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Md. Al-Amin KHANDAKER Yasuyuki NOGAMI
Scalar multiplication over higher degree rational point groups is often regarded as the bottleneck for faster pairing based cryptography. This paper has presented a skew Frobenius mapping technique in the sub-field isomorphic sextic twisted curve of Kachisa-Schaefer-Scott (KSS) pairing friendly curve of embedding degree 18 in the context of Ate based pairing. Utilizing the skew Frobenius map along with multi-scalar multiplication procedure, an efficient scalar multiplication method for KSS curve is proposed in the paper. In addition to the theoretic proposal, this paper has also presented a comparative simulation of the proposed approach with plain binary method, sliding window method and non-adjacent form (NAF) for scalar multiplication. The simulation shows that the proposed method is about 60 times faster than plain implementation of other compared methods.
Yasuyuki NOGAMI Yumi SAKEMI Takumi OKIMOTO Kenta NEKADO Masataka AKANE Yoshitaka MORIKAWA
For ID-based cryptography, not only pairing but also scalar multiplication must be efficiently computable. In this paper, we propose a scalar multiplication method on the circumstances that we work at Ate pairing with Barreto-Naehrig (BN) curve. Note that the parameters of BN curve are given by a certain integer, namely mother parameter. Adhering the authors' previous policy that we execute scalar multiplication on subfield-twisted curve
Yasuyuki NOGAMI Akinori SAITO Yoshitaka MORIKAWA
In many cryptographic applications, a large-order finite field is used as a definition field, and accordingly, many researches on a fast implementation of such a large-order extension field are reported. This paper proposes a definition field Fpm with its characteristic p a pseudo Mersenne number, the modular polynomial f(x) an irreducible all-one polynomial (AOP), and using a suitable basis. In this paper, we refer to this extension field as an all-one polynomial field (AOPF) and to its basis as pseudo polynomial basis (PPB). Among basic arithmetic operations in AOPF, a multiplication between non-zero elements and an inversion of a non-zero element are especially time-consuming. As a fast realization of the former, we propose cyclic vector multiplication algorithm (CVMA), which can be used for possible extension degree m and exploit a symmetric structure of multiplicands in order to reduce the number of operations. Accordingly, CVMA attains a 50% reduction of the number of scalar multiplications as compared to the usually adopted vector multiplication procedure. For fast realization of inversion, we use the Itoh-Tsujii algorithm (ITA) accompanied with Frobenius mapping (FM). Since this paper adopts the PPB, FM can be performed without any calculations. In addition to this feature, ITA over AOPF can be composed with self reciprocal vectors, and by using CVMA this fact can also save computation cost for inversion.