Lenstra elliptic-curve factorization: Difference between revisions

Revision as of 05:56, 22 March 2024

Latest revision as of 17:59, 17 August 2025

(7 intermediate revisions by 6 users not shown)

Line 9:

The Lenstra elliptic-curve factorization method to find a factor of a given natural number <math>n</math> works as follows:

# Pick a random [[elliptic curve]] over <math>\mathbb{Z}/n\mathbb{Z}</math> (the integers modulo <math>n</math>), with equation of the form <math>y^2 = x^3 + ax + b \pmod n</math> together with a non-trivial [[Point (geometry)|point]] <math>P(x_0,y_0)</math> on it.

#:This can be done by first picking random <math>x_0,y_0,a \in \mathbb{Z}/n\mathbb{Z}</math>, and then setting <math>b = y_0^2 - x_0^3 - ax_0\pmod n</math> to assureensure the point is on the curve.

# One can define ''Addition'' of two points on the curve, to define a [[Group (mathematics)|group]]. The addition laws are given in the [[elliptic curve#The group law|article on elliptic curves]].

#:We can form repeated multiples of a point <math>P</math>: <math>[k]P = P + \ldots + P \text{ (k times)}</math>. The addition formulae involve taking the modular slope of a chord joining <math>P</math> and <math>Q</math>, and thus division between residue classes modulo <math>n</math>, performed using the [[extended Euclidean algorithm]]. In particular, division by some <math>v \bmod n</math> includes calculation of the <math>\gcd(v,n)</math>.

Line 23:

If ''p'' and ''q'' are two prime divisors of ''n'', then {{math|1=''y''²&nbsp;=&nbsp;''x''³&nbsp;+}} {{math|1=''ax''&nbsp;+&nbsp;''b''&nbsp;(mod&nbsp;''n'')}} implies the same equation also {{math|1=modulo&nbsp;''p''}} and {{math|1=modulo&nbsp;''q''.}} These two smaller elliptic curves with the <math>\boxplus</math>-addition are now genuine [[group (mathematics)|groups]]. If these groups have ''N''_''p'' and ''N_q'' elements, respectively, then for any point ''P'' on the original curve, by [[Lagrange's theorem (group theory)|Lagrange's theorem]], {{math|1=''k''&nbsp;>&nbsp;0}} is minimal such that <math>kP=\infty</math> on the curve modulo ''p'' implies that ''k'' divides ''N''_''p''; moreover, <math>N_p P=\infty</math>. The analogous statement holds for the curve modulo ''q''. When the elliptic curve is chosen randomly, then ''N''_''p'' and ''N''_''q'' are random numbers close to {{math|1=''p''&nbsp;+&nbsp;1}} and {{math|1=''q''&nbsp;+&nbsp;1,}} respectively (see below). Hence it is unlikely that most of the prime factors of ''N''_''p'' and ''N''_''q'' are the same, and it is quite likely that while computing ''eP'', we will encounter some ''kP'' that is &infin; {{math|1=modulo&nbsp;''p''}} but not {{math|1=modulo&nbsp;''q'',}} or vice versa. When this is the case, ''kP'' does not exist on the original curve, and in the computations we found some ''v'' with either {{math|1=gcd(''v'',''p'')&nbsp;=&nbsp;''p''}} or {{math|1=gcd(''v'',&nbsp;''q'')&nbsp;=&nbsp;''q'',}} but not both. That is, {{math|1=gcd(''v'',&nbsp;''n'')}} gave a non-trivial factor {{math|1=of&nbsp;''n''.}}

ECM is at its core an improvement of the older [[Pollard's p &minus; 1 algorithm|{{math|1=''p''&nbsp;&minus;&nbsp;1}} algorithm]]. The {{math|1=''p''&nbsp;&minus;&nbsp;1}} algorithm finds prime factors ''p'' such that {{math|1=''p''&nbsp;&minus;&nbsp;1}} is [[smooth number|b-powersmooth]] for small values of ''b''. For any ''e'', a multiple of {{math|1=''p''&nbsp;&minus;&nbsp;1,}} and any ''a'' [[relatively prime]] to ''p'', by [[Fermat's little theorem]] we have {{math|1=''a''^''e''&nbsp;&equiv;&nbsp;1 ([[modular arithmetic|mod]] ''p'')}}. Then {{math|1=[[greatest common divisor|gcd]](''a''^''e''&nbsp;&minus;&nbsp;1,&nbsp;''n'')}} is likely to produce a factor of ''n''. However, the algorithm fails when {{math|1=''p''&nbsp;-−&nbsp;1}} has large prime factors, as is the case for numbers containing [[strong prime]]s, for example.

ECM gets around this obstacle by considering the [[group (mathematics)|group]] of a random [[elliptic curve]] over the [[finite field]] '''Z'''_''p'', rather than considering the [[multiplicative group]] of '''Z'''_''p'' which always has order&nbsp;{{math|1=''p''&nbsp;&minus;&nbsp;1.}}

Line 33:

The following example is from {{harvtxt|Trappe|Washington|2006}}, with some details added.

We want to factor {{<math|1=''>n'' = 455839</math>.}} Let's choose the elliptic curve {{<math|1=''>y''^{^2} = ''x''^{^3} + 5''x''5x –- 5</math>,}} with the point {{<math|1=''>P'' = (1, 1)}}</math> on it, and let's try to compute {{the point <math|1=>(10!)''P''</math>.}}

The slope of the tangent line at some point ''<math>A''=(''x'', ''y'')</math> ison {{math|1=''s''the =curve is (3''x''<supmath>\lambda=\frac{3x^2</sup> + 5)/(2''y'')}{2y}\ (\mathrm{mod}\ n)}}</math>. Using ''s''<math>\lambda</math>, we can compute 2''A''point <math>2A</math>. If the value of ''s''<math>\lambda</math> isdoes ofnot theexist, formas ''a/b'' whereresult ''b''of <math>y</math> 1not andhaving gcd(''a'',''b'') = 1, we have to find the [[modular inverse]], of ''b''. If it does not exist,then <math>\gcd(''n'',''b''y)</math> is a non-trivial factor of ''<math>n''</math>.

First, we compute <math>2!P</math>. Using [[Elliptic_curve_point_multiplication#Point_doubling|computepoint 2''P''doubling]]., Wewe have {{<math|1=''s''>\lambda(''P'') = ''s''\lambda(1,1) = 4</math>,}} so the coordinates of {{math|1=2''P''point = (''x′'', ''y′'')}} are {{<math|1=''x′'' = ''s''² – 2''x'' 2P= 14}} and {{math|1=''y′'' = ''s''(''x'' – ''x′'') – '',y''}} {{math|1== 4(1 – 14) – 1 = –53,}} all numbers understood {{math|1=(mod ''n'').}} Just to check that this 2''P'' is indeed on the curve: {{nowrap|1=(–53)<sup>2</supmath> = 2809 = 14³ + 5·14 – 5.}}are

:<math>x'=4^2-2(1)=14</math>

:<math>y'=4(1-14)-1=-53</math>

yielding the point <math>2P=(14,-53)</math>.

Next, we compute <math>3!P</math>. We have <math>\lambda(2P)=\lambda(14,-53)=-593/106\ (\mathrm{mod}\ n)</math>. Since <math>\gcd(106,455839)=1</math>, the modular inverse of 106 exists. Using the [[extended Euclidean algorithm]], we can obtain that <math>\lambda=-593/106=322522\ (\mathrm{mod}\ 455839)</math>.

Then we compute 3(2''P''). We have {{math|1=''s''(2''P'') = ''s''(14,-53) = –593/106 (mod ''n'').}} Using the [[Euclidean algorithm]]: {{nowrap|1=455839 = 4300·106 + 39,}} then {{nowrap |1=106 = 2·39 + 28,}} then {{nowrap|1=39 = 28 + 11,}} then {{nowrap |1=28 = 2·11 + 6,}} then {{nowrap|1=11 = 6 + 5,}} then {{nowrap |1=6 = 5 + 1.}} Hence {{nowrap|1=gcd(455839, 106) = 1,}} and working backwards (a version of the [[extended Euclidean algorithm]]): {{nowrap |1=1 = 6 – 5 = 2·6 – 11 = 2·28 – 5·11}} {{nowrap |1== 7·28 – 5·39 = 7·106 – 19·39 = 81707·106 – 19·455839.}} Hence {{nowrap |1=106⁻¹ = 81707 (mod 455839),}} and {{nowrap |1=–593/106 = –133317 (mod 455839).}} Given this ''s'', we can compute the coordinates of 2(2''P''), just as we did above: {{math|1=4''P'' = (259851, 116255).}} Just to check that this is indeed a point on the curve: {{math|1=''y''² = 54514 = ''x''³ + 5''x'' – 5 (mod 455839).}} After this, we can compute <math>3(2P) = 4P \boxplus 2P</math>.

Given this, we can compute the coordinates of <math>2(2P)</math>, just as we did above. The coordinates of point <math>4P=(x',y')</math> are

We can similarly compute 4!''P'', and so on, but 8!''P'' requires inverting {{nowrap|1=599 (mod 455839).}} The Euclidean algorithm gives that 455839 is divisible by 599, and we have found a {{nowrap|1=factorization 455839 = 599·761.}}

:<math>x'=322522^2-2(14)=259851\pmod{455839}</math>

:<math>y'=322522(14-259851)-(-53)=116255\pmod{455839}</math>

This yields <math>4P=(259851,116255)</math>.

After this, we can compute <math>3(2P) = 4P + 2P</math> using [[Elliptic_curve_point_multiplication#Point_addition|point addition]]. The line joining <math>4P</math> and <math>2P</math> has slope <math>\lambda=116308/259837=206097\ (\mathrm{mod}\ n)</math>, so the coordinates of <math>6P=(x',y')</math> are

The reason that this worked is that the curve {{nowrap|1=(mod 599)}} has {{nowrap|1=640 = 2⁷·5}} points, while {{nowrap|1=(mod 761)}} it has {{nowrap|1=777 = 3·7·37}} points. Moreover, 640 and 777 are the smallest positive integers ''k'' such that {{math|1=''kP'' = &infin;}} on the curve {{nowrap|1=(mod 599)}} and {{math|1=(mod 761),}} respectively. Since {{nowrap|8!}} is a multiple of 640 but not a multiple of 777, we have {{math|1=8!''P'' = &infin;}} on the curve {{nowrap|1=(mod 599),}} but not on the curve {{nowrap|1=(mod 761),}} hence the repeated addition broke down here, yielding the factorization.

:<math>x'=206097^2-14-259851=179685\pmod{455839}</math>

:<math>y'=206097(14-179685)-(-53)=427131\pmod{455839}</math>

yielding the point <math>6P=(179685,427131)</math>

We can similarly compute points <math>4!''P''</math>, <math>5!P</math>, and so on, but computing <math>8!''P''</math> requires inverting {{nowrap|1=599 (mod 455839).}}, Thewhich Euclideanis algorithmnot givespossible thatbecause <math>\gcd(599,455839)\ne1</math>. is divisible by 599Thus, and we have found a factorization {{nowrap|1=factorization 455839 = 599·761.}}.

The reason that this workedworks is that the curve {{nowrap|1=(mod 599)}} has {{nowrap|1=640 = 2⁷·5}} points, while {{nowrap|1=(mod 761)}} it has {{nowrap|1=777 = 3·7·37}} points. Moreover, 640 and 777 are the smallest positive integers ''k'' such that {{math|1=''kP'' = &infin;}} on the curve {{nowrap|1=(mod 599)}} and {{mathnowrap|1=(mod 761),}} respectively. Since {{nowrap|8!}} is a multiple of 640 but not a multiple of 777, we have {{math|1=8!''P'' = &infin;}} on the curve {{nowrap|1=(mod 599),}} but not on the curve {{nowrap|1=(mod 761),}} hence the repeated addition broke down here, yielding the factorization.

==The algorithm with projective coordinates==

Before considering the projective plane over <math>(\Z/n\Z)/\sim,</math> first consider a 'normal' [[projective space]] over ℝ<math>\mathbb{R}</math>: Instead of points, lines through the origin are studied. A line may be represented as a non-zero point <math>(x,y,z)</math>, under an equivalence relation ~ given by: <math>(x,y,z)\sim(x',y',z')</math> ⇔ ∃ '''''c''''' ≠ 0 such that ''x' = '''c'''x'', ''y' = '''c'''y'' and ''z' = '''c'''z''. Under this equivalence relation, the space is called '''the projective plane''' <math>\mathbb{P}^2</math>; points, denoted by <math>(x:y:z)</math>, correspond to lines in a three-dimensional space that pass through the origin. Note that the point <math>(0:0:0)</math> does not exist in this space since to draw a line in any possible direction requires at least one of x',y' or z' ≠ 0. Now observe that almost all lines go through any given reference plane - such as the (''X'',''Y'',1)-plane, whilst the lines precisely parallel to this plane, having coordinates (''X,Y'',0), specify directions uniquely, as 'points at infinity' that are used in the affine (''X,Y'')-plane it lies above.

In the algorithm, only the group structure of an elliptic curve over the field ℝ<math>\mathbb{R}</math> is used. Since we do not necessarily need the field ℝ<math>\mathbb{R}</math>, a finite field will also provide a group structure on an elliptic curve. However, considering the same curve and operation over <math>(\Z/n\Z)/\sim</math> with {{mvar|n}} not a prime does not give a group. The Elliptic Curve Method makes use of the failure cases of the addition law.

We now state the algorithm in projective coordinates. The neutral element is then given by the point at infinity <math>(0:1:0)</math>. Let {{mvar|n}} be a (positive) integer and consider the elliptic curve (a set of points with some structure on it) <math>E(\Z/n\Z)=\{(x:y:z) \in \mathbb{P}^2\ |\ y^2z=x^3+axz^2+bz^3\}</math>.

Line 117 ⟶ 130:

==GMP-ECM and EECM-MPFQ==

The use of Twisted Edwards elliptic curves, as well as other techniques were used by Bernstein et al<ref name=Bernstein2008 /> to provide an optimized implementation of ECM. Its only drawback is that it works on smaller composite numbers than the more general purpose implementation, GMP-ECM of ZimmermanZimmermann.

==Hyperelliptic-curve method (HECM)==

Line 133 ⟶ 146:

*{{cite book |last=Cohen |first=Henri |title=A Course in Computational Algebraic Number Theory |volume=138 |publisher=Springer-Verlag |___location=Berlin |year=1993 |isbn=978-0-387-55640-6 | mr=1228206 | doi=10.1007/978-3-662-02945-9|series=Graduate Texts in Mathematics |s2cid=118037646 }}

*{{cite journal |last=Cosset |first=R. |title=Factorization with genus 2 curves |journal=Mathematics of Computation |volume=79 |issue=270 |year=2010 |pages=1191–1208 |doi=10.1090/S0025-5718-09-02295-9 | mr=2600562|arxiv=0905.2325 |bibcode=2010MaCom..79.1191C |s2cid=914296 }}

*{{cite book |editor1-link=Arjen Lenstra |editor1-last=Lenstra |editor1-first=A. K. |editor2-last=Lenstra Jr. |editor2-first=H. W. |title=The development of the number field sieve |series=Lecture Notes in Mathematics |volume=1554 |publisher=Springer-Verlag |___location=Berlin |year=1993 |pages=11–42 |mr=1321216 | doi=10.1007/BFb0091534|isbn=978-3-540-57013-4 |url=http://infoscience.epfl.ch/record/164684 }}

*{{cite journal |last=Lenstra Jr. |first=H. W. |title=Factoring integers with elliptic curves |journal=[[Annals of Mathematics]] |volume=126 |year=1987 |issue=3 |pages=649–673 |mr=0916721 |doi=10.2307/1971363 |url=https://openaccess.leidenuniv.nl/bitstream/handle/1887/2140/346_079.pdf?sequence=1 |jstor=1971363 |hdl=1887/2140 |hdl-access=free }}

*{{cite book

Line 170 ⟶ 183:

* [http://ecm.gforge.inria.fr/ GMP-ECM] {{Webarchive|url=https://web.archive.org/web/20090912000929/http://ecm.gforge.inria.fr/ |date=2009-09-12 }}, an efficient implementation of ECM.

* [http://www.loria.fr/~zimmerma/records/ecmnet.html ECMNet], an easy client-server implementation that works with several factorization projects.

* [httphttps://www.sourceforge.net/projects/pyecm pyecm], a python implementation of ECM.

* [http://www.rechenkraft.net/yoyo/ Distributed computing project yoyo@Home] Subproject ECM is a program for Elliptic Curve Factorization which is used to find factors for different kinds of numbers.

* [https://web.archive.org/web/20130811025532/http://ardoino.com/2008/03/large-integers-factorization/ Lenstra Elliptic Curve Factorization algorithm source code] Simple C and GMP Elliptic Curve Factorization Algorithm source code.