Integer Factorization Based on Elliptic Curve Method: Towards Better Exploitation of Reconfigurable Hardware

This series of papers is concerned with a probabilistic algorithm for finding small prime factors of an integer. While the algorithm is not practical, it yields an improvement over previous complexity results. The algorithm uses the jacobian varieties of curves of genus 2 in the same way that the elliptic curve method uses elliptic curves. In this first paper in the series a new density theorem is presented for smooth numbers in short intervals. It is a key ingredient of the analysis of the algorithm.

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JKL-ECM: an implementation of ECM using Hessian curves

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157016000231 ◽

2016 ◽

Vol 19 (A) ◽

pp. 83-99

Author(s):

Henriette Heer ◽

Gary McGuire ◽

Oisín Robinson

Keyword(s):

High Performance ◽

Torsion Subgroup ◽

Integer Factorization ◽

Arbitrary Size ◽

Base Points ◽

Performance Curves ◽

Small Parameters ◽

Curve Method ◽

Torsion Subgroups ◽

Elliptic Curve Method

We present JKL-ECM, an implementation of the elliptic curve method of integer factorization which uses certain twisted Hessian curves in a family studied by Jeon, Kim and Lee. This implementation takes advantage of torsion subgroup injection for families of elliptic curves over a quartic number field, in addition to the ‘small parameter’ speedup. We produced thousands of curves with torsion$\mathbb{Z}/6\mathbb{Z}\oplus \mathbb{Z}/6\mathbb{Z}$and small parameters in twisted Hessian form, which admit curve arithmetic that is ‘almost’ as fast as that of twisted Edwards form. This allows JKL-ECM to compete with GMP-ECM for finding large prime factors. Also, JKL-ECM, based on GMP, accepts integers of arbitrary size. We classify the torsion subgroups of Hessian curves over$\mathbb{Q}$and further examine torsion properties of the curves described by Jeon, Kim and Lee. In addition, the high-performance curves with torsion$\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/8\mathbb{Z}$of Bernsteinet al. are completely recovered by the$\mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/8\mathbb{Z}$family of Jeon, Kim and Lee, and hundreds more curves are produced besides, all with small parameters and base points.

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Factoring N = pq 2 with the Elliptic Curve Method

Lecture Notes in Computer Science - Algorithmic Number Theory ◽

10.1007/3-540-45455-1_37 ◽

2002 ◽

pp. 475-490 ◽

Cited By ~ 2

Author(s):

Peter Ebinger ◽

Edlyn Teske

Keyword(s):

Elliptic Curve ◽

Curve Method ◽

Elliptic Curve Method

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Elliptic Curve Method

10.1007/springerreference_238 ◽

2011 ◽

Keyword(s):

Elliptic Curve ◽

Curve Method ◽

Elliptic Curve Method

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Implementation of Embedded Mobile Device Elliptic Curve Algorithm Fast Generating Algorithm

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.694-697.2599 ◽

2013 ◽

Vol 694-697 ◽

pp. 2599-2603

Author(s):

Li Hong Zhang ◽

Shu Qian Chen

Keyword(s):

Embedded Systems ◽

Elliptic Curve ◽

Mobile Device ◽

Operating Efficiency ◽

Elliptic Curve Cryptosystem ◽

Generation Algorithm ◽

Curve Method ◽

Security Guarantees ◽

Elliptic Curve Method ◽

Improved Algorithm

Elliptic curve cryptosystem is used in the process of embedded systems, the selection and generation algorithm of the elliptic curve will directly affect the efficiency of systems. From Elliptic Curve's selection, Elliptic Curve's structure, Elliptic Curve's generation, this paper discussed the realization of a random elliptic curve method of Embedded Mobile Device, the SEA algorithm and its improved algorithm. The results show that this method can achieve a quick implementation of the elliptic curve method to improve the operating efficiency of embedded systems in the same security guarantees.

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