Fast base extension using a redundant modulus in RNS

116

279

Abstract. This paper proposes a fast parallel Montgomery multiplication algorithm based on Residue Number Systems (RNS). It is easy to construct a fast modular exponentiation by applying the algorithm repeatedly. To realize an efficient RNS Montgomery multiplication, the main contribution of this paper is to provide a new RNS base extension algorithm. Cox-Rower Architecture described in this paper is a hardware suitable for the RNS Montgomery multiplication. In this architecture, a base extension algorithm is executed in parallel by plural Rower units controlled by a Cox unit. Each Rower unit is a single-precision modular multiplier-and-accumulator, whereas Cox unit is typically a 7 bit adder. Although the main body of the algorithm processes numbers in an RNS form, efficient procedures to transform RNS to or from a radix representation are also provided. The exponentiation algorithm can, thus, be adapted to an existing standard radix interface of RSA cryptosystem.

Section: Introductionmentioning

confidence: 99%

Cox-Rower Architecture for Fast Parallel Montgomery Multiplication

Kawamura

Koike

Sano

et al. 2000

116

279

“…q | q j q I j e j (mod q) Consequently, the cancellation of ik terms in the public matrix rlk RNS [0] by using this method causes an extra noise growth bounded by (this can be fairly compared to (22) in the case where…”

Section: Lemma 8 Brings the Rest Of The Noisementioning

confidence: 99%

“…m`cannot be tolerated. m sk allows us to perform a complete Shenoy and Kumaresan like conversion [22]. The next lemma describes such kind of conversion for a more general context where the input can be either positive or negative, and can be larger, in absolute value, than M .…”

Section: Reducing (Mod Q) a Ciphertext In Bmentioning

confidence: 99%

A Full RNS Variant of FV Like Somewhat Homomorphic Encryption Schemes

Bajard

Eynard

Hasan

et al. 2017

144

163

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.

“…There are mainly two types of BE algorithms: those based on the CRT relation (Eq. 2) [28,25,18], and those using an intermediate representation called mixed radix system (MRS) [30,7,3]. In hardware, the most efficient BE implementations in state-of-art use the CRT based solution [12,14].…”

Section: Base Extensionmentioning

confidence: 99%

Single Base Modular Multiplication for Efficient Hardware RNS Implementations of ECC

Bigou

Tisserand

2015

Abstract. The paper describes a new RNS modular multiplication algorithm for efficient implementations of ECC over FP . Thanks to the proposition of RNS-friendly Mersenne-like primes, the proposed RNS algorithm requires 2 times less moduli than the state-of-art ones, leading to 4 times less precomputations and about 2 times less operations. FPGA implementations of our algorithm are presented, with area reduced up to 46 %, for a time overhead less than 10 %.