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

Разностные характеристики по модулю 2n композиции нескольких побитовых исключающих или

We study the additive differential probabilities adp® of compositions of k — 1 bitwise XORs. For vectors a1,...,ak+1 G Zn, it is defined as the probability of transformation input differences a1,...,ak to the output difference ak+1 by the function x1 ф ... ф xk, where x1,... ,xk G Zn and k > 2. It is used for differential cryptanalysis of symmetric-key primitives, such as Addition-Rotation-XOR constructions. Several results which are known for adp2® are generalized for adpk®. Some argument symmetries are proven for adpk®. Recurrence formulas which allow us to reduce the dimension of the arguments are obtained. All impossible differentials as well as all differentials of adpk® with the probability 1 are found. For even k, it is proven that max adp® (a1,..., ak ak+1) = adp®(0,..., 0, ak+1 ak+1). Matrices that can a1,...,ak be used for efficient calculating adpk® are constructed. It is also shown that the cases of even and odd k differ significantly

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

The mathematical problems, presented at the Third International Students’ Olympiad in Cryptography NSUCRYPTO’2016, and their solutions are considered. They are related to the construction of algebraic immune vectorial Boolean functions and big Fermat numbers, the 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. The problem is the following: construct F : ^ with maximum possible component algebraic immunity 3 or prove that it does not exist. Alexey Udovenko from University of Luxembourg has found such a function

Математические проблемы и решения Девятой международной олимпиады по криптографии NSUCRYPTO

Every year the International Olympiad in Cryptography Non-Stop University CRYPTO (NSUCRYPTO) offers mathematical problems for university and school students and, moreover, for professionals in the area of cryptography and computer science. The main goal of NSUCRYPTO is to draw attention of students and young researchers to modern cryptography and raise awareness about open problems in the field. We present problems of NSUCRYPTO'22 and their solutions. There are 16 problems on the following topics: ciphers, cryptosystems, protocols, e-money and cryptocurrencies, hash functions, matrices, quantum computing, S-boxes, etc. They vary from easy mathematical tasks that could be solved by school students to open problems that deserve separate discussion and study. So, in this paper, we consider several open problems on three-pass protocols, public and private keys pairs, modifications of discrete logarithm problem, cryptographic permutations, and quantum circuits