Scalar Multiplication Using Frobenius Expansion over Twisted Elliptic Curve for Ate Pairing Based Cryptography
Scalar Multiplication Using Frobenius Expansion over Twisted Elliptic Curve for Ate Pairing Based Cryptography
复制标题
使用扭曲椭圆曲线上的 Frobenius 展开进行标量乘法用于基于 Ate 配对的密码学
DOI：
10.1587/transfun.e92.a.182
复制
发表时间：
2009
影响因子：
0.5
通讯作者：
Y. Morikawa
中科院分区：
文献类型：
--
作者：
Y. Nogami;Yumi Sakemi;Takumi Okimoto;K. Nekado;Masataka Akane;Y. 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 $\\ ilde{E} (\\boldsymbol{F}_{p^2}$) instead of doing on the original curve $E(\\boldsymbol{F}_{p^{12}}$), we at first show sextic twisted subfield Frobenius mapping (ST-SFM) $\\ ilde{\\varphi}$ in $\\ ilde{E} (\\boldsymbol{F}_{p^2})$. On BN curves, note $\\ ilde{\\varphi}$ is identified with the scalar multiplication by p. However a scalar is always smaller than the order r of BN curve for Ate pairing, so ST-SFM does not directly applicable to the above circumstances. We then exploit the expressions of the curve order r and the characteristic p by the mother parameter to derive some radices such that they are expressed as a polynomial of p. Thus, a scalar multiplication [s] can be written by the series of ST-SFMs $\\ ilde{\\varphi}$. In combination with the binary method or multi-exponentiation technique, this paper shows that the proposed method runs about twice or more faster than plain binary method.
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 $\\ ilde{E} (\\boldsymbol{F}_{p^2}$) instead of doing on the original curve $E(\\boldsymbol{F}_{p^{12}}$), we at first show sextic twisted subfield Frobenius mapping (ST-SFM) $\\ ilde{\\varphi}$ in $\\ ilde{E} (\\boldsymbol{F}_{p^2})$. On BN curves, note $\\ ilde{\\varphi}$ is identified with the scalar multiplication by p. However a scalar is always smaller than the order r of BN curve for Ate pairing, so ST-SFM does not directly applicable to the above circumstances. We then exploit the expressions of the curve order r and the characteristic p by the mother parameter to derive some radices such that they are expressed as a polynomial of p. Thus, a scalar multiplication [s] can be written by the series of ST-SFMs $\\ ilde{\\varphi}$. In combination with the binary method or multi-exponentiation technique, this paper shows that the proposed method runs about twice or more faster than plain binary method.