Efficient Cryptosystems From $2^k$ -th Power Residue Symbols

Fabrice Benhamouda (1, 2) , Javier Herranz (3) , Marc Joye (4) , Benoît Libert (5, 6)

1 PSL - Université Paris Sciences et Lettres
2 CASCADE - Construction and Analysis of Systems for Confidentiality and Authenticity of Data and Entities
3 MAK - Departament of Matemàtica Aplicada IV Mathematics Applied to Cryptography
4 NXP Semiconductors
5 ENS de Lyon - École normale supérieure de Lyon
6 ARIC - Arithmetic and Computing

Goldwasser and Micali (1984) highlighted the importance of randomizing the plaintext for public-key encryption and introduced the notion of semantic security. They also realized a cryptosystem meeting this security notion under the standard complexity assumption of deciding quadratic residuosity modulo a composite number. The Goldwasser-Micali cryptosystem is simple and elegant but is quite wasteful in bandwidth when encrypting large messages. A number of works followed to address this issue and proposed various modifications. This paper revisits the original Goldwasser-Micali cryptosystem using 2 k-th power residue symbols. The so-obtained cryptosystems appear as a very natural generalization for k ≥ 2 (the case k 1 corresponds exactly to the Goldwasser-Micali cryptosystem). Advantageously, they are efficient in both bandwidth and speed; in particular, they allow for fast decryption. Further, the cryptosystems described in this paper inherit the useful features of the original cryptosystem (like its homomorphic property) and are shown to be secure under a similar complexity assumption. As a prominent application, this paper describes an efficient lossy trapdoor function based thereon.

Mots clés: standard model; homomorphic encryption; Public-key encryption; quadratic residuosity; Goldwasser-Micali cryptosystem

Domaines: Cryptographie et sécurité [cs.CR]

Fabrice Benhamouda, Javier Herranz, Marc Joye, Benoît Libert. Efficient Cryptosystems From $2^k$ -th Power Residue Symbols. Journal of Cryptology, 2016, ⟨10.1007/s00145-016-9229-5⟩. ⟨hal-01394400⟩