Modern Elliptic Curve Factorization, Part 2

April 27, 2010

The elliptic curve factorization algorithm is remarkably similar to Pollard’s p-1 algorithm:

define p-1(n, B1, B2)         1         define ecf(n, B1, B2, C)
        q = 2         2                 Qx,zC // random point
        for p ∈ π(2 … B1)         3                 for p ∈ π(2 … B1)
                a = ⌊logp B1         4                         a = ⌊logp B1
                q = qpa (mod n)         5                         Q = paC Q
        g = gcd(q, n)         6                 g = gcd(Qz, n)
        if (1 < g < n) return g         7                 if (1 < g < n) return g
        for p ∈ π(B1B2)         8                 for p ∈ π(B1B2)
                q = qp (mod n)         9                         Q = pC Q
          10                         g = g × Qz (mod n)
        g = gcd(q, n)         11                 g = gcd(g, n)
        if (1 < g < n) return g         12                 if (1 < g < n) return g
        return failure         13                 return failure

Lines 3 and 8 loop over the same lists of primes. Line 4 computes the same maximum exponent. Lines 6 and 7, and lines 11 and 12, compute the same greatest common divisor. And line 13 returns the same failure. The structures of the two algorithms are identical. Also, note that both algorithms use variables that retain their meaning from the first stage (lines 2 to 7) to the second stage (lines 8 to 12).

Lines 2, 5, 9 and 10 differ because p-1 is based on modular arithmetic and ecf is based on elliptic arithmetic (note that the ⊗C operator, otimes-sub-C, indicates elliptic multiplication on curve C). The most important difference is on the first line, where ecf has an extra argument C indicating the choice of elliptic curve. If the p-1 method fails, there is nothing to do, but if the ecf method fails, you can try again with a different curve.

Use of Montgomery’s elliptic curve parameterization from the previous exercise is a huge benefit. Lenstra’s original algorithm, which is the first stage of the modern algorithm, required that we check the greatest common divisor at each step, looking for a failure of the elliptic arithmetic. With the modern algorithm, we compute the greatest common divisor only once, at the end, because Montgomery’s parameterization has the property that a failure of the elliptic arithmetic propagates through the calculation. And without inversions, Montgomery’s parameterization is also much faster to calculate. Brent’s parameterization of the randomly-selected curve is also important, as it guarantees that the order of the curve (the number of points) will be a multiple of 12, which has various good mathematical properties that we won’t enumerate here.

There is a simple coding trick that can speed up the second stage. Instead of multiplying each prime times Q, we iterate over i from B1 + 1 to B2, adding 2Q at each step; when i is prime, the current Q can be accumulated into the running solution. Again, we defer the calculation of the greatest common divisor until the end of the iteration.

Your task is to write a function that performs elliptic curve factorization. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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6 Responses to “Modern Elliptic Curve Factorization, Part 2”

  1. Mike said

    Two questions:
    in line 6 of ecf, what is ‘q’
    in line 11 of ecf, is that supposed to be g = gcd(*q*,n) like in the p-1 function?

  2. programmingpraxis said

    In line 6, q should read Q-sub-z (it’s the z parameter of the point Q). I’ll fix that.

    In line 11, g is the g that is initialized on line 6 and accumulated on line 11.

  3. tiwari_mukesh said

    Since no one posted the source code for this so here is mine.How ever running this algorithm and Lenstra implementation, Lenstra seems to be good.

    --y^2 = x^3+ax^2+x mod n
    import Random
    import System.Random
    import Data.Bits
    data Elliptic = Cone Integer Integer Integer deriving (Eq,Show) 
    data Point = Conp Integer Integer  deriving (Eq,Show)
    
    
    addPoints::Point->Point->Point->Integer->Point
    addPoints (Conp 0 0) p2 _ _ = p2
    addPoints p1 (Conp 0 0) _ _ = p1
    addPoints (Conp x_1 z_1) (Conp x_2 z_2) (Conp x_0 z_0) n = 
    	let 
    		x_3= mod ((((x_1-z_1)*(x_2+z_2)+(x_1+z_1)*(x_2-z_2))^2)*z_0) n 
    		z_3= mod ((((x_1-z_1)*(x_2+z_2)-(x_1+z_1)*(x_2-z_2))^2)*x_0) n
    	in (Conp x_3 z_3)
    
    
    
    doublePoint::Point->Elliptic->Point
    doublePoint (Conp x z) (Cone an ad n) = Conp (mod x' n) (mod y' n)
    					where 
    					   x'= mod (4*ad*(x+z)^2*(x-z)^2) n
    					   y'= mod ((4*ad*(x-z)^2+t*an)*t) n 
    					    where 
    					     t=(x+z)^2-(x-z)^2
    
    
    
    --logupperbound inspired by programming praxis
    --http://programmingpraxis.com/2010/05/07/integer-logarithms/
    ilognew::Integer->Integer->Int
    ilognew b n = f (div ubound 2)  ubound where
        ubound = until (\e -> b ^ e > n) (* 2) 1 
        f lo hi | lo>=hi = div (lo+hi) 2 
                | 2 ^ mid <= n = f (mid+1) hi
                | otherwise = f lo mid
                where mid = div (lo + hi) 2
    
    
    --http://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication
    --montogomery ladder addition but do a binary search to find the 2^p>k so all bits from p-1 will be set 
    montMul::Integer->Elliptic->Point->Integer->Point
    montMul _ _ _ 0= (Conp 0 0)
    montMul n eCurve pOint k = tmpFun  (Conp 0 0)  pOint (ilognew 2 k  -1) where 
    	tmpFun r0 r1  0 = if (.&.) k ( shiftL (1::Integer) 0 ) /=0 then  addPoints r0 r1 pOint n  else  doublePoint r0 eCurve
    	tmpFun r0 r1  cnt |(.&.) k ( shiftL (1::Integer) cnt ) /=0  = tmpFun (addPoints r0 r1 pOint n) (doublePoint r1 eCurve)  (cnt-1)
    			  |otherwise = tmpFun (doublePoint r0 eCurve) (addPoints r0 r1 pOint n)  (cnt-1)
    
    
    
    
    
    
    montgomeryCurve::Integer-> IO [Integer]
    montgomeryCurve  n = do  
    			s<-getStdRandom (randomR (6,n-1))
    			let u=mod (s*s-5) n
    			let v=mod (4*s) n
    			let an=mod (((v-u)^3)*(3*u+v)) n
    			let ad=mod (4*u^3*v) n
    		        return [an,ad,mod (u*u*u) n,mod (v*v*v) n]
    
    
    prime = 2:filter rabinMiller  [3,5..]
    
    
    fun2mk :: Integer->(Integer,Integer)
    fun2mk n = let
    		(s,d)= f 0 (n-1) 
    		        where
    			  f s d | mod d 2 ==0 = f (s+1) (div d 2)
    				| otherwise = (s,d)
    	   in (s,d)
    
    
    rabinMiller :: Integer->Bool
    rabinMiller n   | n<2 = False
    		| n==2 = True
    		| even n = False
    		| b0==1 || b0== n' = True
    		| otherwise = iter (tail b)
    		where
    			n'= n-1
    			(k,m)=fun2mk n
    			a= head ( take 1 $ randomRs (2,n-2) (mkStdGen 3) :: [Integer])
    			b0 = mod (a^m) n
    			b= take (fromIntegral k) $ iterate (\x-> mod (x*x) n) b0
    			iter []= False
    			iter (x:xs) | x==1 = False
    				    | x==n'= True
    				    | otherwise = iter xs
    
    
    
    allp::Integer->Integer->[Integer]
    allp p_1 p_2 = takeWhile (<=p_2) $ dropWhile (<p_1) prime
    
    
    
    --http://programmingpraxis.com/2010/04/27/modern-elliptic-curve-factorization-part-2/
    ellipticFac::Integer->Integer->Elliptic->Point->Integer
    ellipticFac  b_1 b_2 (Cone an ad n)  (Conp u v) = 
    	let 
    		c= foldl lcm 1 [1..b_1]
    		(Conp x z)= montMul n (Cone an ad n) (Conp u v) c
    		g=gcd n z
    	in case (and [1<g, g<n]) of 
    		True -> g
    		_ ->   
    		 let 
    		   (Conp x' z')= foldl ( montMul n (Cone an ad n) ) (Conp x z) (allp b_1 b_2)
    		   g'=gcd (mod (g*z') n) n
    	         in case (and [1<g',g'<n]) of 
    		     True -> g'
    		     _ -> n
    									
    
    
    
    
    factor::Integer->Integer->IO [Integer]
    factor 1 _ = return []
    factor n cnt | cnt>=5 = return [n]
    	     | otherwise = do 
    				[an,ad,u,v] <-montgomeryCurve n
    				let p=ellipticFac 10000 20000 (Cone an ad n) (Conp u v)
    				case (and [1<p,p<n]) of 
    					False -> factor n (cnt+1)
    					_ -> do 
    						ps<-factor (div n p) (cnt+1)
    						ls<-factor p (cnt+1)
    						return (ls++ps)
    				
    				
    main = do
    		line<-getLine 
    		let n=read line ::Integer
    		lst<-factor n 0
    		print lst
    		return ()
    
  4. [...] is almost similar to previous example except using Montgomery curve rather than Lenstra curve. Programmingpraxis used Montgomery curve to find the prime factorisation. I personally feel that Lenstra algorithm [...]

  5. Lucas A. Brown said

    A solution in Python, borrowing the functions from http://programmingpraxis.com/2010/04/23/modern-elliptic-curve-factorization-part-1/#comment-1192:

    from random import randrange
    
    # Recursive sieve of Eratosthenes
    def primegen():
        yield 2; yield 3; yield 5; yield 7; yield 11; yield 13
        ps = primegen() # yay recursion
        p = ps.next() and ps.next()
        q, sieve, n = p**2, {}, 13
        while True:
            if n not in sieve:
                if n < q: yield n
                else:
                    next, step = q + 2*p, 2*p
                    while next in sieve: next += step
                    sieve[next] = step
                    p = ps.next()
                    q = p**2
            else:
                step = sieve.pop(n)
                next = n + step
                while next in sieve: next += step
                sieve[next] = step
            n += 2
    
    def add(p1, p2, p0, n): # Add two points p1 and p2 given point P0 = P1-P2 modulo n
        x1,z1 = p1; x2,z2 = p2; x0,z0 = p0
        t1, t2 = (x1-z1)*(x2+z2), (x1+z1)*(x2-z2)
        return (z0*pow(t1+t2,2,n) % n, x0*pow(t1-t2,2,n) % n)
    
    def double(p, A, n): # double point p on A modulo n
        x, z = p; An, Ad = A
        t1, t2 = pow(x+z,2,n), pow(x-z,2,n)
        t = t1 - t2
        return (t1*t2*4*Ad % n, (4*Ad*t2 + t*An)*t % n)
    
    def multiply(m, p, A, n): # multiply point p by m on curve A modulo n
        if m == 0: return (0, 0)
        elif m == 1: return p
        else:
            q = double(p, A, n)
            if m == 2: return q
            b = 1
            while b < m: b *= 2
            b /= 4
            r = p
            while b:
                if m&b: q, r = double(q, A, n), add(q, r, p, n)
                else:   q, r = add(r, q, p, n), double(r, A, n)
                b /= 2
            return r
    
    def ilog(x, b): # greatest integer l such that b**l <= x. # TODO: is there a faster way to compute this?  Binsearch, perhaps?
        l = 0
        while x >= b:
            x /= b
            l += 1
        return l
    
    def ecm(n, B1=10, B2=20):       # TODO: determine the best defaults for B1 and B2 and the best way to increment them and iters
        iters = 1
        while True:
            iters *= 2
            for _ in xrange(iters):
                seed = randrange(6, n)
                u, v = (seed**2 - 5) % n, 4*seed % n
                p = pow(u, 3, n)
                Q, C = (pow(v-u,3,n)*(3*u+v) % n, 4*p*v % n), (p, pow(v,3,n))
                pg = primegen()
                p = pg.next()
                while p <= B1: Q, p = multiply(p**ilog(B1, p), Q, C, n), pg.next()
                g = gcd(Q[1], n)
                if 1 < g < n: return g
                while p <= B2:
                    # "There is a simple coding trick that can speed up the second stage. Instead of multiplying each prime times Q,
                    # we iterate over i from B1 + 1 to B2, adding 2Q at each step; when i is prime, the current Q can be accumulated
                    # into the running solution. Again, we defer the calculation of the greatest common divisor until the end of the
                    # iteration.                                                TODO: implement this trick and compare performance."
                    Q = multiply(p, Q, C, n)
                    g *= Q[1]
                    g %= n
                    p = pg.next()
                g = gcd(g, n)
                if 1 < g < n: return g
                # This seed failed.  Try again with a new one.
            # These bounds failed.  Increase and try again.
            B1 *= 2
            B2 *= 2
    
    print ecm(20639383 * 117876683047)
    

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