relation on the set LA[n] is considered, which allows one to divide this set into equivalence classes, the number of which Kn is much less than the number of elements in LA[n]. For example, |LA[5] | = 1024, K5 = 12. Thus, each binary relation from LA[n] is equivalent to exactly one of the Kn representatives of the equivalence classes and, therefore, has its main properties. But then the study of a wide class of problems can be reduced to considering a relatively small set of them. The process of finding the specified set of equivalence class representatives is illustrated for n = 2,3,4, 5. A method for solving the problem posed is also given, using the representation of binary relations in the form of graphs (the method of selecting the minimum sets of contour representatives in the median ρ), which has exponential computational complexity.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67583218","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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