Lenstra elliptic-curve factorization

Background Lenstra elliptic-curve factorization method uses a elliptic curve mod n (i.e. the number to be factored) and multiplies a random point P on it. The multiplication is based on elliptic curve point multiplication, which in turn is just the repeated addition of elliptic curve points, described in the article on elliptic curves. This addition would form a group in the non-modular case and in the case when n is prime, because \mathbb{Z}/n\mathbb{Z} (the integers modulo n) forms a group when n is prime. When modular numbers are used instead of the whole range of integers, the addition of two points on the same elliptic curve would involve taking the modular slope of a chord joining P and Q, and thus division between residue classes modulo n, performed using the extended Euclidean algorithm. In particular, division by some v \bmod n includes calculation of the \gcd(v,n). Assuming we calculate a slope of the form u/v with \gcd(u,v)=1, then if v = 0 \bmod n, the result of the point addition will be \infty, the point "at infinity" corresponding to the intersection of the "vertical" line joining P(x,y), P'(x,-y) and the curve. However, if \gcd(v,n) \neq 1, n, then the point addition will not produce a meaningful point on the curve; but, more importantly, \gcd(v,n) is a non-trivial factor of n: meaning that we have successfully factored the number. The usual multiplication methods such as multiplication by doubling still apply. Naive successive addition is not required. Process The Lenstra elliptic-curve factorization method to find a factor of a given natural number n works as follows: • Pick a random elliptic curve over \mathbb{Z}/n\mathbb{Z} (the integers modulo n), with equation of the form y^2 = x^3 + ax + b \pmod n together with a non-trivial point P(x_0,y_0) on it. • :This can be done by first picking random x_0,y_0,a \in \mathbb{Z}/n\mathbb{Z}, and then setting b = y_0^2 - x_0^3 - ax_0\pmod n to ensure the point is on the curve. • As discussed above, we have defined addition and multiplication of a point on the curve. With enough repeated additions, we should be able to cause a failure to add, thus finding us a factor. As a result, we compute [k]P on the elliptic curve (\bmod n), where k is a product of many small numbers. • k can be a product of small primes raised to small powers, as in the p-1 algorithm, or the factorial B! for some not too large B. This can be done efficiently, one small factor at a time. Say, to get [B!]P, first compute [2]P, then [3]([2]P), then [4]([3!]P), and so on. B is picked to be small enough so that B-wise point addition can be performed in reasonable time. • Check the result of the addition. • If we finish all the calculations above without encountering non-invertible elements (\bmod n), it means that the elliptic curves' (modulo primes) order is not smooth enough, so we need to try again with a different curve and starting point. • If we encounter a p = \gcd(v,n) \neq 1 we are done: it is a non-trivial factor of n. The time complexity depends on the size of the number's smallest prime factor and can be represented by , where p is the smallest factor of n, or L_p\left[\frac{1}{2},\sqrt{2}\right], in L-notation. ==Explanation==

If p and q are two prime divisors of n, then implies the same equation also and These two smaller elliptic curves with the \boxplus-addition are now genuine groups. If these groups have Np and Nq elements, respectively, then for any point P on the original curve, by Lagrange's theorem, is minimal such that kP=\infty on the curve modulo p implies that k divides Np; moreover, N_p P=\infty. The analogous statement holds for the curve modulo q. When the elliptic curve is chosen randomly, then Np and Nq are random numbers close to and respectively (see below). Hence it is unlikely that most of the prime factors of Np and Nq are the same, and it is quite likely that while computing eP, we will encounter some kP that is ∞ but not 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 or but not both. That is, gave a non-trivial factor ECM is at its core an improvement of the older algorithm. The algorithm finds prime factors p such that is b-powersmooth for small values of b. For any e, a multiple of and any a relatively prime to p, by Fermat's little theorem we have . Then is likely to produce a factor of n. However, the algorithm fails when has large prime factors, as is the case for numbers containing strong primes, for example. ECM gets around this obstacle by considering the group of a random elliptic curve over the finite field Zp, rather than considering the multiplicative group of Zp which always has order  The order of the group of an elliptic curve over Zp varies (quite randomly) between and by Hasse's theorem, and is likely to be smooth for some elliptic curves. Although there is no proof that a smooth group order will be found in the Hasse-interval, by using heuristic probabilistic methods, the Canfield–Erdős–Pomerance theorem with suitably optimized parameter choices, and the L-notation, we can expect to try curves before getting a smooth group order. This heuristic estimate is very reliable in practice. ==Example usage==

The following example is from , with some details added. We want to factor n = 455839. Let's choose the elliptic curve y^2 = x^3 + 5x - 5, with the point P=(1,1) on it, and let's try to compute the point (10!)P. The slope of the tangent line at some point A=(x,y) on the curve is \lambda=\frac{3x^2+5}{2y}\ (\mathrm{mod}\ n). Using \lambda, we can compute point 2A. If the value of \lambda does not exist, as a result of y not having a modular inverse, then \gcd(n,y) is a non-trivial factor of n. First, we compute 2!P. Using point doubling, we have \lambda(P)=\lambda(1,1)=4, so the coordinates of point 2P=(x',y') are :x'=4^2-2(1)=14 :y'=4(1-14)-1=-53 yielding the point 2P=(14,-53). Next, we compute 3!P. We have \lambda(2P)=\lambda(14,-53)=-593/106\ (\mathrm{mod}\ n). Since \gcd(106,455839)=1, the modular inverse of 106 exists. Using the extended Euclidean algorithm, we can obtain that \lambda=-593/106=322522\ (\mathrm{mod}\ 455839). Given this, we can compute the coordinates of 2(2P), just as we did above. The coordinates of point 4P=(x',y') are :x'=322522^2-2(14)=259851\pmod{455839} :y'=322522(14-259851)-(-53)=116255\pmod{455839} This yields 4P=(259851,116255). After this, we can compute 3(2P) = 4P + 2P using point addition. The line joining 4P and 2P has slope \lambda=116308/259837=206097\ (\mathrm{mod}\ n), so the coordinates of 6P=(x',y') are :x'=206097^2-14-259851=179685\pmod{455839} :y'=206097(14-179685)-(-53)=427131\pmod{455839} yielding the point 6P=(179685,427131) We can similarly compute points 4!P, 5!P, and so on, but computing 8!P requires inverting , which is not possible because \gcd(599,455839)=599\ne1. Thus 599 is a divisor of 455839. After a quick division, we have . The reason this works is that the curve has points, while it has points. Moreover, 640 and 777 are the smallest positive integers k such that on the curve and respectively. Since is a multiple of 640 but not a multiple of 777, we have on the curve but not on the curve hence the repeated addition broke down here, yielding the factorization. ==The algorithm, with projective coordinates==

Before considering the projective plane over (\Z/n\Z)/\sim, first consider a 'normal' projective space over \mathbb{R}: Instead of points, lines through the origin are studied. A line may be represented as a non-zero point (x,y,z), under an equivalence relation ~ given by: (x,y,z)\sim(x',y',z') ⇔ ∃ c ≠ 0 such that ''x' = cx, y' = cy and z' = cz''. Under this equivalence relation, the space is called the projective plane \mathbb{P}^2; points, denoted by (x:y:z), correspond to lines in a three-dimensional space that pass through the origin. Note that the point (0:0:0) 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. The corrdinate (x:y:z) corresponds to (x/z:y/z) in affine space. In the algorithm, only the group structure of an elliptic curve over the field \mathbb{R} is used. Since we do not necessarily need the field \mathbb{R}, a finite field will also provide a group structure on an elliptic curve. However, considering the same curve and operation over (\Z/n\Z)/\sim with 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 (0:1:0). Let be a (positive) integer to be factored and consider the elliptic curve (a set of points with some structure on it) E(\Z/n\Z)=\{(x:y:z) \in \mathbb{P}^2\ |\ y^2z=x^3+axz^2+bz^3\}. • Pick x_P,y_P,a \in \Z/n\Z with ≠ 0. • Calculate b = y_P^2 - x_P^3 - ax_P. The elliptic curve is then in Weierstrass form given by y^2 = x^3 + ax + b and by using projective coordinates the elliptic curve is given by the homogeneous equation ZY^2=X^3+aZ^2X+bZ^3. It has the point P=(x_P:y_P:1). • Choose an upperbound B \in \Z for this elliptic curve. • Remark: You will only find factors if the group order of the elliptic curve over \Z/p\Z (denoted by \#E(\Z/p\Z)) is B-smooth, which means that all prime factors of n have to be less or equal to . • Calculate k={\rm lcm}(1,\dots ,B). • Calculate kP := P + P + \cdots + P (multiplication is repeated addition) in the ring E(\Z/n\Z). • If the calculation successfully returns kP = (0:1:0), it means is not -smooth or is prime. Go back to step 2 to pick another curve. • If the calculation fails at some point, it means a non-trivial divisor can be found. It can fail because addition and multiplication are not well-defined if is not prime, but this only occurs when a inversion of a particular residue is tried. In this case the factor is found as \gcd(v,n) as above. In point 5 it is said that under the right circumstances a non-trivial divisor can be found. As pointed out in Lenstra's article (Factoring Integers with Elliptic Curves) the addition needs the assumption \gcd(x_1-x_2,n)=1. If P,Q are not (0:1:0) and distinct (otherwise addition works similarly, but is a little different), then addition works as follows: • To calculate: R = P + Q; P = (x_1:y_1:1), Q = (x_2:y_2:1), • \lambda =(y_1-y_2) (x_1-x_2)^{-1}, • x_3 = \lambda^2 - x_1 - x_2, • y_3 = \lambda(x_1-x_3) - y_1, • R = P + Q = (x_3:y_3:1). If addition fails, this will be due to a failure calculating \lambda. In particular, because (x_1-x_2)^{-1} can not always be calculated if is not prime (and therefore \Z/n\Z is not a field). Without making use of \Z/n\Z being a field, one could calculate: • \lambda'=y_1-y_2, • x_3' = {\lambda'}^2 - x_1(x_1-x_2)^2 - x_2(x_1-x_2)^2, • y_3' = \lambda'(x_1(x_1-x_2)^2-x_3') - y_1(x_1-x_2)^3, • R = P + Q = (x_3'(x_1-x_2):y_3':(x_1-x_2)^3), and simplify if possible. This calculation is always legal and if the gcd of the -coordinate with ≠ (1 or ), so when simplifying fails, a non-trivial divisor of is found. Two-stage variant Analogous to the two-stage variant of Pollard's p − 1 algorithm, Lenstra ECM can too be done in two stages. This allows one to save a time factor of O(log p). • For each prime , B_1 \leq p \leq B_2, • Compute a point (x_p:y_p:z_p) = pQ on . • Compute g := \operatorname{gcd}\left(n, z_p\right). If g \neq 1, output g and exit. • If all primes in range are tried without producing a factor, report failure. It is possible for stage 1 to yield a factor like previously discussed: a non-invertible denominator implies a factor. B_1 is functionally the same as B from the standard version, so it too happens when the group order is B-smooth. In other words, one looks for a prime divisor such that sP is the neutral element of E(\mathbb{Z}/p\mathbb{Z}) in stage 1. The second stage is very similar to the second stage of p-1 and p+1. It is a continuation of the work in stage 1 and can be described using very similar mathematical terms. It relaxes the condition such that one can find a factor when is (B_1, B_2)-smooth, or in other words the largest prime factor of is at most B_2 and the second-smallest is at most B_1. To achieve stage 2, one hopes there is a prime between B_1 and B_2 such that pQ = (0:1:0) \mod p; looking for a failed inversion would produce it after a gcd. Equivalently, one is looking for a prime divisor such that sP has small prime order in E(\mathbb{Z}/q\mathbb{Z}). Checking for a small order of sP is done in stage 2 by computing (ls)P modulo for each prime . This approach was later extended to p-1 and p+1.(Montgomery and Kruppa 2008). ==Twisted Edwards curves==

The use of Edwards curves needs fewer modular multiplications and less time than the use of Montgomery curves or Weierstrass curves (other used methods). Using Edwards curves you can also find more primes. Definition. Let k be a field in which 2 \neq 0, and let a,d \in k\setminus\{0\} with a\neq d. Then the twisted Edwards curve E_{E,a,d} is given by ax^2+y^2=1+dx^2y^2. An Edwards curve is a twisted Edwards curve in which a=1. There are five known ways to build a set of points on an Edwards curve: the set of affine points, the set of projective points, the set of inverted points, the set of extended points and the set of completed points. The set of affine points is given by: :\{(x,y)\in \mathbb{A}^2 : ax^2+y^2=1+dx^2y^2\}. The addition law is given by :(e,f),(g,h) \mapsto \left(\frac{eh+fg}{1+ degfh},\frac{fh-aeg}{1-degfh}\right). The point (0,1) is its neutral element and the inverse of (e,f) is (-e,f). The other representations are defined similar to how the projective Weierstrass curve follows from the affine. Any elliptic curve in Edwards form has a point of order 4. So the torsion group of an Edwards curve over \Q is isomorphic to either \Z/4\Z, \Z/8\Z, \Z/12\Z,\Z/2\Z \times \Z/4\Z or \Z/2\Z\times \Z/8\Z. The most interesting cases for ECM are \Z/12\Z and \Z/2\Z\times \Z/8\Z, since they force the group orders of the curve modulo primes to be divisible by 12 and 16 respectively. The following curves have a torsion group isomorphic to \Z/12\Z: • x^2+y^2=1+dx^2y^2 with point (a,b) where b \notin\{-2,-1/2,0,\pm1\}, a^2=-(b^2+2b) and d=-(2b+1)/(a^2b^2) • x^2+y^2=1+dx^2y^2 with point (a,b) where a=\frac{u^2-1}{u^2+1}, b=-\frac{(u-1)^2}{u^2+1} and d=\frac{(u^2+1)^3(u^2-4u+1)}{(u-1)^6(u+1)^2}, u\notin\{0,\pm1\}. Every Edwards curve with a point of order 3 can be written in the ways shown above. Curves with torsion group isomorphic to \Z/2\Z\times \Z/8\Z and \Z/2\Z\times \Z/4\Z may be more efficient at finding primes. == Software implementations ==

GMP-ECM of Paul Zimmerman is a general-purpose implementation of the Lenstra algorithm based on the GNU Multiple Precision Arithmetic Library. It has been continually updated, with the latest version as of September 2025 being 7.0.6 from July 2024. It allows Montgomery, Weierstrass, and (twisted) Hessian curves. It can run stage 1 for a subset of Montgomery curves on a CUDA GPU, with the earlier implementation by Cyril Bouvier in 2012 having been replaced by Seth Troisi's newer implementation in 2021. Version 7.0.6 also includes an implementation of the HECM method, (described below), the p-1 and p+1 methods, and primality proving using APRCL. GMP-ECM is used in SageMath. Daniel J. Bernstein and coworkers have published a series of implementations based on Twisted Edwards elliptic curves between 2008 and 2010. They all claim to outperform the contemporary version of GMP-ECM, with the latest being EECM-MPFQ of 2008. Two GPU implementations are also available from Bernstein, the newer and faster one being CUDA-EECM of 2009. Prime95 includes an implementation of Lenstra ECM for Montgomery and Edwards curves. It is used for the ECM subproject of Great Internet Mersenne Prime Search, which seeks to factor composite Mersenne numbers no smaller than 21213. It can produce stage 1 output compatible with GMP-ECM as well as consume stage 1 output from GMP-ECM. It is faster than GMP-ECM at stage 1. John Wloka and coworkers published ecmongpu, an implementation of both stage 1 and stage 2 of Lenstra ECM based on Twisted Edwards elliptic curves, in 2020. Their paper report on performance for factoring moduli up to 448 bits long (between 2447 and 2448-1). All software listed above are open-source. In addition, the open-source PARI/GP and proprietary Magma (computer algebra system) also contain "good ECM implementations" according to Paul Zimmerman. is giantint by Richard Crandall. ==Hyperelliptic-curve method (HECM)==

There are recent developments in using hyperelliptic curves to factor integers. Cosset shows in his article (of 2010) that one can build a hyperelliptic curve with genus two (so a curve y^2 = f(x) with of degree 5), which gives the same result as using two "normal" elliptic curves at the same time. By making use of the Kummer surface, calculation is more efficient. The disadvantages of the hyperelliptic curve (versus an elliptic curve) are compensated by this alternative way of calculating. Therefore, Cosset roughly claims that using hyperelliptic curves for factorization is no worse than using elliptic curves. ==Quantum version (GEECM)==

Bernstein, Heninger, Lou, and Valenta suggest GEECM, a quantum version of ECM with Edwards curves. It uses Grover's algorithm to roughly double the length of the primes found compared to standard EECM, assuming a quantum computer with sufficiently many qubits and of comparable speed to the classical computer running EECM. ==References==