dalla Teoria dei Gruppi alla Crittografia

Cryptography is defined as the method to protect information and communication. Throughout the centuries, many cryptographic methods have been developed, and almost all share a common foundation: the resolution of a mathematical problem. Many of these methods are based on applications of number theory; however, in recent decades, cryptographic systems based on group theory have begun to be studied. Cryptography based on groups is indeed a very recent and rapidly growing research field that lies at the intersection of group theory, combinatorics, complexity theory, code theory, and cryptology.

In this course, we present several well-known cryptographic protocols, such as Diffie-Hellman, Ko-Lee, and Anshel-Anshel-Goldfeld. After that, we survey elementary topics in combinatorial group theory and complexity theory. Then, we describe algorithmic problems arising from group theory, together with cryptographic protocols based on these problems. The final part of the course will focus on presenting some classes of groups and their suitability as platforms for both classical and quantum cryptography.