Highly parallel modular multiplication in the residue number system using sum of residues reduction

Phillips, Braden J.; Kong, Yinan; Lim, Zhining

doi:10.1007/s00200-010-0124-2

Cited by 17 publications

(19 citation statements)

References 8 publications

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“…The factor (κ) is a function of γ i , not a constant. Therefore the value of V -which is actually the output of SOR algorithm introduced in [19,20] -is not presenting the true reduction of Z p . In fact:…”

Section: Improved Sum Of Residues (Sor) Algorithmmentioning

confidence: 99%

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Improved Sum of Residues Modular Multiplication Algorithm

Mehrabi

2019

Cryptography

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Modular reduction of large values is a core operation in most common public-key cryptosystems that involves intensive computations in finite fields. Within such schemes, efficiency is a critical issue for the effectiveness of practical implementation of modular reduction. Recently, Residue Number Systems have drawn attention in cryptography application as they provide a good means for extreme long integer arithmetic and their carry-free operations make parallel implementation feasible. In this paper, we present an algorithm to calculate the precise value of “ X mod p ” directly in the RNS representation of an integer. The pipe-lined, non-pipe-lined, and parallel hardware architectures are proposed and implemented on XILINX FPGAs.

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Section: Improved Sum Of Residues (Sor) Algorithmmentioning

confidence: 99%

“…Modular reduction based on the sum of residues (SOR) algorithm was first presented by Phillips et al [19] in 2010. The SOR algorithm hardware implementation was proposed later in [20].…”

Section: Introductionmentioning

confidence: 99%

Improved Sum of Residues Modular Multiplication Algorithm

Mehrabi

2019

Cryptography

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“…As a consequence, divisions and modular reductions are complex and costly operations in RNS. Efficient RNS modular reduction and RNS modular multiplication methods have been proposed in [27,19,1,26] RNS modular multiplication for RSA was studied in [27,19,1]. Full RSA in RNS implementations can be found in [23,2,21].…”

Section: Notations and Definitionsmentioning

confidence: 99%

Improving Modular Inversion in RNS Using the Plus-Minus Method

Bigou

Tisserand

2013

Cryptographic Hardware and Embedded Systems - CHES 2013

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Abstract. The paper describes a new RNS modular inversion algorithm based on the extended Euclidean algorithm and the plus-minus trick. In our algorithm, comparisons over large RNS values are replaced by cheap computations modulo 4. Comparisons to an RNS version based on Fermat's little theorem were carried out. The number of elementary modular operations is significantly reduced: a factor 12 to 26 for multiplications and 6 to 21 for additions. Virtex 5 FPGAs implementations show that for a similar area, our plus-minus RNS modular inversion is 6 to 10 times faster.

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“…The modular multiplication, one of the most important arithmetic operation in asymmetric cryptography, is significantly more costly than a simple multiplication. Thus, many algorithms and optimizations have been proposed for RNS modular multiplication, see [26,1,18,2,14,24,12,9,27].…”

Section: Introductionmentioning

confidence: 99%

Single Base Modular Multiplication for Efficient Hardware RNS Implementations of ECC

Bigou

Tisserand

2015

Lecture Notes in Computer Science

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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 %.

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Highly parallel modular multiplication in the residue number system using sum of residues reduction

Cited by 17 publications

References 8 publications

Improved Sum of Residues Modular Multiplication Algorithm

Improved Sum of Residues Modular Multiplication Algorithm

Improving Modular Inversion in RNS Using the Plus-Minus Method

Single Base Modular Multiplication for Efficient Hardware RNS Implementations of ECC

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