Parallel GPU Implementation of ƞT Pairing over Fields of Characteristic Two

Abstract

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...