A note on the properties of associated Boolean functions of quadratic APN functions

Let F be a quadratic APN function in n variables. The associated Boolean function yf in 2n variables (yF(a, b) = 1 if a = 0 and equation F(x) + F(x + a) = b has solutions) has the form yF(a, b) = Ф,р(a) • b + ^F(a) +1 for appropriate functions Ф,р : Fn Fn and ^f : Fn F2. We summarize the known results and prove new ones regarding properties of Ф,р and ^F. For instance, we prove that degree of Ф,р is either n or less or equal to n - 2. Based on computation experiments, we formulate a conjecture that degree of any component function of Ф,р is n — 2. We show that this conjecture is based on two other conjectures of independent interest

Properties of associated Boolean functions of quadratic APN functions

For a function F : Fn Fn, it is defined the associated Boolean function yF in 2n variables as follows: yF(a, b) = 1 if a = 0 and equation F(x) + F(x + a) = b has solutions. A vectorial Boolean function F from F2n to F2n is called almost perfect nonlinear (APN) if equation F(x) + F(x + a) = b has at most 2 solutions for all vectors a, b 6 F2n, where a is nonzero. In case when F is a quadratic APN function its associated function has the form yF(a, b) = Фр(a) • b + ^F(a) + 1 for appropriate functions Фр : Fn Fn and : Fn F2. We study properties of functions Фр and ^F, in particular their degrees

Models of Ecumenical Dialogue in the Discourse of the Russian Orthodox Church of the Moscow Patriarchate

The article is devoted to the models of ecumenical dialogue in the Russian context on the example of the Russian Orthodox Church of the Moscow Patriarchate. The wide-spread and flourishing discussions about ecumenism in the religious studies and theological fields constitute the problem of this study. The author investigates the problem with the aim to identify existing models of an ecumenical dialogue. Following from this purpose the most significant and discussed concepts of ecumenical dialogue in Russia are noted. The emphasis is placed on the widespread concept of two ecumenisms, most fully disclosed by A. V. Shishkov, and related discussions in the scientific and theological fields. Also, a discussion in the church discourse was revealed, which is based on the concepts of representatives of the Russian Orthodox Church: prot. M. Kozlov, mitr. Hilarion (Alfeyev), fr. Georges Florovsky, archm. Sophronia (Sakharova) and others. To outline the broad picture, there was established the position of the Roman Catholic Church as well as its messages addressed to Orthodoxy. In addition, three main trends in the Russian Orthodox Church in relation to ecumenism are identified and described. The first is anti-ecumenism, its members are against any dialogue with non-Orthodoxy. The second is moderate ecumenism, where participants advocate the idea of acceptable and unacceptable forms of ecumenism without making edits to the Church doctrine, and the third are modernist ecumenists who are ready for changes in the teaching of the Church for the sake of the dialogue. Then, there was a transition to the term ecumenism in the context of modern conceptions about the phenomenon and also in the context of the history of the ecumenical movement of the 20th century. The author makes a conclusion about a fundamental importance of the idea of the unity of Christians for the identification of models of ecumenical dialogue. The unity can be determined as unity in one Church, as well as unity within a common space of communication without crossing borders and making one united church. It is noted that the existing concepts and terminology do not reflect this aspect, thus the author suggests the possibility of using such terms as adaptive and integrative dialogue, and lays out prospects for solving the problem

ACTIVITY OF THE CITY METHODICAL ASSOCIATION OF TEACHERS OF THE COURSE «FUNDAMENTALS OF RELIGIOUS CULTURES AND SECULAR ETHICS»

The article presents the experience of the City Methodological Association of Teachers of the module «Fundamentals of Religious Culture and Secular Ethics»; the directions of activity and interaction with representatives of the pedagogical community and the Russian Orthodox Church are disclosed.В статье представлен опыт работы Городского методического объединения учителей модуля «Основы религиозной культуры и светской этики»; раскрыты направления деятельности и взаимодействия с представителями педагогического сообщества и русской православной церкви

Mathematical methods in solutions of the problems from the Third International Students' Olympiad in Cryptography

The mathematical problems and their solutions of the Third International Students' Olympiad in Cryptography NSUCRYPTO'2016 are presented. We consider mathematical problems related to the construction of algebraic immune vectorial Boolean functions and big Fermat numbers, problems about secrete sharing schemes and pseudorandom binary sequences, biometric cryptosystems and the blockchain technology, etc. Two open problems in mathematical cryptography are also discussed and a solution for one of them proposed by a participant during the Olympiad is described. It was the first time in the Olympiad history

Informatization of Extracurricular Activities of Schoolchildren. the Use of Information Technology in the Organization of City Events in the Serov Urban District

В статье описывается опыт работы объединения учителей, преподающих курс «Основы религиозных культур и светской этики» в школах Серовского городского округа. А также приводятся результаты анкетирования пятиклассников по вопросам идеалов и ценностей, на которые ориентируются современные школьники.The article describes the experience of the union of teachers teaching the course "Fundamentals of Religious Cultures and Secular Ethics" in schools in the Serov urban district. And also the results of a questionnaire survey of fifth-graders on the ideals and values that modern schoolchildren are guided by are presented

The Seventh International Olympiad in Cryptography: problems and solutions

The International Olympiad in Cryptography NSUCRYPTO is the unique Olympiad containing scientific mathematical problems for professionals, school and university students from any country. Its aim is to involve young researchers in solving curious and tough scientific problems of modern cryptography. In 2020, it was held for the seventh time. Prizes and diplomas were awarded to 84 participants in the first round and 49 teams in the second round from 32 countries. In this paper, problems and their solutions of NSUCRYPTO'2020 are presented. We consider problems related to attacks on ciphers and hash functions, protocols, permutations, primality tests, etc. We discuss several open problems on JPEG encoding, Miller -- Rabin primality test, special bases in the vector space, AES-GCM. The problem of a modified Miller -- Rabin primality test was solved during the Olympiad. The problem for finding special bases was partially solved