Initial checkin of version 1.1

1 files changed, 448 insertions, 0 deletions

diff --git a/rsa/__init__.py b/rsa/__init__.py
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+"""RSA module
+
+Module for calculating large primes, and RSA encryption, decryption,
+signing and verification. Includes generating public and private keys.
+"""
+
+__author__ = "Sybren Stuvel, Marloes de Boer and Ivo Tamboer"
+__date__ = "2008-04-23"
+
+# NOTE: Python's modulo can return negative numbers. We compensate for
+# this behaviour using the abs() function
+
+import math
+import sys
+import random   # For picking semi-random numbers
+import types
+from cPickle import dumps, loads
+import base64
+import zlib
+
+RANDOM_DEV="/dev/urandom"
+has_broken_randint = False
+
+try:
+    random.randint(1, 10000000000000000000000000L)
+except:
+    has_broken_randint = True
+    print "This system's random.randint() can't handle large numbers"
+    print "Random integers will all be read from %s" % RANDOM_DEV
+
+
+def log(x, base = 10):
+    return math.log(x) / math.log(base)
+
+def gcd(p, q):
+    """Returns the greatest common divisor of p and q
+
+
+    >>> gcd(42, 6)
+    6
+    """
+    if p<q: return gcd(q, p)
+    if q == 0: return p
+    return gcd(q, abs(p%q))
+
+def bytes2int(bytes):
+    """Converts a list of bytes or a string to an integer
+
+    >>> (128*256 + 64)*256 + + 15
+    8405007
+    >>> l = [128, 64, 15]
+    >>> bytes2int(l)
+    8405007
+    """
+
+    if not (type(bytes) is types.ListType or type(bytes) is types.StringType):
+        raise TypeError("You must pass a string or a list")
+
+    # Convert byte stream to integer
+    integer = 0
+    for byte in bytes:
+        integer *= 256
+        if type(byte) is types.StringType: byte = ord(byte)
+        integer += byte
+
+    return integer
+
+def int2bytes(number):
+    """Converts a number to a string of bytes
+    
+    >>> bytes2int(int2bytes(123456789))
+    123456789
+    """
+
+    if not (type(number) is types.LongType or type(number) is types.IntType):
+        raise TypeError("You must pass a long or an int")
+
+    string = ""
+
+    while number > 0:
+        string = "%s%s" % (chr(number & 0xFF), string)
+        number /= 256
+    
+    return string
+
+def fast_exponentiation(a, p, n):
+    """Calculates r = a^p mod n
+    """
+    result = a % n
+    remainders = []
+    while p != 1:
+        remainders.append(p & 1)
+        p = p >> 1
+    while remainders:
+        rem = remainders.pop()
+        result = ((a ** rem) * result ** 2) % n
+    return result
+
+def read_random_int(nbits):
+    """Reads a random integer from RANDOM_DEV of approximately nbits
+    bits rounded up to whole bytes"""
+
+    nbytes = ceil(nbits/8)
+
+    # Read 'nbits' bits of random data
+    fd = open(RANDOM_DEV)
+    randomdata = fd.read(nbytes)
+    fd.close()
+
+    if len(randomdata) != nbytes:
+        raise Exception("Unable to read enough random bytes")
+
+    return bytes2int(randomdata)
+
+def ceil(x):
+    """Returns int(math.ceil(x))"""
+
+    return int(math.ceil(x))
+    
+def randint(minvalue, maxvalue):
+    """Returns a random integer x with minvalue <= x <= maxvalue"""
+
+    if not has_broken_randint:
+        return random.randint(minvalue, maxvalue)
+
+    # Safety - get a lot of random data even if the range is fairly
+    # small
+    min_nbits = 32
+
+    # The range of the random numbers we need to generate
+    range = maxvalue - minvalue
+
+    # Which is this number of bytes
+    rangebytes = ceil(log(range, 2) / 8)
+
+    # Convert to bits, but make sure it's always at least min_nbits*2
+    rangebits = max(rangebytes * 8, min_nbits * 2)
+    
+    # Take a random number of bits between min_nbits and rangebits
+    nbits = random.randint(min_nbits, rangebits)
+    
+    return (read_random_int(nbits) % range) + minvalue
+
+def fermat_little_theorem(p):
+    """Returns 1 if p may be prime, and something else if p definitely
+    is not prime"""
+
+    a = randint(1, p-1)
+    return fast_exponentiation(a, p-1, p)
+
+def jacobi(a, b):
+    """Calculates the value of the Jacobi symbol (a/b)
+    """
+
+    if a % b == 0:
+        return 0
+    result = 1
+    while a > 1:
+        if a & 1:
+            if ((a-1)*(b-1) >> 2) & 1:
+                result = -result
+            b, a = a, b % a
+        else:
+            if ((b ** 2 - 1) >> 3) & 1:
+                result = -result
+            a = a >> 1
+    return result
+
+def jacobi_witness(x, n):
+    """Returns False if n is an Euler pseudo-prime with base x, and
+    True otherwise.
+    """
+
+    j = jacobi(x, n) % n
+    f = fast_exponentiation(x, (n-1)/2, n)
+
+    if j == f: return False
+    return True
+
+def randomized_primality_testing(n, k):
+    """Calculates whether n is composite (which is always correct) or
+    prime (which is incorrect with error probability 2**-k)
+
+    Returns False if the number if composite, and True if it's
+    probably prime.
+    """
+
+    q = 0.5     # Property of the jacobi_witness function
+
+    # t = int(math.ceil(k / log(1/q, 2)))
+    t = ceil(k / log(1/q, 2))
+    for i in range(t+1):
+        x = randint(1, n-1)
+        if jacobi_witness(x, n): return False
+    
+    return True
+
+def is_prime(number):
+    """Returns True if the number is prime, and False otherwise.
+
+    >>> is_prime(42)
+    0
+    >>> is_prime(41)
+    1
+    """
+
+    """
+    if not fermat_little_theorem(number) == 1:
+        # Not prime, according to Fermat's little theorem
+        return False
+    """
+
+    if randomized_primality_testing(number, 5):
+        # Prime, according to Jacobi
+        return True
+    
+    # Not prime
+    return False
+
+    
+def getprime(nbits):
+    """Returns a prime number of max. 'math.ceil(nbits/8)*8' bits. In
+    other words: nbits is rounded up to whole bytes.
+
+    >>> p = getprime(8)
+    >>> is_prime(p-1)
+    0
+    >>> is_prime(p)
+    1
+    >>> is_prime(p+1)
+    0
+    """
+
+    nbytes = int(math.ceil(nbits/8))
+
+    while True:
+        integer = read_random_int(nbits)
+
+        # Make sure it's odd
+        integer |= 1
+
+        # Test for primeness
+        if is_prime(integer): break
+
+        # Retry if not prime
+
+    return integer
+
+def are_relatively_prime(a, b):
+    """Returns True if a and b are relatively prime, and False if they
+    are not.
+
+    >>> are_relatively_prime(2, 3)
+    1
+    >>> are_relatively_prime(2, 4)
+    0
+    """
+
+    d = gcd(a, b)
+    return (d == 1)
+
+def find_p_q(nbits):
+    """Returns a tuple of two different primes of nbits bits"""
+
+    p = getprime(nbits)
+    while True:
+        q = getprime(nbits)
+        if not q == p: break
+    
+    return (p, q)
+
+def extended_euclid_gcd(a, b):
+    """Returns a tuple (d, i, j) such that d = gcd(a, b) = ia + jb
+    """
+
+    if b == 0:
+        return (a, 1, 0)
+
+    q = abs(a % b)
+    r = long(a / b)
+    (d, k, l) = extended_euclid_gcd(b, q)
+
+    return (d, l, k - l*r)
+
+# Main function: calculate encryption and decryption keys
+def calculate_keys(p, q, nbits):
+    """Calculates an encryption and a decryption key for p and q, and
+    returns them as a tuple (e, d)"""
+
+    n = p * q
+    phi_n = (p-1) * (q-1)
+
+    while True:
+        # Make sure e has enough bits so we ensure "wrapping" through
+        # modulo n
+        e = getprime(max(8, nbits/2))
+        if are_relatively_prime(e, n) and are_relatively_prime(e, phi_n): break
+
+    (d, i, j) = extended_euclid_gcd(e, phi_n)
+
+    if not d == 1:
+        raise Exception("e (%d) and phi_n (%d) are not relatively prime" % (e, phi_n))
+
+    if not (e * i) % phi_n == 1:
+        raise Exception("e (%d) and i (%d) are not mult. inv. modulo phi_n (%d)" % (e, i, phi_n))
+
+    return (e, i)
+
+
+def gen_keys(nbits):
+    """Generate RSA keys of nbits bits. Returns (p, q, e, d).
+    """
+
+    while True:
+        (p, q) = find_p_q(nbits)
+        (e, d) = calculate_keys(p, q, nbits)
+
+        # For some reason, d is sometimes negative. We don't know how
+        # to fix it (yet), so we keep trying until everything is shiny
+        if d > 0: break
+
+    return (p, q, e, d)
+
+def gen_pubpriv_keys(nbits):
+    """Generates public and private keys, and returns them as (pub,
+    priv).
+
+    The public key consists of a dict {e: ..., , n: ....). The private
+    key consists of a dict {d: ...., p: ...., q: ....).
+    """
+    
+    (p, q, e, d) = gen_keys(nbits)
+
+    return ( {'e': e, 'n': p*q}, {'d': d, 'p': p, 'q': q} )
+
+def encrypt_int(message, ekey, n):
+    """Encrypts a message using encryption key 'ekey', working modulo
+    n"""
+
+    if type(message) is types.IntType:
+        return encrypt_int(long(message), ekey, n)
+
+    if not type(message) is types.LongType:
+        raise TypeError("You must pass a long or an int")
+
+    if math.floor(log(message, 2)) > math.floor(log(n, 2)):
+        raise OverflowError("The message is too long")
+
+    return fast_exponentiation(message, ekey, n)
+
+def decrypt_int(cyphertext, dkey, n):
+    """Decrypts a cypher text using the decryption key 'dkey', working
+    modulo n"""
+
+    return encrypt_int(cyphertext, dkey, n)
+
+def sign_int(message, dkey, n):
+    """Signs 'message' using key 'dkey', working modulo n"""
+
+    return decrypt_int(message, dkey, n)
+
+def verify_int(signed, ekey, n):
+    """verifies 'signed' using key 'ekey', working modulo n"""
+
+    return encrypt_int(signed, ekey, n)
+
+def picklechops(chops):
+    """Pickles and base64encodes it's argument chops"""
+
+    value = zlib.compress(dumps(chops))
+    encoded = base64.encodestring(value)
+    return encoded.strip()
+
+def unpicklechops(string):
+    """base64decodes and unpickes it's argument string into chops"""
+
+    return loads(zlib.decompress(base64.decodestring(string)))
+
+def chopstring(message, key, n, funcref):
+    """Splits 'message' into chops that are at most as long as n,
+    converts these into integers, and calls funcref(integer, key, n)
+    for each chop.
+
+    Used by 'encrypt' and 'sign'.
+    """
+
+    msglen = len(message)
+    mbits = msglen * 8
+    nbits = int(math.floor(log(n, 2)))
+    nbytes = nbits / 8
+    blocks = msglen / nbytes
+
+    if msglen % nbytes > 0:
+        blocks += 1
+
+    cypher = []
+    
+    for bindex in range(blocks):
+        offset = bindex * nbytes
+        block = message[offset:offset+nbytes]
+        value = bytes2int(block)
+        cypher.append(funcref(value, key, n))
+
+    return picklechops(cypher)
+
+def gluechops(chops, key, n, funcref):
+    """Glues chops back together into a string.  calls
+    funcref(integer, key, n) for each chop.
+
+    Used by 'decrypt' and 'verify'.
+    """
+    message = ""
+
+    chops = unpicklechops(chops)
+    
+    for cpart in chops:
+        mpart = funcref(cpart, key, n)
+        message += int2bytes(mpart)
+    
+    return message
+
+def encrypt(message, key):
+    """Encrypts a string 'message' with the public key 'key'"""
+    
+    return chopstring(message, key['e'], key['n'], encrypt_int)
+
+def sign(message, key):
+    """Signs a string 'message' with the private key 'key'"""
+    
+    return chopstring(message, key['d'], key['p']*key['q'], decrypt_int)
+
+def decrypt(cypher, key):
+    """Decrypts a cypher with the private key 'key'"""
+
+    return gluechops(cypher, key['d'], key['p']*key['q'], decrypt_int)
+
+def verify(cypher, key):
+    """Verifies a cypher with the public key 'key'"""
+
+    return gluechops(cypher, key['e'], key['n'], encrypt_int)
+
+# Do doctest if we're not imported
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()
+
+__all__ = ["gen_pubpriv_keys", "encrypt", "decrypt", "sign", "verify"]
+