pyecm-2.0.2/0000755000175000017500000000000012211747237012204 5ustar martinmartinpyecm-2.0.2/CHANGELOG.txt0000644000175000017500000000257312211747237014243 0ustar martinmartin2.0.2 - Now supports gmpy2, as well as gmpy. 2.0.1 - Fixed minor bug with factoring negative numbers on the command-line. 2.0 - Made asymptotic improvements at the expense of a constant factor. It should now be faster for numbers greater than 30 or so digits but slower for numbers less than that. - Added ability to type expressions instead of plain numbers. For example, the program will now factor "2^30-1" instead of making you type "1073741823". - Added capability to run pyecm in parallel. - Added command line switches, such as a verbose mode and an ability to select sigma (an internal parameter) randomly. - Improved compatibility with installations lacking gmpy and/or psyco. 1.2.2: - Made pyecm a tiny bit faster - Fixed the bug where the version is reported incorrectly as 1.2 - Included the GPL with the source 1.2.1: - Fixed a silly typo in fastprime. This should make it a little bit faster. - Removed silly spaces after lines 1.2: - pyecm now uses gmpy if it's installed on your system (highly recommended: HUGE speed increase when used). If it's not installed, it continues as normal. 1.1.2: - Silly bug with not using psyco fixed 1.1.1: - Minor bug fixes - Minor speed improvement 1.1: - More speed improvements 1.0: - Massive speed improvements 0.2: - Many bugfixes - Speed improvements 0.1: - Initial release pyecm-2.0.2/pyecm.py0000644000175000017500000012051012211747237013672 0ustar martinmartin#!/usr/bin/env python ''' You should install psyco and gmpy if you want maximal speed. Filename: pyecm Authors: Eric Larson , Martin Kelly , License: GNU GPL (see for more information. Description: Factors a number using the Elliptic Curve Method, a fast algorithm for numbers < 50 digits. We are using curves in Suyama's parametrization, but points are in affine coordinates, and the curve is in Wierstrass form. The idea is to do many curves in parallel to take advantage of batch inversion algorithms. This gives asymptotically 7 modular multiplications per bit. WARNING: pyecm is NOT a general-purpose number theory or elliptic curve library. Many of the functions have confusing calling syntax, and some will rather unforgivingly crash or return bad output if the input is not formatted exactly correctly. That said, there are a couple of functions that you CAN safely import into another program. These are: factors, isprime. However, be sure to read the documentation for each function that you use. ''' import math import sys import random try: import psyco psyco.full() PSYCO_EXISTS = True except ImportError: PSYCO_EXISTS = False try: # Try to use gmpy from gmpy2 import isqrt as sqrt from gmpy2 import iroot as root from gmpy2 import gcd, invert, mpz, next_prime import gmpy2 GMPY_EXISTS = True except ImportError: try: from gmpy import gcd, invert, mpz, next_prime, sqrt, root GMPY_EXISTS = True except ImportError: GMPY_EXISTS = False if not GMPY_EXISTS: PRIMES = (5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 167) GMPY_EXISTS = False def gcd(a, b): '''Computes the Greatest Common Divisor of a and b using the standard quadratic time improvement to the Euclidean Algorithm. Returns the GCD of a and b.''' if b == 0: return a elif a == 0: return b count = 0 if a < 0: a = -a if b < 0: b = -b while not ((a & 1) | (b & 1)): count += 1 a >>= 1 b >>= 1 while not a & 1: a >>= 1 while not b & 1: b >>= 1 if b > a: b,a = a,b while b != 0 and a != b: a -= b while not (a & 1): a >>= 1 if b > a: b, a = a, b return a << count def invert(a, b): '''Computes the inverse of a modulo b. b must be odd. Returns the inverse of a (mod b).''' if a == 0 or b == 0: return 0 truth = False if a < 0: truth = True a = -a b_orig = b alpha = 1 beta = 0 while not a & 1: if alpha & 1: alpha += b_orig alpha >>= 1 a >>= 1 if b > a: a, b = b, a alpha, beta = beta, alpha while b != 0 and a != b: a -= b alpha -= beta while not a & 1: if alpha & 1: alpha += b_orig alpha >>= 1 a >>= 1 if b > a: a,b = b,a alpha, beta = beta, alpha if a == b: a -= b alpha -= beta a, b = b, a alpha, beta = beta, alpha if a != 1: return 0 if truth: alpha = b_orig - alpha return alpha def next_prime(n): '''Finds the next prime after n. Returns the next prime after n.''' n += 1 if n <= 167: if n <= 23: if n <= 3: return 3 - (n <= 2) n += (n & 1) ^ 1 return n + (((4 - (n % 3)) >> 1) & 2) n += (n & 1) ^ 1 inc = n % 3 n += ((4 - inc) >> 1) & 2 inc = 6 - ((inc + ((2 - inc) & 2)) << 1) while 0 in (n % 5, n % 7, n % 11): n += inc inc = 6 - inc return n n += (n & 1) ^ 1 inc = n % 3 n += ((4 - inc) >> 1) & 2 inc = 6 - ((inc + ((2 - inc) & 2)) << 1) should_break = False while 1: for prime in PRIMES: if not n % prime: should_break = True break if should_break: should_break = False n += inc inc = 6 - inc continue p = 1 for i in xrange(int(math.log(n) / LOG_2), 0, -1): p <<= (n >> i) & 1 p = (p * p) % n if p == 1: return n n += inc inc = 6 - inc def mpz(n): '''A dummy function to ensure compatibility with those that do not have gmpy. Returns n.''' return n def root(n, k): '''Finds the floor of the kth root of n. This is a duplicate of gmpy's root function. Returns a tuple. The first item is the floor of the kth root of n. The second is 1 if the root is exact (as in, sqrt(16)) and 0 if it is not.''' low = 0 high = n + 1 while high > low + 1: mid = (low + high) >> 1 mr = mid**k if mr == n: return (mid, 1) if mr < n: low = mid if mr > n: high = mid return (low, 0) def sqrt(n): return root(n, 2)[0] # We're done importing. Now for some constants. if GMPY_EXISTS: INV_C = 1.4 else: if PSYCO_EXISTS: INV_C = 7.3 else: INV_C = 13.0 LOG_2 = math.log(2) LOG_4 = math.log(4) LOG_3_MINUS_LOG_LOG_2 = math.log(3) - math.log(LOG_2) LOG_4_OVER_9 = LOG_4 / 9 _3_OVER_LOG_2 = 3 / LOG_2 _5_LOG_10 = 5 * math.log(10) _7_OVER_LOG_2 = 7 / LOG_2 BIG = 2.0**512 BILLION = 10**9 # Something big that fits into an int. MULT = math.log(3) / LOG_2 ONE = mpz(1) SMALL = 2.0**(-30) SMALLEST_COUNTEREXAMPLE_FASTPRIME = 2047 T = (type(mpz(1)), type(1), type(1L)) DUMMY = 'dummy' # Dummy value throughout the program VERSION = '2.0.2' _12_LOG_2_OVER_49 = 12 * math.log(2) / 49 RECORD = 1162795072109807846655696105569042240239 class ts: '''Does basic manipulations with Taylor Series (centered at 0). An example call to ts: a = ts(7, 23, [1<<23, 2<<23, 3<<23]) -- now, a represents 1 + 2x + 3x^2. Here, computations will be done to degree 7, with accuracy 2^(-23). Input coefficients must be integers.''' def __init__(self, degree, acc, p): self.acc = acc self.coefficients = p[:degree + 1] while len(self.coefficients) <= degree: self.coefficients.append(0) def add(self, a, b): '''Adds a and b''' b_ = b.coefficients[:] a_ = a.coefficients[:] self.coefficients = [] while len(b_) > len(a_): a_.append(0) while len(b_) < len(a_): b_.append(0) for i in xrange(len(a_)): self.coefficients.append(a_[i] + b_[i]) self.acc = a.acc def ev(self, x): '''Returns a(x)''' answer = 0 for i in xrange(len(self.coefficients) - 1, -1, -1): answer *= x answer += self.coefficients[i] return answer def evh(self): '''Returns a(1/2)''' answer = 0 for i in xrange(len(self.coefficients) - 1, -1, -1): answer >>= 1 answer += self.coefficients[i] return answer def evmh(self): '''Returns a(-1/2)''' answer = 0 for i in xrange(len(self.coefficients) - 1, -1, -1): answer = - answer >> 1 answer += self.coefficients[i] return answer def int(self): '''Replaces a by an integral of a''' self.coefficients = [0] + self.coefficients for i in xrange(1, len(self.coefficients)): self.coefficients[i] /= i def lindiv(self, a): '''a.lindiv(k) -- sets a/(x-k/2) for integer k''' for i in xrange(len(self.coefficients) - 1): self.coefficients[i] <<= 1 self.coefficients[i] /= a self.coefficients[i + 1] -= self.coefficients[i] self.coefficients[-1] <<= 1 self.coefficients[-1] /= a def neg(self): '''Sets a to -a''' for i in xrange(len(self.coefficients)): self.coefficients[i] = - self.coefficients[i] def set(self, a): '''a.set(b) sets a to b''' self.coefficients = a.coefficients[:] self.acc = a.acc def simp(self): '''Turns a into a type of Taylor series that can be fed into ev, but cannot be computed with further.''' for i in xrange(len(self.coefficients)): shift = max(0, int(math.log(abs(self.coefficients[i]) + 1) / LOG_2) - 1000) self.coefficients[i] = float(self.coefficients[i] >> shift) shift = self.acc - shift for _ in xrange(shift >> 9): self.coefficients[i] /= BIG self.coefficients[i] /= 2.0**(shift & 511) if abs(self.coefficients[i] / self.coefficients[0]) <= SMALL: self.coefficients = self.coefficients[:i] break # Functions are declared in alphabetical order except when dependencies force them to be at the end. def add(p1, p2, n): '''Adds first argument to second (second argument is not preserved). The arguments are points on an elliptic curve. The first argument may be a tuple instead of a list. The addition is thus done pointwise. This function has bizzare input/output because there are fast algorithms for inverting a bunch of numbers at once. Returns a list of the addition results.''' inv = range(len(p1)) for i in xrange(len(p1)): inv[i] = p1[i][0] - p2[i][0] inv = parallel_invert(inv, n) if not isinstance(inv, list): return inv for i in xrange(len(p1)): m = ((p1[i][1] - p2[i][1]) * inv[i]) % n p2[i][0] = (m * m - p1[i][0] - p2[i][0]) % n p2[i][1] = (m * (p1[i][0] - p2[i][0]) - p1[i][1]) % n return p2 def add_sub_x_only(p1, p2, n): '''Given a pair of lists of points p1 and p2, computes the x-coordinates of p1[i] + p2[i] and p1[i] - p2[i] for each i. Returns two lists, the first being the sums and the second the differences.''' sums = range(len(p1)) difs = range(len(p1)) for i in xrange(len(p1)): sums[i] = p2[i][0] - p1[i][0] sums = parallel_invert(sums, n) if not isinstance(sums, list): return (sums, None) for i in xrange(len(p1)): ms = ((p2[i][1] - p1[i][1]) * sums[i]) % n md = ((p2[i][1] + p1[i][1]) * sums[i]) % n sums[i] = (ms * ms - p1[i][0] - p2[i][0]) % n difs[i] = (md * md - p1[i][0] - p2[i][0]) % n sums = tuple(sums) difs = tuple(difs) return (sums, difs) def atdn(a, d, n): '''Calculates a to the dth power modulo n. Returns the calculation's result.''' x = 1 pos = int(math.log(d) / LOG_2) while pos >= 0: x = (x * x) % n if (d >> pos) & 1: x *= a pos -= 1 return x % n def copy(p): '''Copies a list using only deep copies. Returns a copy of p.''' answer = [] for i in p: answer.append(i[:]) return answer def could_be_prime(n): '''Performs some trials to compute whether n could be prime. Run time is O(N^3 / (log N)^2) for N bits. Returns whether it is possible for n to be prime (True or False). ''' if n < 2: return False if n == 2: return True if not n & 1: return False product = ONE log_n = int(math.log(n)) + 1 bound = int(math.log(n) / (LOG_2 * math.log(math.log(n))**2)) + 1 if bound * log_n >= n: bound = 1 log_n = int(sqrt(n)) prime_bound = 0 prime = 3 for _ in xrange(bound): p = [] prime_bound += log_n while prime <= prime_bound: p.append(prime) prime = next_prime(prime) if p != []: p = prod(p) product = (product * p) % n return gcd(n, product) == 1 def double(p, n): '''Doubles each point in the input list. Much like the add function, we take advantage of fast inversion. Returns the doubled list.''' inv = range(len(p)) for i in xrange(len(p)): inv[i] = p[i][1] << 1 inv = parallel_invert(inv, n) if not isinstance(inv, list): return inv for i in xrange(len(p)): x = p[i][0] m = (x * x) % n m = ((m + m + m + p[i][2]) * inv[i]) % n p[i][0] = (m * m - x - x) % n p[i][1] = (m * (x - p[i][0]) - p[i][1]) % n return p def fastprime(n): '''Tests for primality of n using an algorithm that is very fast, O(N**3 / log(N)) (assuming quadratic multiplication) where n has N digits, but ocasionally inaccurate for n >= 2047. Returns the primality of n (True or False).''' if not could_be_prime(n): return False if n == 2: return True j = 1 d = n >> 1 while not d & 1: d >>= 1 j += 1 p = 1 pos = int(math.log(d) / LOG_2) while pos >= 0: p = (p * p) % n p <<= (d >> pos) & 1 pos -= 1 if p in (n - 1, n + 1): return True for _ in xrange(j): p = (p * p) % n if p == 1: return False elif p == n - 1: return True return False def greatest_n(phi_max): '''Finds the greatest n such that phi(n) < phi_max. Returns the greatest n such that phi(n) < phi_max.''' phi_product = 1 product = 1 prime = 1 while phi_product <= phi_max: prime = next_prime(prime) phi_product *= prime - 1 product *= prime n_max = (phi_max * product) / phi_product phi_values = range(n_max) prime = 2 while prime <= n_max: for i in xrange(0, n_max, prime): phi_values[i] -= phi_values[i] / prime prime = next_prime(prime) for i in xrange(n_max - 1, 0, -1): if phi_values[i] <= phi_max: return i def inv_const(n): '''Finds a constant relating the complexity of multiplication to that of modular inversion. Returns the constant for a given n.''' return int(INV_C * math.log(n)**0.42) def naf(d): '''Finds a number's non-adjacent form, reverses the bits, replaces the -1's with 3's, and interprets the result base 4. Returns the result interpreted as if in base 4.''' g = 0L while d: g <<= 2 g ^= ((d & 2) & (d << 1)) ^ (d & 1) d += (d & 2) >> 1 d >>= 1 return g def parallel_invert(l, n): '''Inverts all elements of a list modulo some number, using 3(n-1) modular multiplications and one inversion. Returns the list with all elements inverted modulo 3(n-1).''' l_ = l[:] for i in xrange(len(l)-1): l[i+1] = (l[i] * l[i+1]) % n try: inv = invert(l[-1], n) except ZeroDivisionError: inv = 0 if inv == 0: return gcd(l[-1], n) for i in xrange(len(l)-1, 0, -1): l[i] = (inv * l[i-1]) % n inv = (inv * l_[i]) % n l[0] = inv return l def prod(p): '''Multiplies all elements of a list together. The order in which the elements are multiplied is chosen to take advantage of Python's Karatsuba Multiplication Returns the product of everything in p.''' jump = 1 while jump < len(p): for i in xrange(0, len(p) - jump, jump << 1): p[i] *= p[i + jump] p[i + jump] = None jump <<= 1 return p[0] def rho_ev(x, ts): '''Evaluates Dickman's rho function, which calculates the asymptotic probability as N approaches infinity (for a given x) that all of N's factors are bounded by N^(1/x).''' return ts[int(x)].ev(x - int(x) - 0.5) def rho_ts(n): '''Makes a list of Taylor series for the rho function centered at 0.5, 1.5, 2.5 ... n + 0.5. The reason this is necessary is that the radius of convergence of rho is small, so we need lots of Taylor series centered at different places to correctly evaluate it. Returns a list of Taylor series.''' f = ts(10, 10, []) answer = [ts(10, 10, [1])] for _ in xrange(n): answer.append(ts(10, 10, [1])) deg = 5 acc = 50 + n * int(1 + math.log(1 + n) + math.log(math.log(3 + n))) r = 1 rho_series = ts(1, 10, [0]) while r != rho_series.coefficients[0]: deg = (deg + (deg << 2)) / 3 r = rho_series.coefficients[0] rho_series = ts(deg, acc, [(1L) << acc]) center = 0.5 for i in xrange(1, n+1): f.set(rho_series) center += 1 f.lindiv(int(2*center)) f.int() f.neg() d = ts(deg, acc, [rho_series.evh() - f.evmh()]) f.add(f, d) rho_series.set(f) f.simp() answer[i].set(f) rho_series.simp() return answer def sub_sub_sure_factors(f, u, curve_parameter): '''Finds all factors that can be found using ECM with a smoothness bound of u and sigma and give curve parameters. If that fails, checks for being a prime power and does Fermat factoring as well. Yields factors.''' while not (f & 1): yield 2 f >>= 1 while not (f % 3): yield 3 f /= 3 if isprime(f): yield f return log_u = math.log(u) u2 = int(_7_OVER_LOG_2 * u * log_u / math.log(log_u)) primes = [] still_a_chance = True log_mo = math.log(f + 1 + sqrt(f << 2)) g = gcd(curve_parameter, f) if g not in (1, f): for factor in sub_sub_sure_factors(g, u, curve_parameter): yield factor for factor in sub_sub_sure_factors(f/g, u, curve_parameter): yield factor return g2 = gcd(curve_parameter**2 - 5, f) if g2 not in (1, f): for factor in sub_sub_sure_factors(g2, u, curve_parameter): yield factor for factor in sub_sub_sure_factors(f / g2, u, curve_parameter): yield factor return if f in (g, g2): yield f while still_a_chance: p1 = get_points([curve_parameter], f) for prime in primes: p1 = multiply(p1, prime, f) if not isinstance(p1, list): if p1 != f: for factor in sub_sub_sure_factors(p1, u, curve_parameter): yield factor for factor in sub_sub_sure_factors(f/p1, u, curve_parameter): yield factor return else: still_a_chance = False break if not still_a_chance: break prime = 1 still_a_chance = False while prime < u2: prime = next_prime(prime) should_break = False for _ in xrange(int(log_mo / math.log(prime))): p1 = multiply(p1, prime, f) if not isinstance(p1, list): if p1 != f: for factor in sub_sub_sure_factors(p1, u, curve_parameter): yield factor for factor in sub_sub_sure_factors(f/p1, u, curve_parameter): yield factor return else: still_a_chance = True primes.append(prime) should_break = True break if should_break: break for i in xrange(2, int(math.log(f) / LOG_2) + 2): r = root(f, i) if r[1]: for factor in sub_sub_sure_factors(r[0], u, curve_parameter): for _ in xrange(i): yield factor return a = 1 + sqrt(f) bsq = a * a - f iter = 0 while bsq != sqrt(bsq)**2 and iter < 3: a += 1 iter += 1 bsq += a + a - 1 if bsq == sqrt(bsq)**2: b = sqrt(bsq) for factor in sub_sub_sure_factors(a - b, u, curve_parameter): yield factor for factor in sub_sub_sure_factors(a + b, u, curve_parameter): yield factor return yield f return def sub_sure_factors(f, u, curve_params): '''Factors n as far as possible using the fact that f came from a mainloop call. Yields factors of n.''' if len(curve_params) == 1: for factor in sub_sub_sure_factors(f, u, curve_params[0]): yield factor return c1 = curve_params[:len(curve_params) >> 1] c2 = curve_params[len(curve_params) >> 1:] if mainloop(f, u, c1) == 1: for factor in sub_sure_factors(f, u, c2): yield factor return if mainloop(f, u, c2) == 1: for factor in sub_sure_factors(f, u, c1): yield factor return for factor in sub_sure_factors(f, u, c1): if isprime(factor): yield factor else: for factor_of_factor in sub_sure_factors(factor, u, c2): yield factor_of_factor return def subtract(p1, p2, n): '''Given two points on an elliptic curve, subtract them pointwise. Returns the resulting point.''' inv = range(len(p1)) for i in xrange(len(p1)): inv[i] = p2[i][0] - p1[i][0] inv = parallel_invert(inv, n) if not isinstance(inv, list): return inv for i in xrange(len(p1)): m = ((p1[i][1] + p2[i][1]) * inv[i]) % n p2[i][0] = (m * m - p1[i][0] - p2[i][0]) % n p2[i][1] = (m * (p1[i][0] - p2[i][0]) + p1[i][1]) % n return p2 def congrats(f, veb): '''Prints a congratulations message when a record factor is found. This only happens if the second parameter (verbosity) is set to True. Returns nothing.''' if veb and f > RECORD: print 'Congratulations! You may have found a record factor via pyecm!' print 'Please email the Mainloop call to Eric Larson ' return def sure_factors(n, u, curve_params, veb, ra, ov, tdb, pr): '''Factor n as far as possible with given smoothness bound and curve parameters, including possibly (but very rarely) calling ecm again. Yields factors of n.''' f = mainloop(n, u, curve_params) if f == 1: return if veb: print 'Found factor:', f print 'Mainloop call was:', n, u, curve_params if isprime(f): congrats(f, veb) yield f n /= f if isprime(n): yield n if veb: print '(factor processed)' return for factor in sub_sure_factors(f, u, curve_params): if isprime(factor): congrats(f, veb) yield factor else: if veb: print 'entering new ecm loop to deal with stubborn factor:', factor for factor_of_factor in ecm(factor, True, ov, veb, tdb, pr): yield factor_of_factor n /= factor if isprime(n): yield n if veb: print '(factor processed)' return def to_tuple(p): '''Converts a list of two-element lists into a list of two-element tuples. Returns a list.''' answer = [] for i in p: answer.append((i[0], i[1])) return tuple(answer) def mainloop(n, u, p1): ''' Input: n -- an integer to (try) to factor. u -- the phase 1 smoothness bound p1 -- a list of sigma parameters to try Output: A factor of n. (1 is returned on faliure). Notes: 1. Other parameters, such as the phase 2 smoothness bound are selected by the mainloop function. 2. This function uses batch algorithms, so if p1 is not long enough, there will be a loss in efficiency. 3. Of course, if p1 is too long, then the mainloop will have to use more memory. [The memory is polynomial in the length of p1, log u, and log n].''' k = inv_const(n) log_u = math.log(u) log_log_u = math.log(log_u) log_n = math.log(n) u2 = int(_7_OVER_LOG_2 * u * log_u / log_log_u) ncurves = len(p1) w = int(math.sqrt(_3_OVER_LOG_2 * ncurves / k) - 0.5) number_of_primes = int((ncurves << w) * math.sqrt(LOG_4_OVER_9 * log_n / k) / log_u) # Lagrange multipliers! number_of_primes = min(number_of_primes, int((log_n / math.log(log_n))**2 * ncurves / log_u), int(u / log_u)) number_of_primes = max(number_of_primes, 1) m = math.log(number_of_primes) + log_log_u w = min(w, int((m - 2 * math.log(m) + LOG_3_MINUS_LOG_LOG_2) / LOG_2)) w = max(w, 1) max_order = n + sqrt(n << 2) + 1 # By Hasse's theorem. det_bound = ((1 << w) - 1 + ((w & 1) << 1)) / 3 log_mo = math.log(max_order) p = range(number_of_primes) prime = mpz(2) p1 = get_points(p1, n) if not isinstance(p1, list): return p1 for _ in xrange(int(log_mo / LOG_2)): p1 = double(p1, n) if not isinstance(p1, list): return p1 for i in xrange(1, det_bound): prime = (i << 1) + 1 if isprime(prime): for _ in xrange(int(log_mo / math.log(prime))): p1 = multiply(p1, prime, n) if not isinstance(p1, list): return p1 while prime < sqrt(u) and isinstance(p1, list): for i in xrange(number_of_primes): prime = next_prime(prime) p[i] = prime ** max(1, int(log_u / math.log(prime))) p1 = fast_multiply(p1, prod(p), n, w) if not isinstance(p1, list): return p1 while prime < u and isinstance(p1, list): for i in xrange(number_of_primes): prime = next_prime(prime) p[i] = prime p1 = fast_multiply(p1, prod(p), n, w) if not isinstance(p1, list): return p1 del p small_jump = int(greatest_n((1 << (w + 2)) / 3)) small_jump = max(120, small_jump) big_jump = 1 + (int(sqrt((5 << w) / 21)) << 1) total_jump = small_jump * big_jump big_multiple = max(total_jump << 1, ((int(next_prime(prime)) - (total_jump >> 1)) / total_jump) * total_jump) big_jump_2 = big_jump >> 1 small_jump_2 = small_jump >> 1 product = ONE psmall_jump = multiply(p1, small_jump, n) if not isinstance(psmall_jump, list): return psmall_jump ptotal_jump = multiply(psmall_jump, big_jump, n) if not isinstance(ptotal_jump, list): return ptotal_jump pgiant_step = multiply(p1, big_multiple, n) if not isinstance(pgiant_step, list): return pgiant_step small_multiples = [None] for i in xrange(1, small_jump >> 1): if gcd(i, small_jump) == 1: tmp = multiply(p1, i, n) if not isinstance(tmp, list): return tmp for i in xrange(len(tmp)): tmp[i] = tmp[i][0] small_multiples.append(tuple(tmp)) else: small_multiples.append(None) small_multiples = tuple(small_multiples) big_multiples = [None] for i in xrange(1, (big_jump + 1) >> 1): tmp = multiply(psmall_jump, i, n) if not isinstance(tmp, list): return tmp big_multiples.append(to_tuple(tmp)) big_multiples = tuple(big_multiples) psmall_jump = to_tuple(psmall_jump) ptotal_jump = to_tuple(ptotal_jump) while big_multiple < u2: big_multiple += total_jump center_up = big_multiple center_down = big_multiple pgiant_step = add(ptotal_jump, pgiant_step, n) if not isinstance(pgiant_step, list): return pgiant_step prime_up = next_prime(big_multiple - small_jump_2) while prime_up < big_multiple + small_jump_2: s = small_multiples[abs(int(prime_up) - big_multiple)] for j in xrange(ncurves): product *= pgiant_step[j][0] - s[j] product %= n prime_up = next_prime(prime_up) for i in xrange(1, big_jump_2 + 1): center_up += small_jump center_down -= small_jump pmed_step_up, pmed_step_down = add_sub_x_only(big_multiples[i], pgiant_step, n) if pmed_step_down == None: return pmed_step_up while prime_up < center_up + small_jump_2: s = small_multiples[abs(int(prime_up) - center_up)] for j in xrange(ncurves): product *= pmed_step_up[j] - s[j] product %= n prime_up = next_prime(prime_up) prime_down = next_prime(center_down - small_jump_2) while prime_down < center_down + small_jump_2: s = small_multiples[abs(int(prime_down) - center_down)] for j in xrange(ncurves): product *= pmed_step_down[j] - s[j] product %= n prime_down = next_prime(prime_down) if gcd(product, n) != 1: return gcd(product, n) return 1 def fast_multiply(p, d, n, w): '''Multiplies each element of p by d. Multiplication is on an elliptic curve. Both d and

must be odd. Also,

may not be divisible by anything less than or equal to 2 * (2**w + (-1)**w) / 3 + 1. Returns the list p multiplied by d.''' mask = (1 << (w << 1)) - 1 flop = mask / 3 g = naf(d) >> 4 precomp = {} m = copy(p) p = double(p, n) for i in xrange((flop >> w) + (w & 1)): key = naf((i << 1) + 1) precomp[key] = to_tuple(m) precomp[((key & flop) << 1) ^ key] = precomp[key] m = add(p, m, n) while g > 0: if g & 1: t = g & mask sh = 1 + int(math.log(t) / LOG_4) for _ in xrange(sh): p = double(p, n) if g & 2: p = subtract(precomp[t], p, n) else: p = add(precomp[t], p, n) g >>= (sh << 1) if not isinstance(p, list): return p else: p = double(p, n) g >>= 2 return p def get_points(p1, n): '''Outputs points in Weierstrass form, given input in Suyama parametrization. Returns the points.''' p1 = list(p1) invs = p1[:] ncurves = len(p1) for j in xrange(ncurves): sigma = mpz(p1[j]) u = (sigma**2 - 5) % n v = sigma << 2 i = (((u * u) % n) * ((v * u << 2) % n)) % n p1[j] = [u, v, i] invs[j] = (i * v) % n invs = parallel_invert(invs, n) if not isinstance(invs, list): return invs for j in xrange(ncurves): u, v, i = p1[j] inv = invs[j] a = (((((((v - u)**3 % n) * v) % n) * (u + u + u + v)) % n) * inv - 2) % n # <-- This line is a thing of beauty x_0 = (((((u * i) % n) * inv) % n) ** 3) % n # And this one gets second place b = ((((x_0 + a) * x_0 + 1) % n) * x_0) % n x_0 = (b * x_0) % n y_0 = (b**2) % n while a % 3: a += n x_0 = (x_0 + a * b / 3) % n c = (y_0 * ((1 - a**2 / 3) % n)) % n p1[j] = [x_0, y_0, c] return p1 def isprime(n): ''' Tests for primality of n trying first fastprime and then a slower but accurate algorithm. Time complexity is O(N**3) (assuming quadratic multiplication), where n has N digits. Returns the primality of n (True or False).''' if not fastprime(n): return False elif n < SMALLEST_COUNTEREXAMPLE_FASTPRIME: return True do_loop = False j = 1 d = n >> 1 a = 2 bound = int(0.75 * math.log(math.log(n)) * math.log(n)) + 1 while not d & 1: d >>= 1 j += 1 while a < bound: a = next_prime(a) p = atdn(a, d, n) if p == 1 or p == n - 1: continue for _ in xrange(j): p = (p * p) % n if p == 1: return False elif p == n - 1: do_loop = True break if do_loop: do_loop = False continue return False return True def multiply(p1, d, n): '''Multiplies each element of a list by a number, without using too much overhead. Returns a list p multiplied through by d.''' pos = int(math.log(d) / LOG_2) - 1 p = copy(p1) while pos >= 0: p = double(p, n) if not isinstance(p, list): return p if (d >> pos) & 1: p = add(p1, p, n) if not isinstance(p, list): return p pos -= 1 return p def ecm(n, ra, ov, veb, tdb, pr): # DOCUMENTATION '''Input: n -- An integer to factor veb -- If True, be verbose ra -- If True, select sigma values randomly ov -- How asymptotically fast the calculation is pr -- What portion of the total processing power this run gets Output: Factors of n, via a generator. Notes: 1. A good value of ov for typical numbers is somewhere around 10. If this parameter is too high, overhead and memory usage grow. 2. If ra is set to False and veb is set to True, then results are reproducible. If ra is set to True, then one number may be done in parallel on disconnected machines (at only a small loss of efficiency, which is less if pr is set correctly).''' if veb: looking_for = 0 k = inv_const(n) if ra: sigma = 6 + random.randrange(BILLION) else: sigma = 6 for factor in sure_factors(n, k, range(sigma, sigma + k), veb, ra, ov, tdb, pr): yield factor n /= factor if n == 1: return if ra: sigma += k + random.randrange(BILLION) else: sigma += k x_max = 0.5 * math.log(n) / math.log(k) t = rho_ts(int(x_max)) prime_probs = [] nc = 1 + int(_12_LOG_2_OVER_49 * ov * ov * k) eff_nc = nc / pr for i in xrange(1 + (int(math.log(n)) >> 1)): if i < math.log(tdb): prime_probs.append(0) else: prime_probs.append(1.0/i) for i in xrange(len(prime_probs)): p_success = rho_ev((i - 2.65) / math.log(k), t) p_fail = max(0, (1 - p_success * math.log(math.log(k)))) ** (k / pr) prime_probs[i] = p_fail * prime_probs[i] / (p_fail * prime_probs[i] + 1 - prime_probs[i]) while n != 1: low = int(k) high = n while high > low + 1: u = (high + low) >> 1 sum = 0 log_u = math.log(u) for i in xrange(len(prime_probs)): log_p = i - 2.65 log_u = math.log(u) quot = log_p / log_u sum += prime_probs[i] * (rho_ev(quot - 1, t) - rho_ev(quot, t) * log_u) if sum < 0: high = u else: low = u if ra: sigma += nc + random.randrange(BILLION) else: sigma += nc for factor in sure_factors(n, u, range(sigma, sigma + nc), veb, ra, ov, tdb, pr): yield factor n /= factor for i in xrange(len(prime_probs)): p_success = rho_ev((i - 2.65) / math.log(u), t) p_fail = max(0, (1 - p_success * math.log(math.log(u)))) ** eff_nc prime_probs[i] = p_fail * prime_probs[i] / (p_fail * prime_probs[i] + 1 - prime_probs[i]) prime_probs = prime_probs[:1 + (int(math.log(n)) >> 1)] if veb and n != 1: m = max(prime_probs) for i in xrange(len(prime_probs)): if prime_probs[i] == m: break new_looking_for = (int(i / _5_LOG_10) + 1) new_looking_for += new_looking_for << 2 if new_looking_for != looking_for: looking_for = new_looking_for print 'Searching for primes around', looking_for, 'digits' return def factors(n, veb, ra, ov, pr): '''Generates factors of n. Strips small primes, then feeds to ecm function. Input: n -- An integer to factor veb -- If True, be verbose ra -- If True, select sigma values randomly ov -- How asymptotically fast the calculation is pr -- What portion of the total processing power this run gets Output: Factors of n, via a generator. Notes: 1. A good value of ov for typical numbers is somewhere around 10. If this parameter is too high, overhead and memory usage grow. 2. If ra is set to False and veb is set to True, then results are reproducible. If ra is set to True, then one number may be done in parallel on disconnected machines (at only a small loss of efficiency, which is less if pr is set correctly).''' if type(n) not in T: raise ValueError, 'Number given must be integer or long.' if not 0 < pr <= 1: yield 'Error: pr must be between 0 and 1' return while not n & 1: n >>= 1 yield 2 n = mpz(n) k = inv_const(n) prime = 2 trial_division_bound = max(10 * k**2, 100) while prime < trial_division_bound: prime = next_prime(prime) while not n % prime: n /= prime yield prime if isprime(n): yield n return if n == 1: return for factor in ecm(n, ra, ov, veb, trial_division_bound, pr): yield factor ### End of algorithm code; beginning of interface code ## def is_switch(s): '''Tests whether the input string is a switch (e.g. "-v" or "--help"). Returns True or False.''' for i in xrange(len(s)): if s[i] != '-': break if i == 0: # s not begin with "-" return False for char in s[i:]: if not char.isalpha(): if char == '=': # Switches like "--portion=" are acceptable return True else: return False return True def parse_switch(s, switch): '''Parses a switch in the form '--string=num' and returns num or calls help() if the string is invalid. Returns the num in '--string=num'.''' try: return float(s[len(switch) + 3:]) except ValueError: help() def valid_input(s): '''Tests the input string for validity as a mathematical expressions. Returns True or False.''' valid = ('(', ')', '0', '1', '2', '3', '4', '5', '6', '7', '8', '9', '+', '-', '*', '/', '^', ' ', '\t') for char in s: if char not in valid: return False return True def help(): print '''\ Usage: pyecm [OPTION] [expression to factor] Factor numbers using the Elliptic Curve Method. --portion=num Does only part of the work for factoring, corresponding to what fraction of the total work the machine is doing. Useful for working in parallel. For example, if there are three machines: 1GHz, 1GHz, and 2GHz, print should be set to 0.25 for the 1GHz machines and 0.5 for the 2GHz machine. Implies -r and -v. -r is needed to avoid duplicating work and -v is needed to report results. --ov=num Sets the value of the internal parameter ov, which determines the trade-off between memory and time usage. Do not touch if you do not know what you are doing. Please read all the documentation and understand the full implications of the parameter before using this switch. -n, --noverbose Terse. On by default. Needed to cancel the -v from the --portion or --random switches. If both -n and -v are specified, the one specified last takes precedence. -r, --random Chooses random values for sigma, an internal parameter in the calculation. Implies -v; if you're doing something random, you want to know what's happening. -v, --verbose Explains what is being done with intermediate calculations and results. With no integers to factor given via command-line, read standard input. Please report bugs to Eric Larson .''' sys.exit() def command_line(veb, ra, ov, pr): l = len(sys.argv) for i in xrange(1, l): if not is_switch(sys.argv[i]): break for j in xrange(i, l): # Start with the first non-switch if j != i: # Pretty printing print response = sys.argv[j] if valid_input(response): response = response.replace('^', '**') try: n = eval(response) int(n) except (SyntaxError, TypeError, ValueError): help() else: help() print 'Factoring %d:' % n if n < 0: print -1 n = -n if n == 0: print '0 does not have a well-defined factorization.' continue elif n == 1: print 1 continue if ov == DUMMY: ov = 2*math.log(math.log(n)) for factor in factors(n, veb, ra, ov, pr): print factor def interactive(veb, ra, ov, pr): print 'pyecm v. %s (interactive mode):' % VERSION print 'Type "exit" at any time to quit.' print response = raw_input() while response != 'exit' and response != 'quit': if valid_input(response): response = response.replace('^', '**') try: n = eval(response) int(n) except (SyntaxError, TypeError, ValueError): help() else: help() print 'Factoring number %d:' % n if n < 0: print -1 n = -n if n == 0: print '0 does not have a well-defined factorization.' print response = raw_input() continue elif n == 1: print 1 print response = raw_input() continue if ov == DUMMY: ov = 2*math.log(math.log(n)) for factor in factors(n, veb, ra, ov, pr): print factor print response = raw_input() def main(): ra = veb = False pr = 1.0 ov = DUMMY for item in sys.argv[1:]: if item == '--help': help() elif item == '--noverbose': veb = False elif item == '--random': ra = veb = True elif item == '--verbose': veb = True elif item[:10] == '--portion=': pr = parse_switch(item, 'portion') ra = veb = True elif item[:5] == '--ov=': ov = parse_switch(item, 'ov') elif len(item) >= 2 and item[0] == '-' and item[1] != '-': # Short switch for char in item: if char == 'h': help() elif char == 'n': veb = False elif char == 'r': ra = veb = True elif char == 'v': veb = True else: if not valid_input(item): print 'I am confused about the following: "%s". Here\'s the help page:' % item print help() if len(sys.argv) > 1 and not is_switch(sys.argv[-1]): command_line(veb, ra, ov, pr) else: interactive(veb, ra, ov, pr) if __name__ == '__main__': try: main() except (EOFError, KeyboardInterrupt): sys.exit() pyecm-2.0.2/README.txt0000644000175000017500000000113012211747237013675 0ustar martinmartinHOW TO RUN PYECM: ALL PLATFORMS: - First, make sure python (>= 2.3) is installed and working. You can always get the latest release from http://python.org - For speed increases, you may optionally install gmpy and/or psyco. Both are highly recommended, especially gmpy, as they result in massive speed increases. UNIX, LINUX, MAC: - You can either use python pyecm.py to run it or change the filename to pyecm and execute it as a script. WINDOWS: - Just run pyecm.py as you would any other python program. PROJECT LINKS: gmpy (http://code.google.com/p/gmpy/) psyco (http://psyco.sourceforge.net/) pyecm-2.0.2/COPYING.txt0000644000175000017500000004311012211747237014054 0ustar martinmartin GNU GENERAL PUBLIC LICENSE Version 2, June 1991 Copyright (C) 1989, 1991 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software--to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation's software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.) 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These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Program (or any work based on the Program), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Program or works based on it. 6. Each time you redistribute the Program (or any work based on the Program), the recipient automatically receives a license from the original licensor to copy, distribute or modify the Program subject to these terms and conditions. You may not impose any further restrictions on the recipients' exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License. 7. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all. For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program. If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances. It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice. This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License. 8. If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License. 9. The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. Each version is given a distinguishing version number. If the Program specifies a version number of this License which applies to it and "any later version", you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation. 10. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally. NO WARRANTY 11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. END OF TERMS AND CONDITIONS How to Apply These Terms to Your New Programs If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms. To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found. Copyright (C) This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Also add information on how to contact you by electronic and paper mail. If the program is interactive, make it output a short notice like this when it starts in an interactive mode: Gnomovision version 69, Copyright (C) year name of author Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details. The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than `show w' and `show c'; they could even be mouse-clicks or menu items--whatever suits your program. You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the program, if necessary. Here is a sample; alter the names: Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovision' (which makes passes at compilers) written by James Hacker. , 1 April 1989 Ty Coon, President of Vice This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Library General Public License instead of this License.