-- | -- Module: Math.NumberTheory.Primes.Factorisation.Montgomery -- Copyright: (c) 2011 Daniel Fischer -- Licence: MIT -- Maintainer: Daniel Fischer -- Stability: Provisional -- Portability: Non-portable (GHC extensions) -- -- Factorisation of 'Integer's by the elliptic curve algorithm after Montgomery. -- The algorithm is explained at -- -- and -- -- -- The implementation is not very optimised, so it is not suitable for factorising numbers -- with only huge prime divisors. However, factors of 20-25 digits are normally found in -- acceptable time. The time taken depends, however, strongly on how lucky the curve-picking -- is. With luck, even large factors can be found in seconds; on the other hand, finding small -- factors (about 10 digits) can take minutes when the curve-picking is bad. -- -- Given enough time, the algorithm should be able to factor numbers of 100-120 digits, but it -- is best suited for numbers of up to 50-60 digits. {-# LANGUAGE CPP, BangPatterns, MagicHash #-} {-# OPTIONS_HADDOCK hide #-} module Math.NumberTheory.Primes.Factorisation.Montgomery ( -- * Complete factorisation functions -- ** Functions with input checking factorise , defaultStdGenFactorisation -- ** Functions without input checking , factorise' , stepFactorisation , defaultStdGenFactorisation' -- * Partial factorisation , smallFactors , stdGenFactorisation , curveFactorisation -- ** Single curve worker , montgomeryFactorisation , findParms ) where #include "MachDeps.h" import GHC.Base #if __GLASGOW_HASKELL__ < 705 import GHC.Word -- Moved to GHC.Types #endif import Data.Array.Base import System.Random import Control.Monad.State.Strict import Control.Applicative import Data.Bits import Data.Maybe import Math.NumberTheory.Logarithms import Math.NumberTheory.Logarithms.Internal import Math.NumberTheory.Powers.General (highestPower, largePFPower) import Math.NumberTheory.Powers.Squares (integerSquareRoot') import Math.NumberTheory.Primes.Sieve.Eratosthenes import Math.NumberTheory.Primes.Sieve.Indexing import Math.NumberTheory.Primes.Testing.Probabilistic import Math.NumberTheory.Utils -- | @'factorise' n@ produces the prime factorisation of @n@, including -- a factor of @(-1)@ if @n < 0@. @'factorise' 0@ is an error and the -- factorisation of @1@ is empty. Uses a 'StdGen' produced in an arbitrary -- manner from the bit-pattern of @n@. factorise :: Integer -> [(Integer,Int)] factorise n | n < 0 = (-1,1):factorise (-n) | n == 0 = error "0 has no prime factorisation" | n == 1 = [] | otherwise = factorise' n -- | Like 'factorise', but without input checking, hence @n > 1@ is required. factorise' :: Integer -> [(Integer,Int)] factorise' n = defaultStdGenFactorisation' (mkStdGen $ fromInteger n `xor` 0xdeadbeef) n -- | @'stepFactorisation'@ is like 'factorise'', except that it doesn't use a -- pseudo random generator but steps through the curves in order. -- This strategy turns out to be surprisingly fast, on average it doesn't -- seem to be slower than the 'StdGen' based variant. stepFactorisation :: Integer -> [(Integer,Int)] stepFactorisation n = let (sfs,mb) = smallFactors 100000 n in sfs ++ case mb of Nothing -> [] Just r -> curveFactorisation (Just 10000000000) bailliePSW (\m k -> (if k < (m-1) then k else error "Curves exhausted",k+1)) 6 Nothing r -- | @'defaultStdGenFactorisation'@ first strips off all small prime factors and then, -- if the factorisation is not complete, proceeds to curve factorisation. -- For negative numbers, a factor of @-1@ is included, the factorisation of @1@ -- is empty. Since @0@ has no prime factorisation, a zero argument causes -- an error. defaultStdGenFactorisation :: StdGen -> Integer -> [(Integer,Int)] defaultStdGenFactorisation sg n | n == 0 = error "0 has no prime factorisation" | n < 0 = (-1,1) : defaultStdGenFactorisation sg (-n) | n == 1 = [] | otherwise = defaultStdGenFactorisation' sg n -- | Like 'defaultStdGenFactorisation', but without input checking, so -- @n@ must be larger than @1@. defaultStdGenFactorisation' :: StdGen -> Integer -> [(Integer,Int)] defaultStdGenFactorisation' sg n = let (sfs,mb) = smallFactors 100000 n in sfs ++ case mb of Nothing -> [] Just m -> stdGenFactorisation (Just 10000000000) sg Nothing m ---------------------------------------------------------------------------------------------------- -- Factorisation wrappers -- ---------------------------------------------------------------------------------------------------- -- | A wrapper around 'curveFactorisation' providing a few default arguments. -- The primality test is 'bailliePSW', the @prng@ function - naturally - -- 'randomR'. This function also requires small prime factors to have been -- stripped before. stdGenFactorisation :: Maybe Integer -- ^ Lower bound for composite divisors -> StdGen -- ^ Standard PRNG -> Maybe Int -- ^ Estimated number of digits of smallest prime factor -> Integer -- ^ The number to factorise -> [(Integer,Int)] -- ^ List of prime factors and exponents stdGenFactorisation primeBound sg digits n = curveFactorisation primeBound bailliePSW (\m -> randomR (6,m-2)) sg digits n -- | @'curveFactorisation'@ is the driver for the factorisation. Its performance (and success) -- can be influenced by passing appropriate arguments. If you know that @n@ has no prime divisors -- below @b@, any divisor found less than @b*b@ must be prime, thus giving @Just (b*b)@ as the -- first argument allows skipping the comparatively expensive primality test for those. -- If @n@ is such that all prime divisors must have a specific easy to test for structure, a -- custom primality test can improve the performance (normally, it will make very little -- difference, since @n@ has not many divisors, and many curves have to be tried to find one). -- More influence has the pseudo random generator (a function @prng@ with @6 <= fst (prng k s) <= k-2@ -- and an initial state for the PRNG) used to generate the curves to try. A lucky choice here can -- make a huge difference. So, if the default takes too long, try another one; or you can improve your -- chances for a quick result by running several instances in parallel. -- -- @'curveFactorisation'@ requires that small prime factors have been stripped before. Also, it is -- unlikely to succeed if @n@ has more than one (really) large prime factor. curveFactorisation :: Maybe Integer -- ^ Lower bound for composite divisors -> (Integer -> Bool) -- ^ A primality test -> (Integer -> g -> (Integer,g)) -- ^ A PRNG -> g -- ^ Initial PRNG state -> Maybe Int -- ^ Estimated number of digits of the smallest prime factor -> Integer -- ^ The number to factorise -> [(Integer,Int)] -- ^ List of prime factors and exponents curveFactorisation primeBound primeTest prng seed mbdigs n | ptest n = [(n,1)] | otherwise = evalState (fact n digits) seed where digits = fromMaybe 8 mbdigs mult 1 xs = xs mult j xs = [(p,j*k) | (p,k) <- xs] dbl (u,v) = (mult 2 u, mult 2 v) ptest = case primeBound of Just bd -> \k -> k <= bd || primeTest k Nothing -> primeTest rndR k = state (\gen -> prng k gen) perfPw = case primeBound of Nothing -> highestPower Just bd -> largePFPower (integerSquareRoot' bd) fact m digs = do let (b1,b2,ct) = findParms digs (pfs,cfs) <- repFact m b1 b2 ct if null cfs then return pfs else do nfs <- forM cfs $ \(k,j) -> mult j <$> fact k (if null pfs then digs+4 else digs) return (mergeAll $ pfs:nfs) repFact m b1 b2 count = case perfPw m of (_,1) -> workFact m b1 b2 count (b,e) | ptest b -> return ([(b,e)],[]) | otherwise -> do (as,bs) <- workFact b b1 b2 count return $ (mult e as, mult e bs) workFact m b1 b2 count | count < 0 = return ([],[(m,1)]) | otherwise = do s <- rndR m case montgomeryFactorisation m b1 b2 s of Nothing -> workFact m b1 b2 (count-1) Just d -> do let !cof = m `quot` d case gcd cof d of 1 -> do (dp,dc) <- if ptest d then return ([(d,1)],[]) else repFact d b1 b2 (count-1) (cp,cc) <- if ptest cof then return ([(cof,1)],[]) else repFact cof b1 b2 (count-1) return (merge dp cp, dc ++ cc) g -> do let d' = d `quot` g c' = cof `quot` g (dp,dc) <- if ptest d' then return ([(d',1)],[]) else repFact d' b1 b2 (count-1) (cp,cc) <- if ptest c' then return ([(c',1)],[]) else repFact c' b1 b2 (count-1) (gp,gc) <- if ptest g then return ([(g,2)],[]) else dbl <$> repFact g b1 b2 (count-1) return (mergeAll [dp,cp,gp], dc ++ cc ++ gc) ---------------------------------------------------------------------------------------------------- -- The workhorse -- ---------------------------------------------------------------------------------------------------- -- | @'montgomeryFactorisation' n b1 b2 s@ tries to find a factor of @n@ using the -- curve and point determined by the seed @s@ (@6 <= s < n-1@), multiplying the -- point by the least common multiple of all numbers @<= b1@ and all primes -- between @b1@ and @b2@. The idea is that there's a good chance that the order -- of the point in the curve over one prime factor divides the multiplier, but the -- order over another factor doesn't, if @b1@ and @b2@ are appropriately chosen. -- If they are too small, none of the orders will probably divide the multiplier, -- if they are too large, all probably will, so they should be chosen to fit -- the expected size of the smallest factor. -- -- It is assumed that @n@ has no small prime factors. -- -- The result is maybe a nontrivial divisor of @n@. montgomeryFactorisation :: Integer -> Word -> Word -> Integer -> Maybe Integer montgomeryFactorisation n b1 b2 s = go p5 (list primeStore) where l2 = wordLog2' b1 b1i = toInteger b1 (^~) :: Word -> Int -> Word w ^~ i = w ^ i (e, p0) = montgomeryData n s dbl pt = double n e pt dbln 0 !pt = pt dbln k pt = dbln (k-1) (dbl pt) p2 = dbln l2 p0 #if WORD_SIZE_IN_BITS == 64 mul a b c = (a*b) `quot` c -- can't overflow, work on Int #else mul a b c = fromInteger ((toInteger a * b) `quot` c) -- might overflow if Int is used #endif adjust bd ml w | w <= bd = adjust bd ml (w*ml) | otherwise = w l3 = mul l2 190537 301994 w3 = 3 ^~ l3 pw3 = adjust (b1 `quot` 3) 3 w3 p3 = multiply n e pw3 p2 l5 = mul l2 1936274 4495889 w5 = 5 ^~ l5 pw5 = adjust (b1 `quot` 5) 5 w5 p5 = multiply n e pw5 p3 go (P _ 0) _ = Nothing go !pt@(P _ z) (pr:prs) | pr <= b1 = let !lp = integerLogBase' (fromIntegral pr) b1i in go (multiply n e (pr ^~ lp) pt) prs | otherwise = case gcd n z of 1 -> lgo (multiply n e pr pt) prs g -> Just g go (P _ z) _ = case gcd n z of 1 -> Nothing g -> Just g lgo (P _ 0) _ = Nothing lgo !pt@(P _ z) (pr:prs) | pr <= b2 = lgo (multiply n e pr pt) prs | otherwise = case gcd n z of 1 -> Nothing g -> Just g lgo (P _ z) _ = case gcd n z of 1 -> Nothing g -> Just g ---------------------------------------------------------------------------------------------------- -- Helpers, Curves and elliptic arithmetics -- ---------------------------------------------------------------------------------------------------- -- A Montgomery curve is given by y^2 = x^3 + (A_n / A_d - 2)*x^2 + x (mod n). -- We store A_n and 4*A_d, since A_n occurs with the factor 4 in all formulae. data Curve = C !Integer !Integer -- Point in the projective plane, will be on the curve -- A coordinate transformation eliminates the y-coordinate, hence -- we store only x and z data Point = P !Integer !Integer -- Get curve and point to start -- Input should satisfy 6 <= s < n-1 montgomeryData :: Integer -> Integer -> (Curve, Point) montgomeryData n s = (C an ad4, P x z) where u = (s*s-5) `mod` n v = (4*s) `mod` n d = (v-u) x = (u*u*u) `mod` n z = (v*v*v) `mod` n an = ((d*d)*(d*(3*u+v))) `mod` n ad4 = (16*x*v) `mod` n -- Addition on the curve, given the modulus n and three points, -- p0, p1 and p2, with p0 = p2 - p1, calculate the point p1 + p2. -- Note that the addition does not depend on the curve. add :: Integer -> Point -> Point -> Point -> Point add n (P x0 z0) (P x1 z1) (P x2 z2) = P x3 z3 where a = (x1-z1)*(x2+z2) b = (x1+z1)*(x2-z2) c = a+b d = a-b x3 = (c*c*z0) `rem` n z3 = (d*d*x0) `rem` n -- Double a point on the curve. double :: Integer -> Curve -> Point -> Point double n (C an ad4) (P x z) = P x' z' where r = x+z s = x-z u = r*r v = s*s t = u-v x' = (ad4*u*v) `rem` n z' = ((ad4*v+t*an)*t) `rem` n -- Multiply a point on the curve by a Word. -- Within Word range, we can use the faster variant going -- from high-order bits to low-order. multiply :: Integer -> Curve -> Word -> Point -> Point multiply n cve (W# w##) p = case wordLog2# w## of l# -> go (l# -# 1#) p (double n cve p) where go 0# !p0 !p1 = case w## `and#` 1## of 0## -> double n cve p0 _ -> add n p p0 p1 go i# p0 p1 = case (uncheckedShiftRL# w## i#) `and#` 1## of 0## -> go (i# -# 1#) (double n cve p0) (add n p p0 p1) _ -> go (i# -# 1#) (add n p p0 p1) (double n cve p1) {- Not (yet) needed -- Multiply a point on the curve by an Integer. multIgr :: Integer -> Curve -> Integer -> Point -> Point multIgr n cve k p = go k where go 1 = (p, double n cve p) go m = case m `quotRem` 2 of (q,r) -> let !(!s, l) = go q in case r of 0 -> (double n cve s, add n p s l) _ -> (add n p s l, double n cve l) -} -- primes, compactly stored as a bit sieve primeStore :: [PrimeSieve] primeStore = psieveFrom 7 -- generate list of primes from arrays list :: [PrimeSieve] -> [Word] list sieves = concat [[off + toPrim i | i <- [0 .. li], unsafeAt bs i] | PS vO bs <- sieves, let { (_,li) = bounds bs; off = fromInteger vO; }] -- | @'smallFactors' bound n@ finds all prime divisors of @n > 1@ up to @bound@ by trial division and returns the -- list of these together with their multiplicities, and a possible remaining factor which may be composite. smallFactors :: Integer -> Integer -> ([(Integer,Int)], Maybe Integer) smallFactors bd n = case shiftToOddCount n of (0,m) -> go m prms (k,m) -> (2,k) <: if m == 1 then ([],Nothing) else go m prms where prms = tail (primeStore >>= primeList) x <: ~(l,b) = (x:l,b) go m (p:ps) | m < p*p = ([(m,1)], Nothing) | bd < p = ([], Just m) | otherwise = case splitOff p m of (0,_) -> go m ps (k,r) | r == 1 -> ([(p,k)], Nothing) | otherwise -> (p,k) <: go r ps go m [] = ([(m,1)], Nothing) -- helpers: merge sorted lists merge :: [(Integer,Int)] -> [(Integer,Int)] -> [(Integer,Int)] merge xxs@(x@(p,k):xs) yys@(y@(q,m):ys) = case compare p q of LT -> x : merge xs yys EQ -> (p,k+m) : merge xs ys GT -> y : merge xxs ys merge xs [] = xs merge _ ys = ys mergeAll :: [[(Integer,Int)]] -> [(Integer,Int)] mergeAll [] = [] mergeAll [xs] = xs mergeAll (xs:ys:zss) = merge (merge xs ys) (mergeAll zss) -- Parameters for the factorisation, the two b-parameters for montgomery and the number of tries -- to use these, depending on the size of the factor we are looking for. -- The numbers are roughly based on the parameters listed on Dario Alpern's ECM site. testParms :: [(Int,Word,Word,Int)] testParms = [ (12, 400, 10000, 10), (15, 2000, 50000, 25), (20, 11000, 150000, 90) , (25, 50000, 500000, 300), (30, 250000, 1500000, 700) , (35, 1000000, 4000000, 1800), (40, 3000000, 12000000, 5100) , (45, 11000000, 45000000, 10600), (50, 43000000, 200000000, 19300) , (55, 80000000, 400000000,30000), (60, 120000000, 700000000, 50000) ] findParms :: Int -> (Word, Word, Int) findParms digs = go (100, 1000, 7) testParms where go p ((d,b1,b2,ct):rest) | digs < d = p | otherwise = go (b1,b2,ct) rest go p [] = p