ECM using Edwards curves

ECM using Edwards curves

Cached

Download Links

by Daniel J. Bernstein , Peter Birkner , Tanja Lange , Christiane Peters

Citations:

BibTeX

@MISC{Bernstein_ecmusing,
    author = {Daniel J. Bernstein and Peter Birkner and Tanja Lange and Christiane Peters},
    title = {ECM using Edwards curves},
    year = {}
}

Bookmark

OpenURL

Abstract

Abstract. This paper introduces GMP-EECM, a fast implementation of the elliptic-curve method of factoring integers. GMP-EECM is based on, but faster than, the well-known GMP-ECM software. The main changes are as follows: (1) use Edwards curves instead of Montgomery curves; (2) use twisted inverted Edwards coordinates; (3) use signedsliding-window addition chains; (4) batch primes to increase the window size; (5) choose curves with small parameters a, d, X1, Y1, Z1; (6) choose curves with larger torsion.

Citations

149	Speeding the Pollard and elliptic curve methods of factorization - Montgomery - 1987
121	The arithmetic of elliptic curves, Graduate Texts - Silverman - 1986
89	Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Adv - Chudnovsky, Chudnovsky
78	Rational Isogenies of Prime Degree - Mazur - 1978
61	Theorems of factorization and primality testing - Pollard - 1974
60	Playoust: The Magma algebra system. I. The user language - Bosma, Cannon, et al. - 1997
41	Some integer factorization algorithms using elliptic curves - Brent - 1986
36	An FFT Extension of the Elliptic Curve Method of Factorization - Montgomery - 1992
31	A normal form for elliptic curves - Edwards
28	Finding suitable curves for the elliptic curve method of factorization - Atkin, Morain - 1993
26	Efficient exponentiation using precomputation and vector addition chains - Rooij - 1994
17	Fast multiplication and its applications - Bernstein - 2008
14	Tanja Lange, Explicit-formulas database (2007). URL: http:// hyperelliptic.org/EFD. Citations in this document: §2 - Bernstein
14	The mpFq library and implementing curve-based key exchanges - Gaudry, Thomé - 2007
13	Fast genus 2 arithmetic based on Theta functions, 2005, to appear - Gaudry
12	A practical analysis of the elliptic curve factoring algorithm - Silverman, Wagstaff
11	Factoring integers with elliptic curves’, Annals of Math., 126 - Lenstra - 1987
10	Tanja Lange, Faster addition and doubling on elliptic curves - Bernstein
7	Twisted Edwards curves revisited - Hisil, Wong, et al.
5	Tanja Lange, Inverted Edwards coordinates - Bernstein
4	Subtleties in the distribution of the numbers of points on elliptic curves over a finite prime field - McKee - 1999
4	Tanja Lange, Christiane Peters, Twisted Edwards curves - Bernstein, Birkner, et al.
4	On the frequency of numbers containing primes of a certain relative magnitude - Dickman - 1930
3	Hsiao-Feng Lu (editors), Applied algebra, algebraic algorithms and errorcorrecting codes, 17th international symposium, AAECC-17 - Boztas
3	Arbitrarily tight bounds on the distribution of smooth integers - Bernstein
3	Pohst (editors), Algorithmic number theory, proceedings of the 7th international symposium (ANTS-VII) held at the Technische Universität - Hess, Pauli, et al.
2	Tanja Lange, Christiane Peters, Optimizing doublebase elliptic-curve single-scalar multiplication - Bernstein, Birkner
2	Tanja Lange, A complete set of addition laws for incomplete Edwards curves - Bernstein
2	Stevenhagen (editors), Algorithmic number theory: lattices, number fields, curves and cryptography - Buhler, Peter - 2008
2	Factorization with genus 2 curves - Cosset
2	Pandu Rangan, Moti Yung (editors - Srinathan, C
1	Elliptic curves and primality proving, Mathematics of Computation 61 - Atkin, Morain - 1993
1	Tanja Lange, Analysis and optimization of elliptic-curve single-scalar multiplication - Bernstein
1	theory for the millennium. I: papers from the conference held at the University of Illinois at Urbana-Champaign - Bennett, Berndt, et al.
1	Scaled remainder trees (2004). URL: http://cr.yp.to/papers. html#scaledmod. Citations in this document: §5.3 - Bernstein
1	Tanja Lange, Starfish on strike - Bernstein, Birkner
1	Shparlinski (editors), Finite fields and applications: papers from the 8th international conference held - Mullen, Panario, et al.
1	Informal preliminary report (8), cited in [18] as personal communication and in [36 - Suyama - 1985
1	50 largest factors found by ECM. URL: http://www.loria.fr/ ~zimmerma/records/top50.html. Citations in this document: §1 - Zimmermann
1	20 Years of ECM, in ANTS VII [28] (2006), 525–542. Citations in this document - Zimmermann, Dodson
1	Améliorations de la multiplication et de la factorisation d’entier - Kruppa
1	20 years of ECM, in ANTS VII [28] (2006), 525–542. URL: http://www.loria.fr/~zimmerma/papers/40760525.pdf. Citations in this document - Zimmermann, Dodson