Hard mathematical problems used in Cryptography

Seminar in Mathematics
Abstract: The security of public key cryptography relies on hardness assumptions coming from a variety of mathematical problems. One of the two main candidates originally considered to construct public-key cryptosystems is modular exponentiation with its hard inverse operation, computing discrete logarithms. More recently, with the emergence of new technologies such as quantum computers, cryptographers have focused on new hard problems coming from different mathematical objects, euclidean lattices for example. The main goal of my research is to evaluate the hardness of such mathematical problems in order to correctly assess the security of cryptographic schemes widely deployed and used in the real world.
More precisely, I have focused on the complexity of algorithms solving the discrete logarithm problem and the use of RingLWE (Learning With Errors) to construct fully homomorphic encryption schemes. I will cover results from both these directions during this talk.