Elliptic curve method using complex multiplication method

Yusuke Aikawa, Koji Nuida, Masaaki Shirase

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)74-80
Number of pages7
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number1
Publication statusPublished - Jan 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Signal Processing
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering
  • Applied Mathematics


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