### Abstract

In this paper, we particularly deal with no F_{p}-rational two-torsion elliptic curves, where F_{p} is the prime field of the characteristic p. First we introduce a shift product-based polynomial transform. Then, we show that the parities of (#E - 1)/2 and (#E′ - 1)/2 are reciprocal to each other, where #E and #E′ are the orders of the two candidate curves obtained at the last step of complex multiplication (CM)-based algorithm. Based on this property, we propose a method to check the parity by using the shift product-based polynomial transform. For a 160 bits prime number as the characteristic, the proposed method carries out the parity check 25 or more times faster than the conventional checking method when 4 divides the characteristic minus 1. Finally, this paper shows that the proposed method can make CM-based algorithm that looks up a table of precomputed class polynomials more than 10 percent faster.

Original language | English |
---|---|

Pages (from-to) | 745-760 |

Number of pages | 16 |

Journal | ETRI Journal |

Volume | 28 |

Issue number | 6 |

Publication status | Published - Dec 2006 |

### Fingerprint

### Keywords

- CM method
- Irreducible cubic polynomial
- Quadratic power residue/non-residue

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Computer Networks and Communications

### Cite this

*ETRI Journal*,

*28*(6), 745-760.

**A method for distinguishing the two candidate elliptic curves in the complex multiplication method.** / Nogami, Yasuyuki; Obara, Mayumi; Morikawa, Yoshitaka.

Research output: Contribution to journal › Article

*ETRI Journal*, vol. 28, no. 6, pp. 745-760.

}

TY - JOUR

T1 - A method for distinguishing the two candidate elliptic curves in the complex multiplication method

AU - Nogami, Yasuyuki

AU - Obara, Mayumi

AU - Morikawa, Yoshitaka

PY - 2006/12

Y1 - 2006/12

N2 - In this paper, we particularly deal with no Fp-rational two-torsion elliptic curves, where Fp is the prime field of the characteristic p. First we introduce a shift product-based polynomial transform. Then, we show that the parities of (#E - 1)/2 and (#E′ - 1)/2 are reciprocal to each other, where #E and #E′ are the orders of the two candidate curves obtained at the last step of complex multiplication (CM)-based algorithm. Based on this property, we propose a method to check the parity by using the shift product-based polynomial transform. For a 160 bits prime number as the characteristic, the proposed method carries out the parity check 25 or more times faster than the conventional checking method when 4 divides the characteristic minus 1. Finally, this paper shows that the proposed method can make CM-based algorithm that looks up a table of precomputed class polynomials more than 10 percent faster.

AB - In this paper, we particularly deal with no Fp-rational two-torsion elliptic curves, where Fp is the prime field of the characteristic p. First we introduce a shift product-based polynomial transform. Then, we show that the parities of (#E - 1)/2 and (#E′ - 1)/2 are reciprocal to each other, where #E and #E′ are the orders of the two candidate curves obtained at the last step of complex multiplication (CM)-based algorithm. Based on this property, we propose a method to check the parity by using the shift product-based polynomial transform. For a 160 bits prime number as the characteristic, the proposed method carries out the parity check 25 or more times faster than the conventional checking method when 4 divides the characteristic minus 1. Finally, this paper shows that the proposed method can make CM-based algorithm that looks up a table of precomputed class polynomials more than 10 percent faster.

KW - CM method

KW - Irreducible cubic polynomial

KW - Quadratic power residue/non-residue

UR - http://www.scopus.com/inward/record.url?scp=33845396854&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33845396854&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33845396854

VL - 28

SP - 745

EP - 760

JO - ETRI Journal

JF - ETRI Journal

SN - 1225-6463

IS - 6

ER -