Extended Euclidean Algorithm and CRT Algorithm

In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based on the source code of the NZMATH, a number theory oriented calculation system developed by Tokyo Metropolitan University [8].This work was supported by JSPS KAKENHI 21240001 and 22300285Okazaki Hiroyuki - Shinshu University, Nagano, JapanAoki Yosiki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.NZMATH development Group. http://tnt.math.se.tmu.ac.jp/nzmath/.Donald E. Knuth. Art of Computer Programming. Volume 2: Seminumerical Algorithms, 3rd Edition, Addison-Wesley Professional, 1997.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

On Experience of Using Distance Learning Technologies for Teaching Cryptology

Part 3: Tools and Applications for TeachingInternational audienceThe necessity of using Distance Learning (DL) for teaching cryptology is analyzed. The modern features of applying different DL approaches to solve this task are extracted. The NRNU MEPhI’s experience in creating mass-oriented DL project called Cryptowiki.net is described; its structure and assignments implemented by the students of cryptologic courses are shown. The related works are presented. Cryptowiki.net’s difference from the analogs is stressed out. The main findings of the research are formulated in conclusion

Τα Στοχαστικά Μαθηματικά στα Προγράμματα Σπουδών διαφόρων χωρών

Η σημασία της ανάπτυξης στατιστικού γραμματισμού και της στατιστικής σκέψης έχει αποτελέσει στόχο στα σύγχρονα Προγράμματα Σπουδών σε όλο τον κόσμο. Η παρούσα μελέτη επιχειρεί να μελετήσει τον τρόπο με τον οποίο εμφανίζονται η στατιστική σκέψη και ο στατιστικός γραμματισμός σε κάποια Προγράμματα Σπουδών Αγγλοσαξονικών χωρών όπου τα Στοχαστικά Μαθη-ματικά, δηλαδή η Στατιστική και οι Πιθανότητες, έχουν εισαχθεί πάνω από δύο δεκαετίες στην υποχρεωτική εκπαίδευση.The importance of developing statistical literacy and statistical thinking has been a goal in contemporary curricula worldwide. This study attempts to study how they appear statistical thinking and statistical literacy in some curricula of Anglo-Saxon countries where stochastics, i.e. Statistics and Probability, intro-duced more than two decades of compulsory schooling