Linearly Homomorphic Encryption from DDH

Guilhem Castagnos 1, 2 Fabien Laguillaumie 3
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
3 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : We design a linearly homomorphic encryption scheme whose security relies on the hardness of the decisional Diffie-Hellman problem. Our 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, our 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 whose 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 works is that, for the first time, the security does not depend on the hardness of the factorization of integers. As a consequence, under some conditions, the prime p can be scaled to fit the application needs.
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Guilhem Castagnos, Fabien Laguillaumie. Linearly Homomorphic Encryption from DDH. The Cryptographer's Track at the RSA Conference 2015, Apr 2015, San Francisco, United States. ⟨10.1007/978-3-319-16715-2_26⟩. ⟨hal-01213284⟩

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