# Homomorphic Encryption for Multiplications and Pairing Evaluation

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1 ARIC - Arithmetic and Computing Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l-Informatique du Parallélisme 2 IMB - Institut de Mathématiques de Bordeaux

Abstract : We propose a generic approach to design homomorphic encryption schemes, which extends Gj\o steen-s framework. From this generic method, we deduce a new homomorphic encryption scheme in a com\-posite-order subgroup of points of an elliptic curve which admits a pairing $e: \G\times \G ightarrow \G t$. This scheme has some interesting theoretical and practical properties: it allows an arbitrary number of multiplications in the groups $\G$ \emph{and} $\G t$, \emph{as well as} a pairing evaluation on the underlying plaintexts. We prove the semantic security under chosen plaintext attack of our scheme under a generalized subgroup membership assumption, and we also prove that it \emph{cannot} achieve {\sf ind-cca1} security. We eventually propose an original application to shared decryption. On the theoretical side, this scheme is an example of cryptosystem which can be naturally implemented with groups of prime order, as the homomorphic properties require only a \emph{projecting} pairing using Freeman-s terminology. However the application to shared decryption also relies on the fact that the pairing is \emph{cancelling} and therefore does not survive this conversion.

Autor: Fabien Laguillaumie - Guilhem Castagnos -

Fuente: https://hal.archives-ouvertes.fr/

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