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.

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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.

The Canetti-Krawczyk (CK) and extended Canetti-Krawczyk (eCK) security models, are widely used to provide security arguments for key agreement protocols. We discuss security shades in the (e)CK models, and some practical attacks unconsidered in (e)CK-security arguments. We propose a strong security model which encompasses the eCK one. We also propose a new protocol, called Strengthened MQV (SMQV), which in addition to provide the same efficiency as the (H)MQV protocols, is particularly suited for distributed implementations wherein a tamper-proof device is used to store long-lived keys, while session keys are used on an untrusted host machine. The SMQV protocol meets our security definition under the Gap DiffieHellman assumption and the Random Oracle model.

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International audienceThe modular exponentiation on large numbers is computationally intensive. An effective way for performing this operation consists in using Montgomery exponentiation in the Residue Number System (RNS). This paper presents an algorithmic and architectural study of such exponentiation approach. From the algorithmic point of view, new and state-of-the-art opportunities that come from the reorganization of operations and precomputations are considered. From the architectural perspective, the design opportunities offered by well-known computer arithmetic techniques are studied, with the aim of developing an efficient arithmetic cell architecture. Furthermore, since the use of efficient RNS bases with a low Hamming weight are being considered with ever more interest, four additional cell architectures specifically tailored to these bases are developed and the tradeoff between benefits and drawbacks is carefully explored. An overall comparison among all the considered algorithmic approaches and cell architectures is presented, with the aim of providing the reader with an extensive overview of the Montgomery exponentiation opportunities in RNS

The selection of the elements of the bases in an RNS modular multiplication method is crucial and has a great impact in the overall performance.This work proposes specific sets of optimal RNS moduli with elements of Hamming weight three whose inverses used in the MRS reconstruction have very small Hamming weight. This property is exploited in RNS bases conversions, to completely remove and replace the products by few additions/subtractions and shifts, reducing the time complexity of modular multiplication.These bases are specially crafted to computation with operands of sizes 256 or more and are suitable for cryptographic applications such as the ECC protocols. Disciplines Physical Sciences and Mathematics