The Seventh International Olympiad in Cryptography NSUCRYPTO: 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.