More than 30 years ago, Buchmann and Williams proposed using ideal class groups of imaginary quadratic fields in cryptography with a Diffie-Hellman style key exchange protocol. After several twists, there has been in recent years a new interest in this area. This rebirth is mainly due to two features. First, class groups of imaginary quadratic fields allow the design of cryptographic protocols that do not require a trusted setup. This particularity has been used for example to build cryptographic accumulators and verifiable delay functions. Secondly, using these groups, we proposed in 2015 a versatile encryption scheme, linearly homomorphic modulo a prime that has found many applications, for instance in secure two-party computation.
In this talk, I will give an overview of cryptography based on class groups of imaginary quadratic fields and discuss recent developments.