RNS-modulo reduction upon a restricted base value set and its applicability to RSA cryptography

Schwemmlein, J.; Posch, Karl C.; Posch, Reinhard

doi:10.1016/s0167-4048(99)80061-3

Cited by 11 publications

(6 citation statements)

References 10 publications

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“…Our proposal has an advantage over the Posch, et al's [7] [8] in that the base extension in step 8 is error-free. This makes extra steps for error correction unnecessary in our algorithm.…”

Section: Rns Montgomery Multiplication Algorithmmentioning

confidence: 97%

See 1 more Smart Citation

Cox-Rower Architecture for Fast Parallel Montgomery Multiplication

Kawamura

Koike

Sano

et al. 2000

Lecture Notes in Computer Science

115

279

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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.

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“…Our proposal has an advantage over the Posch, et al's [7] [8] in that the base extension in step 8 is error-free. This makes extra steps for error correction unnecessary in our algorithm.…”

Section: Rns Montgomery Multiplication Algorithmmentioning

confidence: 97%

“…According to [8], these are a reasonable choice for deep sub-micron CMOS technologies such as 0.35 -0.18 µm. If a binary exponentiation is replaced by a 4-bit window method, R is improved to 1.1 [Mbps] with a penalty of approximately 4 kByte RAM increase.…”

Section: Performancementioning

confidence: 99%

Cox-Rower Architecture for Fast Parallel Montgomery Multiplication

Kawamura

Koike

Sano

et al. 2000

Lecture Notes in Computer Science

115

279

View full text Add to dashboard Cite

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“…However, modular reduction in RNS was not easy to carry out before it was replaced with Montgomery reduction (M-red) in [1]. Following proposal of the RNS M-red, promising results have been obtained for the RSA algorithm [2][3][4][5][6], Elliptic Curve Cryptosystem [7][8][9][10][11][12][13][14], Pairing-based Cryptosystem [15,16], modular inversion [17], Lattice-based Cryptosystem [18], and an architectural study [19]. In parallel with these applications, improvements in the RNS M-red algorithm have been proposed [3,5,6,9,[13][14][15].…”

Section: Introductionmentioning

confidence: 99%

RNS Montgomery reduction algorithms using quadratic residuosity

et al. 2018

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The residue number system (RNS) is a method for representing an integer as an n-tuple of its residues with respect to a given base. Since RNS has inherent parallelism, it is actively researched to implement a faster processing system for public-key cryptography. This paper proposes new RNS Montgomery reduction algorithms, Q-RNSs, the main part of which is twice a matrix multiplication. Letting n be the size of a base set, the number of unit modular multiplications in the proposed algorithms is evaluated as (2n 2 + n). This is achieved by posing a new restriction on the RNS base, namely, that its elements should have a certain quadratic residuosity. This makes it possible to remove some multiplication steps from conventional algorithms, and thus the new algorithms are simpler and have higher regularity compared with conventional ones. From our experiments, it is confirmed that there are sufficient candidates for RNS bases meeting the quadratic residuosity requirements.

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“…In [6], Paillier presents a 2n-bit modular multiplication emulated with 8 → 6 calls to a modular multiplier of bitsize n (plus other, negligible operations). Paillier's algorithm, inspired by Montgomery's technique [5], strongly relies on Residue Number Systems (RNS) [7,9] for representing data and performing partial operations on them. It simplifies earlier, more intricate approaches making use of mixed base representations [1,2].…”

Section: Introductionmentioning

confidence: 99%

“…In the most favorable case, we emulate a double-size modular multiplication with no more than 3 Euclidean multiplications, which leads to a speedup factor of 7−3 7 ≈ 57%. In addition, when Euclidean multiplications themselves must be emulated in software from modular multiplications, we show how to use these directly without referring to RNS-based approaches [1,2,7,9]. More precisely, we propose a simple alternative to these works which keeps numbers under a radix representation and runs as fast as 2 Euclidean multiplications.…”

Section: Introductionmentioning

confidence: 99%

Faster Double-Size Modular Multiplication from Euclidean Multipliers

Chevallier-Mames¹,

Joye²,

Paillier³

2003

Lecture Notes in Computer Science

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Abstract.A novel technique for computing a 2n-bit modular multiplication using n-bit arithmetic was introduced at CHES 2002 by Fischer and Seifert. Their technique makes use of an Euclidean division based instruction returning not only the remainder but also the integer quotient resulting from a modular multiplication, i.e., on input x, y and z, both xy/z and xy mod z are returned. A second algorithm making use of a special modular 'multiply-and-accumulate' instruction was also proposed. In this paper, we improve on these algorithms and propose more advanced computational strategies with fewer calls to these basic operations, bringing in a speed-up factor up to 57%. Besides, when Euclidean multiplications themselves have to be emulated in software, we propose a specific modular multiplication based algorithm which surpasses original algorithms in performance by 71%.

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RNS-modulo reduction upon a restricted base value set and its applicability to RSA cryptography

Cited by 11 publications

References 10 publications

Cox-Rower Architecture for Fast Parallel Montgomery Multiplication

Cox-Rower Architecture for Fast Parallel Montgomery Multiplication

RNS Montgomery reduction algorithms using quadratic residuosity

Faster Double-Size Modular Multiplication from Euclidean Multipliers

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