Abstract
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 E∼(F p2) instead of doing on the original curve E(F p12), we at first show sextic twisted subfield Frobenius mapping (ST-SFM) φ∼ in E∼(Fp2). On BN curves, note φ∼ 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 φ∼. 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.
| Original language | English |
|---|---|
| Pages (from-to) | 182-189 |
| Number of pages | 8 |
| Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |
| Volume | E92-A |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Jan 2009 |
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Keywords
- Ate pairing
- BN curve
- Frobenius mapping
- Scalar multiplication
- Twisted subfield computation
ASJC Scopus subject areas
- Electrical and Electronic Engineering
- Computer Graphics and Computer-Aided Design
- Applied Mathematics
- Signal Processing
Cite this
Scalar multiplication using frobenius expansion over twisted elliptic curve for ate pairing based cryptography. / Nogami, Yasuyuki; Sakemi, Yumi; Okimoto, Takumi; Nekado, Kenta; Akane, Masataka; Morikawa, Yoshitaka.
In: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol. E92-A, No. 1, 01.2009, p. 182-189.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Scalar multiplication using frobenius expansion over twisted elliptic curve for ate pairing based cryptography
AU - Nogami, Yasuyuki
AU - Sakemi, Yumi
AU - Okimoto, Takumi
AU - Nekado, Kenta
AU - Akane, Masataka
AU - Morikawa, Yoshitaka
PY - 2009/1
Y1 - 2009/1
N2 - 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 E∼(F p2) instead of doing on the original curve E(F p12), we at first show sextic twisted subfield Frobenius mapping (ST-SFM) φ∼ in E∼(Fp2). On BN curves, note φ∼ 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 φ∼. 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.
AB - 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 E∼(F p2) instead of doing on the original curve E(F p12), we at first show sextic twisted subfield Frobenius mapping (ST-SFM) φ∼ in E∼(Fp2). On BN curves, note φ∼ 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 φ∼. 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.
KW - Ate pairing
KW - BN curve
KW - Frobenius mapping
KW - Scalar multiplication
KW - Twisted subfield computation
UR - http://www.scopus.com/inward/record.url?scp=77749294674&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77749294674&partnerID=8YFLogxK
U2 - 10.1587/transfun.E92.A.182
DO - 10.1587/transfun.E92.A.182
M3 - Article
VL - E92-A
SP - 182
EP - 189
JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
SN - 0916-8508
IS - 1
ER -