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 # # number.py : Number-theoretic functions # # Part of the Python Cryptography Toolkit # # Distribute and use freely; there are no restrictions on further # dissemination and usage except those imposed by the laws of your # country of residence. This software is provided "as is" without # warranty of fitness for use or suitability for any purpose, express # or implied. Use at your own risk or not at all. # __revision__ = "\$Id: number.py,v 1.13 2003/04/04 18:21:07 akuchling Exp \$" bignum = long try: from Crypto.PublicKey import _fastmath except ImportError: _fastmath = None # Commented out and replaced with faster versions below ## def long2str(n): ## s='' ## while n>0: ## s=chr(n & 255)+s ## n=n>>8 ## return s ## import types ## def str2long(s): ## if type(s)!=types.StringType: return s # Integers will be left alone ## return reduce(lambda x,y : x*256+ord(y), s, 0L) def size (N): """size(N:long) : int Returns the size of the number N in bits. """ bits, power = 0,1L while N >= power: bits += 1 power = power << 1 return bits def getRandomNumber(N, randfunc): """getRandomNumber(N:int, randfunc:callable):long Return an N-bit random number.""" S = randfunc(N/8) odd_bits = N % 8 if odd_bits != 0: char = ord(randfunc(1)) >> (8-odd_bits) S = chr(char) + S value = bytes_to_long(S) value |= 2L ** (N-1) # Ensure high bit is set assert size(value) >= N return value def GCD(x,y): """GCD(x:long, y:long): long Return the GCD of x and y. """ x = abs(x) ; y = abs(y) while x > 0: x, y = y % x, x return y def inverse(u, v): """inverse(u:long, u:long):long Return the inverse of u mod v. """ u3, v3 = long(u), long(v) u1, v1 = 1L, 0L while v3 > 0: q=u3 / v3 u1, v1 = v1, u1 - v1*q u3, v3 = v3, u3 - v3*q while u1<0: u1 = u1 + v return u1 # Given a number of bits to generate and a random generation function, # find a prime number of the appropriate size. def getPrime(N, randfunc): """getPrime(N:int, randfunc:callable):long Return a random N-bit prime number. """ number=getRandomNumber(N, randfunc) | 1 while (not isPrime(number)): number=number+2 return number def isPrime(N): """isPrime(N:long):bool Return true if N is prime. """ if N == 1: return 0 if N in sieve: return 1 for i in sieve: if (N % i)==0: return 0 # Use the accelerator if available if _fastmath is not None: return _fastmath.isPrime(N) # Compute the highest bit that's set in N N1 = N - 1L n = 1L while (n> 1L # Rabin-Miller test for c in sieve[:7]: a=long(c) ; d=1L ; t=n while (t): # Iterate over the bits in N1 x=(d*d) % N if x==1L and d!=1L and d!=N1: return 0 # Square root of 1 found if N1 & t: d=(x*a) % N else: d=x t = t >> 1L if d!=1L: return 0 return 1 # Small primes used for checking primality; these are all the primes # less than 256. This should be enough to eliminate most of the odd # numbers before needing to do a Rabin-Miller test at all. sieve=[2, 3, 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, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251] # Improved conversion functions contributed by Barry Warsaw, after # careful benchmarking import struct def long_to_bytes(n, blocksize=0): """long_to_bytes(n:long, blocksize:int) : string Convert a long integer to a byte string. If optional blocksize is given and greater than zero, pad the front of the byte string with binary zeros so that the length is a multiple of blocksize. """ # after much testing, this algorithm was deemed to be the fastest s = '' n = long(n) pack = struct.pack while n > 0: s = pack('>I', n & 0xffffffffL) + s n = n >> 32 # strip off leading zeros for i in range(len(s)): if s[i] != '\000': break else: # only happens when n == 0 s = '\000' i = 0 s = s[i:] # add back some pad bytes. this could be done more efficiently w.r.t. the # de-padding being done above, but sigh... if blocksize > 0 and len(s) % blocksize: s = (blocksize - len(s) % blocksize) * '\000' + s return s def bytes_to_long(s): """bytes_to_long(string) : long Convert a byte string to a long integer. This is (essentially) the inverse of long_to_bytes(). """ acc = 0L unpack = struct.unpack length = len(s) if length % 4: extra = (4 - length % 4) s = '\000' * extra + s length = length + extra for i in range(0, length, 4): acc = (acc << 32) + unpack('>I', s[i:i+4]) return acc # For backwards compatibility... import warnings def long2str(n, blocksize=0): warnings.warn("long2str() has been replaced by long_to_bytes()") return long_to_bytes(n, blocksize) def str2long(s): warnings.warn("str2long() has been replaced by bytes_to_long()") return bytes_to_long(s)