GPU-Accelerated Montgomery Exponentiation

Fleissner, Sebastian

doi:10.1007/978-3-540-72584-8_28

Cited by 28 publications

(13 citation statements)

References 11 publications

(9 reference statements)

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“…Fleissner proposed a 192 bits Montgomery exponentiation algorithm, indicated as "GPU-MonExp", which was executed on NVIDIA 7800GTX GPU using OpenGL shading language [15]. The performance tests had shown that its implementation run 136-168 times faster than the standard Montgomery exponentiation algorithm.…”

Section: Related Workmentioning

confidence: 99%

Parallelizing RSA Algorithm on Multicore CPU and GPU

Fadhil¹,

Younis²

2014

IJCA

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Public key algorithms are extensively known to be slower than symmetric key alternatives in the a r e a of cryptographic algorithms for the reason of their basis in modular arithmetic. The most public key algorithm widely used is the RSA. Therefore, how to enhance the speed of RSA algorithm has been the research significant topic in the computer security as well as in computing fields. With remarkable increase in the computing capability of the modern Graphics Processing Unit's (GPUs) as a co-processor of the CPU, one can significantly benefit from the Single Instruction Multiple Thread (SIMT) style of computing. This paper proposes a hybrid system to parallelize the RSA for multicore CPU and many cores GPUs with variable key size. In doing so, three variants implementation for the RSA algorithm are done to facilitate the performance comparison against Crypto++ library and sequential counterpart. The GPU implementation gained approximately 23 speed up factor over the sequential CPU implementation; while the multithread CPU implementation gained only 6 speed up factor over the sequential CPU implementation as far as the latency is concerned. Furthermore, additional speedup could be gained as far as the throughput is concerned; the throughput gained for 1024 bits is ~1800 msg/sec; as for 2048 bits is ~250 msg/sec. Due to overlapping of multithread operation whenever free resources are available. The experiments are conducted on a laptop with Intel Core I7-2670QM, 2.20 GHz CPU and Nvidia GeForce GT630M GPU. Results reveal that the GPU is appropriate to speed up the RSA algorithm.

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Section: Related Workmentioning

confidence: 99%

Parallelizing RSA Algorithm on Multicore CPU and GPU

Fadhil¹,

Younis²

2014

IJCA

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“…Efforts to speed up modular arithmetic using the power of GPUs are reported in [2], [5]. In [5], the authors implemented three modular arithmetic operations on a GPU: addition, subtraction, and multiplication.…”

Section: Related Workmentioning

confidence: 99%

“…As there are many parameters for pairings, by implementing pairings on a GPU, we can exploit parallelization methods to extend the program code flexibly. In the case that the field characteristic defining the curve and pairing is large, we can compute elements of the field in parallel as modular arithmetic on prime fields using a GPU [2]- [5]. On the other hand, if the characteristic is small, we can implement arithmetic on the field efficiently using a GPU and polynomial bases.…”

Section: Introductionmentioning

confidence: 99%

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Parallel GPU Implementation of ƞT Pairing over Fields of Characteristic Two

Ito¹,

Inomata²,

Fujikawa³

2014

IJCCE

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Abstract-Pairings on hyperelliptic curves have been applied to many cryptographic schemes, and it is important to exploit methods that increase the speed of various pairings and their curves. Additionally, multiple pairings should be performed efficiently in some cryptographic application such as attribute-based encryption or functional encryption. We propose an efficient extension field construction method that defines a curve and its pairing. We also implemented the parallel arithmetic on extension fields and multiple pairings in parallel and reported experimental timing results. We achieved timing of 12.7ms and 52.0ms per pairing when computed 1248 pairings by using GPU Tesla K20c. We took the extension degree of base field = which is greater than the parameter = , that was appropriate for the pairing at the 128-bit security level. By normalization of experimental result, we achieved a certain level of speeding up of the pairing compared to the state-of-the-art CPU implementation. In addition, we achieved scalability with the extension degree of base field in our parallel implementation by performing Karatsuba multiplications between multiple elements of extension field in parallel.Index Terms-ƞ T pairing, multiple pairings, GPU implementation, CUDA, karatsuba method, DLP in finite field of small characteristic, security level. I. INTRODUCTIONKoblitz [1] suggested a hyperelliptic cryptosystem us-ing Jacobians of hyperelliptic curves as arithmetic generalizations on groups of elliptic curves. Arithmetic on Jacobians of hyperelliptic curves is more complex than on elliptic curve groups. Alternatively, we can use smaller finite fields; i.e., we can employ smaller size keys by using higher genus curves to achieve the same level of security.Pairings on elliptic curves or higher genus curves have attracted significant attention and have been applied to many cryptographic schemes, such as ID-based cryptography. Generally, calculation methods for pairings are complex and the cost of pairings is considerably higher than that of arithmetic on curves. In addition, the cost is significantly higher when using algebraic curves of higher genus.Modern graphics processing unit (GPU) technology for general purposes, based on GPU computation has advanced significantly, while the use thereof in high level cryptography implementations has increased rapidly. There has been much research on increasing the speed of multiple-precision Manuscript received December 30, 2013; revised February 27, 2014. M. Ishii is with Nara Institute of Science and Technology, Nara, Japan (e-mail: masahiro-i@is.naist.jp).A. Inomata is with Initiative Center, Nara Institute of Science and Technology, Nara, Japan (e-mail: atsuo@itc.naist.jp).K. Fujikawa is with Information Initiative Center, Nara Institute of Science and Technology, Nara, Japan (e-mail: fujikawa@itc.naist.jp).arithmetic or arithmetic on finite fields using GPUs, which is explored further in Section II.In this study, we consider the parallelization of arithmetic on extension fields. The pa...

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“…[17] and [8] first considered modular multiplication (i.e., the multiplication operation over a finite field F q where q is a large prime number) over GPUs. At that time, GPUs were still designed for processing graphics only and therefore a large effort was required for researchers to map their programs to the GPU architecture.…”

Section: Related Workmentioning

confidence: 99%

Acceleration of Composite Order Bilinear Pairing on Graphics Hardware

Zhang

Xue

Wong

et al. 2012

Information and Communications Security

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Abstract. Recently, composite-order bilinear pairing has been shown to be useful in many cryptographic constructions. However, it is time-costly to evaluate. This is because the composite order should be at least 1024bit and, hence, the elliptic curve group order n and base field become too large, rendering the bilinear pairing algorithm itself too slow to be practical (e.g., the Miller loop is Ω(n)). Thus, composite-order computation easily becomes the bottleneck of a cryptographic construction, especially, in the case where many pairings need to be evaluated at the same time. The existing solution to this problem that converts composite-order pairings to prime-order ones is only valid for certain constructions. In this paper, we leverage the huge number of threads available on Graphics Processing Units (GPUs) to speed up composite-order pairing computation. We investigate suitable SIMD algorithms for base field, extension field, elliptic curve and bilinear pairing computation as well as mapping these algorithms into GPUs with careful considerations. Experimental results show that our method achieves a record of 8.7ms per pairing on a 1024bit security level, which is a 20-fold speedup compared to state-of-the-art CPU implementation. This result also opens the road to adopting higher security levels and using rich-resource parallel platforms, which for example are available in cloud computing. In fact, we can achieve more than 24 times speedup on a 2048bit security level and a record of 7 × 10 −6 USD per pairing on the Amazon cloud computing environment.

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GPU-Accelerated Montgomery Exponentiation

Cited by 28 publications

References 11 publications

Parallelizing RSA Algorithm on Multicore CPU and GPU

Parallelizing RSA Algorithm on Multicore CPU and GPU

Parallel GPU Implementation of ƞT Pairing over Fields of Characteristic Two

Acceleration of Composite Order Bilinear Pairing on Graphics Hardware

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