Section: New Results

Cryptology with quadratic fields

Participant : Guilhem Castagnos.

In [22] Guilhem Castagnos and Fabien Laguillaumie design a linearly homomorphic encryption scheme the security of which relies on the hardness of the decisional Diffie-Hellman problem. The approach requires some special features of the underlying group. In particular, its order is unknown and it contains a subgroup in which the discrete logarithm problem is tractable. Therefore, their instantiation holds in the class group of a non-maximal order of an imaginary quadratic field. Its algebraic structure makes it possible to obtain such a linearly homomorphic scheme in which the message space is the whole set of integers modulo a prime $p$ and which supports an unbounded number of additions modulo $p$ from the ciphertexts. A notable difference with previous work is that, for the first time, the security does not depend on the hardness of the factorisation of integers. As a consequence, under some conditions, the prime $p$ can be scaled to fit the application needs. This paper has beenpresented at the cryptographer's track at the RSA Conference 2015.