Paper 2015/047

Linearly Homomorphic Encryption from DDH

Guilhem Castagnos and Fabien Laguillaumie


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.

Note: An extended abstract of this paper will be published in the proceedings of CT-RSA 2015. This is the full version.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Major revision. Proc. of CT-RSA 2015
Linearly Homomorphic EncryptionOrders of Quadratic FieldsDiffie-Hellman Assumptions
Contact author(s)
guilhem castagnos @ math u-bordeaux1 fr
2015-01-26: last of 2 revisions
2015-01-22: received
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      author = {Guilhem Castagnos and Fabien Laguillaumie},
      title = {Linearly Homomorphic Encryption from DDH},
      howpublished = {Cryptology ePrint Archive, Paper 2015/047},
      year = {2015},
      note = {\url{}},
      url = {}
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