Abstract. Since Gentry's breakthrough work in 2009, homomorphic cryptography has received a widespread attention. Implementation of a fully homomorphic cryptographic scheme is however still highly expensive. Somewhat Homomorphic Encryption (SHE) schemes, on the other hand, allow only a limited number of arithmetical operations in the encrypted domain, but are more practical. Many SHE schemes have been proposed, among which the most competitive ones rely on (Ring-) Learning With Error (RLWE) and operations occur on high-degree polynomials with large coe cients. This work focuses in particular on the Chinese Remainder Theorem representation (a.k.a. Residue Number Systems) applied to large coe cients. In SHE schemes like that of Fan and Vercauteren (FV), such a representation remains hardly compatible with procedures involving coe cient-wise division and rounding required in decryption and homomorphic multiplication. This paper suggests a way to entirely eliminate the need for multi-precision arithmetic, and presents techniques to enable a full RNS implementation of FV-like schemes. For dimensions between 2 11 and 2 15 , we report speed-ups from 5⇥ to 20⇥ for decryption, and from 2⇥ to 4⇥ for multiplication.

In this paper we propose an efficient hardware implementation of RSA based on the Residue Number System (RNS) which allows for fast parallel arithmetic. We propose RNS versions of Montgomery multiplication and exponentiation algorithms and illustrate the efficiency of our approach with two implementations of RSA. For the very first time a very attractive conversion-free RSA encryption/decryption scheme is proposed. Compared to previously proposed methods our solution requires less elementary operations and is very promising.

No abstract

Abstract. In this paper we show how the usage of Residue Number Systems (RNS) can easily be turned into a natural defense against many side-channel attacks (SCA). We introduce a Leak Resistant Arithmetic (LRA), and present its capacities to defeat timing, power (SPA, DPA) and electromagnetic (EMA) attacks.