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In 2017, Shirase proposed a variant of Elliptic Curve Method combined with Complex Multiplication method for generating certain special kinds of elliptic curves. His algorithm can efficiently factorize a given composite integer when it has a prime factor p of the form 4p=1+Dv^{2} for some integer v, where -D is an auxiliary input integer called a discriminant. However, there is a disadvantage that the previous method works only for restricted cases where the class polynomial associated to -D has degree at most two. In this paper, we propose a generalization of the previous algorithm to the cases of class polynomials having arbitrary degrees, which enlarges the class of composite integers factorizable by our algorithm. We also extend the algorithm to more various cases where we have 4p=t^{2}+Dv^{2} and p+1-t is a smooth integer.
Yusuke AIKAWA
Hokkaido University
Koji NUIDA
The University of Tokyo/Information Technology Research Institute,Future University Hakodate
Masaaki SHIRASE
Future University Hakodate
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Yusuke AIKAWA, Koji NUIDA, Masaaki SHIRASE, "Elliptic Curve Method Using Complex Multiplication Method" in IEICE TRANSACTIONS on Fundamentals,
vol. E102-A, no. 1, pp. 74-80, January 2019, doi: 10.1587/transfun.E102.A.74.
Abstract: In 2017, Shirase proposed a variant of Elliptic Curve Method combined with Complex Multiplication method for generating certain special kinds of elliptic curves. His algorithm can efficiently factorize a given composite integer when it has a prime factor p of the form 4p=1+Dv^{2} for some integer v, where -D is an auxiliary input integer called a discriminant. However, there is a disadvantage that the previous method works only for restricted cases where the class polynomial associated to -D has degree at most two. In this paper, we propose a generalization of the previous algorithm to the cases of class polynomials having arbitrary degrees, which enlarges the class of composite integers factorizable by our algorithm. We also extend the algorithm to more various cases where we have 4p=t^{2}+Dv^{2} and p+1-t is a smooth integer.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E102.A.74/_p
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@ARTICLE{e102-a_1_74,
author={Yusuke AIKAWA, Koji NUIDA, Masaaki SHIRASE, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Elliptic Curve Method Using Complex Multiplication Method},
year={2019},
volume={E102-A},
number={1},
pages={74-80},
abstract={In 2017, Shirase proposed a variant of Elliptic Curve Method combined with Complex Multiplication method for generating certain special kinds of elliptic curves. His algorithm can efficiently factorize a given composite integer when it has a prime factor p of the form 4p=1+Dv^{2} for some integer v, where -D is an auxiliary input integer called a discriminant. However, there is a disadvantage that the previous method works only for restricted cases where the class polynomial associated to -D has degree at most two. In this paper, we propose a generalization of the previous algorithm to the cases of class polynomials having arbitrary degrees, which enlarges the class of composite integers factorizable by our algorithm. We also extend the algorithm to more various cases where we have 4p=t^{2}+Dv^{2} and p+1-t is a smooth integer.},
keywords={},
doi={10.1587/transfun.E102.A.74},
ISSN={1745-1337},
month={January},}
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TY - JOUR
TI - Elliptic Curve Method Using Complex Multiplication Method
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 74
EP - 80
AU - Yusuke AIKAWA
AU - Koji NUIDA
AU - Masaaki SHIRASE
PY - 2019
DO - 10.1587/transfun.E102.A.74
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E102-A
IS - 1
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - January 2019
AB - In 2017, Shirase proposed a variant of Elliptic Curve Method combined with Complex Multiplication method for generating certain special kinds of elliptic curves. His algorithm can efficiently factorize a given composite integer when it has a prime factor p of the form 4p=1+Dv^{2} for some integer v, where -D is an auxiliary input integer called a discriminant. However, there is a disadvantage that the previous method works only for restricted cases where the class polynomial associated to -D has degree at most two. In this paper, we propose a generalization of the previous algorithm to the cases of class polynomials having arbitrary degrees, which enlarges the class of composite integers factorizable by our algorithm. We also extend the algorithm to more various cases where we have 4p=t^{2}+Dv^{2} and p+1-t is a smooth integer.
ER -