def inplace_pow(self, exponent, modulus=None):
if modulus is None:
if exponent < 0:
raise ValueError("Exponent must not be negative")
# Normal exponentiation
if exponent > 256:
raise ValueError("Exponent is too big")
_gmp.mpz_pow_ui(
self._mpz_p,
self._mpz_p, # Base
c_ulong(int(exponent)))
else:
# Modular exponentiation
if not isinstance(modulus, Integer):
modulus = Integer(modulus)
if not modulus:
raise ZeroDivisionError("Division by zero")
if modulus.is_negative():
raise ValueError("Modulus must be positive")
if isinstance(exponent, int):
if exponent < 0:
raise ValueError("Exponent must not be negative")
if exponent < 65536:
_gmp.mpz_powm_ui(self._mpz_p, self._mpz_p,
c_ulong(exponent), modulus._mpz_p)
return self
exponent = Integer(exponent)
elif exponent.is_negative():
raise ValueError("Exponent must not be negative")
_gmp.mpz_powm(self._mpz_p, self._mpz_p, exponent._mpz_p,
modulus._mpz_p)
return self
def __isub__(self, term):
if isinstance(term, int):
if 0 <= term < 65536:
_gmp.mpz_sub_ui(self._mpz_p, self._mpz_p, c_ulong(term))
return self
if -65535 < term < 0:
_gmp.mpz_add_ui(self._mpz_p, self._mpz_p, c_ulong(-term))
return self
term = Integer(term)
_gmp.mpz_sub(self._mpz_p, self._mpz_p, term._mpz_p)
return self
def __imul__(self, term):
if isinstance(term, int):
if 0 <= term < 65536:
_gmp.mpz_mul_ui(self._mpz_p, self._mpz_p, c_ulong(term))
return self
if -65535 < term < 0:
_gmp.mpz_mul_ui(self._mpz_p, self._mpz_p, c_ulong(-term))
_gmp.mpz_neg(self._mpz_p, self._mpz_p)
return self
term = Integer(term)
_gmp.mpz_mul(self._mpz_p, self._mpz_p, term._mpz_p)
return self
def multiply_accumulate(self, a, b):
"""Increment the number by the product of a and b."""
if not isinstance(a, Integer):
a = Integer(a)
if isinstance(b, int):
if 0 < b < 65536:
_gmp.mpz_addmul_ui(self._mpz_p, a._mpz_p, c_ulong(b))
return self
if -65535 < b < 0:
_gmp.mpz_submul_ui(self._mpz_p, a._mpz_p, c_ulong(-b))
return self
b = Integer(b)
_gmp.mpz_addmul(self._mpz_p, a._mpz_p, b._mpz_p)
return self
def seek(self, position):
"""Seek to a certain position in the key stream.
:param integer position:
The absolute position within the key stream, in bytes.
"""
position, offset = divmod(position, 64)
block_low = position & 0xFFFFFFFF
block_high = position >> 32
result = _raw_chacha20_lib.chacha20_seek(self._state.get(),
c_ulong(block_high),
c_ulong(block_low), offset)
if result:
raise ValueError("Error %d while seeking with ChaCha20" % result)
def __irshift__(self, pos):
if pos < 0:
raise ValueError("negative shift count")
if pos > 65536:
if self < 0:
return -1
else:
return 0
_gmp.mpz_tdiv_q_2exp(self._mpz_p, self._mpz_p, c_ulong(int(pos)))
return self
def fail_if_divisible_by(self, small_prime):
"""Raise an exception if the small prime is a divisor."""
if isinstance(small_prime, int):
if 0 < small_prime < 65536:
if _gmp.mpz_divisible_ui_p(self._mpz_p, c_ulong(small_prime)):
raise ValueError("The value is composite")
return
small_prime = Integer(small_prime)
if _gmp.mpz_divisible_p(self._mpz_p, small_prime._mpz_p):
raise ValueError("The value is composite")
def get_bit(self, n):
"""Return True if the n-th bit is set to 1.
Bit 0 is the least significant."""
if self < 0:
raise ValueError("no bit representation for negative values")
if n < 0:
raise ValueError("negative bit count")
if n > 65536:
return 0
return bool(_gmp.mpz_tstbit(self._mpz_p, c_ulong(int(n))))
def gcd(self, term):
"""Compute the greatest common denominator between this
number and another term."""
result = Integer(0)
if isinstance(term, int):
if 0 < term < 65535:
_gmp.mpz_gcd_ui(result._mpz_p, self._mpz_p, c_ulong(term))
return result
term = Integer(term)
_gmp.mpz_gcd(result._mpz_p, self._mpz_p, term._mpz_p)
return result
def __ilshift__(self, pos):
if not 0 <= pos < 65536:
raise ValueError("Incorrect shift count")
_gmp.mpz_mul_2exp(self._mpz_p, self._mpz_p, c_ulong(int(pos)))
return self
def __lshift__(self, pos):
result = Integer(0)
if not 0 <= pos < 65536:
raise ValueError("Incorrect shift count")
_gmp.mpz_mul_2exp(result._mpz_p, self._mpz_p, c_ulong(int(pos)))
return result
class Integer(object):
"""A fast, arbitrary precision integer"""
_zero_mpz_p = new_mpz()
_gmp.mpz_init_set_ui(_zero_mpz_p, c_ulong(0))
def __init__(self, value):
"""Initialize the integer to the given value."""
self._mpz_p = new_mpz()
self._initialized = False
if isinstance(value, float):
raise ValueError("A floating point type is not a natural number")
self._initialized = True
if isinstance(value, int):
_gmp.mpz_init(self._mpz_p)
result = _gmp.gmp_sscanf(tobytes(str(value)), b("%Zd"),
self._mpz_p)
if result != 1:
raise ValueError("Error converting '%d'" % value)
else:
_gmp.mpz_init_set(self._mpz_p, value._mpz_p)
# Conversions
def __int__(self):
# buf will contain the integer encoded in decimal plus the trailing
# zero, and possibly the negative sign.
# dig10(x) < log10(x) + 1 = log2(x)/log2(10) + 1 < log2(x)/3 + 1
buf_len = _gmp.mpz_sizeinbase(self._mpz_p, 2) // 3 + 3
buf = create_string_buffer(buf_len)
_gmp.gmp_snprintf(buf, c_size_t(buf_len), b("%Zd"), self._mpz_p)
return int(get_c_string(buf))
def __str__(self):
return str(int(self))
def __repr__(self):
return "Integer(%s)" % str(self)
def to_bytes(self, block_size=0):
"""Convert the number into a byte string.
This method encodes the number in network order and prepends
as many zero bytes as required. It only works for non-negative
values.
:Parameters:
block_size : integer
The exact size the output byte string must have.
If zero, the string has the minimal length.
:Returns:
A byte string.
:Raise ValueError:
If the value is negative or if ``block_size`` is
provided and the length of the byte string would exceed it.
"""
if self < 0:
raise ValueError("Conversion only valid for non-negative numbers")
buf_len = (_gmp.mpz_sizeinbase(self._mpz_p, 2) + 7) // 8
if buf_len > block_size > 0:
raise ValueError("Number is too big to convert to byte string"
"of prescribed length")
buf = create_string_buffer(buf_len)
_gmp.mpz_export(
buf,
null_pointer, # Ignore countp
1, # Big endian
c_size_t(1), # Each word is 1 byte long
0, # Endianess within a word - not relevant
c_size_t(0), # No nails
self._mpz_p)
return bchr(0) * max(0, block_size - buf_len) + get_raw_buffer(buf)
@staticmethod
def from_bytes(byte_string):
"""Convert a byte string into a number.
:Parameters:
byte_string : byte string
The input number, encoded in network order.
It can only be non-negative.
:Return:
The ``Integer`` object carrying the same value as the input.
"""
result = Integer(0)
_gmp.mpz_import(
result._mpz_p,
c_size_t(len(byte_string)), # Amount of words to read
1, # Big endian
c_size_t(1), # Each word is 1 byte long
0, # Endianess within a word - not relevant
c_size_t(0), # No nails
byte_string)
return result
# Relations
def _apply_and_return(self, func, term):
if not isinstance(term, Integer):
term = Integer(term)
return func(self._mpz_p, term._mpz_p)
def __eq__(self, term):
if not isinstance(term, (Integer, int)):
return False
return self._apply_and_return(_gmp.mpz_cmp, term) == 0
def __ne__(self, term):
if not isinstance(term, (Integer, int)):
return True
return self._apply_and_return(_gmp.mpz_cmp, term) != 0
def __lt__(self, term):
return self._apply_and_return(_gmp.mpz_cmp, term) < 0
def __le__(self, term):
return self._apply_and_return(_gmp.mpz_cmp, term) <= 0
def __gt__(self, term):
return self._apply_and_return(_gmp.mpz_cmp, term) > 0
def __ge__(self, term):
return self._apply_and_return(_gmp.mpz_cmp, term) >= 0
def __bool__(self):
return _gmp.mpz_cmp(self._mpz_p, self._zero_mpz_p) != 0
def is_negative(self):
return _gmp.mpz_cmp(self._mpz_p, self._zero_mpz_p) < 0
# Arithmetic operations
def __add__(self, term):
result = Integer(0)
if not isinstance(term, Integer):
term = Integer(term)
_gmp.mpz_add(result._mpz_p, self._mpz_p, term._mpz_p)
return result
def __sub__(self, term):
result = Integer(0)
if not isinstance(term, Integer):
term = Integer(term)
_gmp.mpz_sub(result._mpz_p, self._mpz_p, term._mpz_p)
return result
def __mul__(self, term):
result = Integer(0)
if not isinstance(term, Integer):
term = Integer(term)
_gmp.mpz_mul(result._mpz_p, self._mpz_p, term._mpz_p)
return result
def __floordiv__(self, divisor):
if not isinstance(divisor, Integer):
divisor = Integer(divisor)
if _gmp.mpz_cmp(divisor._mpz_p, self._zero_mpz_p) == 0:
raise ZeroDivisionError("Division by zero")
result = Integer(0)
_gmp.mpz_fdiv_q(result._mpz_p, self._mpz_p, divisor._mpz_p)
return result
def __mod__(self, divisor):
if not isinstance(divisor, Integer):
divisor = Integer(divisor)
comp = _gmp.mpz_cmp(divisor._mpz_p, self._zero_mpz_p)
if comp == 0:
raise ZeroDivisionError("Division by zero")
if comp < 0:
raise ValueError("Modulus must be positive")
result = Integer(0)
_gmp.mpz_mod(result._mpz_p, self._mpz_p, divisor._mpz_p)
return result
def inplace_pow(self, exponent, modulus=None):
if modulus is None:
if exponent < 0:
raise ValueError("Exponent must not be negative")
# Normal exponentiation
if exponent > 256:
raise ValueError("Exponent is too big")
_gmp.mpz_pow_ui(
self._mpz_p,
self._mpz_p, # Base
c_ulong(int(exponent)))
else:
# Modular exponentiation
if not isinstance(modulus, Integer):
modulus = Integer(modulus)
if not modulus:
raise ZeroDivisionError("Division by zero")
if modulus.is_negative():
raise ValueError("Modulus must be positive")
if isinstance(exponent, int):
if exponent < 0:
raise ValueError("Exponent must not be negative")
if exponent < 65536:
_gmp.mpz_powm_ui(self._mpz_p, self._mpz_p,
c_ulong(exponent), modulus._mpz_p)
return self
exponent = Integer(exponent)
elif exponent.is_negative():
raise ValueError("Exponent must not be negative")
_gmp.mpz_powm(self._mpz_p, self._mpz_p, exponent._mpz_p,
modulus._mpz_p)
return self
def __pow__(self, exponent, modulus=None):
result = Integer(self)
return result.inplace_pow(exponent, modulus)
def __abs__(self):
result = Integer(0)
_gmp.mpz_abs(result._mpz_p, self._mpz_p)
return result
def sqrt(self, modulus=None):
"""Return the largest Integer that does not
exceed the square root"""
if modulus is None:
if self < 0:
raise ValueError("Square root of negative value")
result = Integer(0)
_gmp.mpz_sqrt(result._mpz_p, self._mpz_p)
else:
if modulus <= 0:
raise ValueError("Modulus must be positive")
modulus = int(modulus)
result = Integer(
SlowInteger._tonelli_shanks(int(self) % modulus, modulus))
return result
def __iadd__(self, term):
if isinstance(term, int):
if 0 <= term < 65536:
_gmp.mpz_add_ui(self._mpz_p, self._mpz_p, c_ulong(term))
return self
if -65535 < term < 0:
_gmp.mpz_sub_ui(self._mpz_p, self._mpz_p, c_ulong(-term))
return self
term = Integer(term)
_gmp.mpz_add(self._mpz_p, self._mpz_p, term._mpz_p)
return self
def __isub__(self, term):
if isinstance(term, int):
if 0 <= term < 65536:
_gmp.mpz_sub_ui(self._mpz_p, self._mpz_p, c_ulong(term))
return self
if -65535 < term < 0:
_gmp.mpz_add_ui(self._mpz_p, self._mpz_p, c_ulong(-term))
return self
term = Integer(term)
_gmp.mpz_sub(self._mpz_p, self._mpz_p, term._mpz_p)
return self
def __imul__(self, term):
if isinstance(term, int):
if 0 <= term < 65536:
_gmp.mpz_mul_ui(self._mpz_p, self._mpz_p, c_ulong(term))
return self
if -65535 < term < 0:
_gmp.mpz_mul_ui(self._mpz_p, self._mpz_p, c_ulong(-term))
_gmp.mpz_neg(self._mpz_p, self._mpz_p)
return self
term = Integer(term)
_gmp.mpz_mul(self._mpz_p, self._mpz_p, term._mpz_p)
return self
def __imod__(self, divisor):
if not isinstance(divisor, Integer):
divisor = Integer(divisor)
comp = _gmp.mpz_cmp(divisor._mpz_p, divisor._zero_mpz_p)
if comp == 0:
raise ZeroDivisionError("Division by zero")
if comp < 0:
raise ValueError("Modulus must be positive")
_gmp.mpz_mod(self._mpz_p, self._mpz_p, divisor._mpz_p)
return self
# Boolean/bit operations
def __and__(self, term):
result = Integer(0)
if not isinstance(term, Integer):
term = Integer(term)
_gmp.mpz_and(result._mpz_p, self._mpz_p, term._mpz_p)
return result
def __or__(self, term):
result = Integer(0)
if not isinstance(term, Integer):
term = Integer(term)
_gmp.mpz_ior(result._mpz_p, self._mpz_p, term._mpz_p)
return result
def __rshift__(self, pos):
result = Integer(0)
if pos < 0:
raise ValueError("negative shift count")
if pos > 65536:
if self < 0:
return -1
else:
return 0
_gmp.mpz_tdiv_q_2exp(result._mpz_p, self._mpz_p, c_ulong(int(pos)))
return result
def __irshift__(self, pos):
if pos < 0:
raise ValueError("negative shift count")
if pos > 65536:
if self < 0:
return -1
else:
return 0
_gmp.mpz_tdiv_q_2exp(self._mpz_p, self._mpz_p, c_ulong(int(pos)))
return self
def __lshift__(self, pos):
result = Integer(0)
if not 0 <= pos < 65536:
raise ValueError("Incorrect shift count")
_gmp.mpz_mul_2exp(result._mpz_p, self._mpz_p, c_ulong(int(pos)))
return result
def __ilshift__(self, pos):
if not 0 <= pos < 65536:
raise ValueError("Incorrect shift count")
_gmp.mpz_mul_2exp(self._mpz_p, self._mpz_p, c_ulong(int(pos)))
return self
def get_bit(self, n):
"""Return True if the n-th bit is set to 1.
Bit 0 is the least significant."""
if self < 0:
raise ValueError("no bit representation for negative values")
if n < 0:
raise ValueError("negative bit count")
if n > 65536:
return 0
return bool(_gmp.mpz_tstbit(self._mpz_p, c_ulong(int(n))))
# Extra
def is_odd(self):
return _gmp.mpz_tstbit(self._mpz_p, 0) == 1
def is_even(self):
return _gmp.mpz_tstbit(self._mpz_p, 0) == 0
def size_in_bits(self):
"""Return the minimum number of bits that can encode the number."""
if self < 0:
raise ValueError("Conversion only valid for non-negative numbers")
return _gmp.mpz_sizeinbase(self._mpz_p, 2)
def size_in_bytes(self):
"""Return the minimum number of bytes that can encode the number."""
return (self.size_in_bits() - 1) // 8 + 1
def is_perfect_square(self):
return _gmp.mpz_perfect_square_p(self._mpz_p) != 0
def fail_if_divisible_by(self, small_prime):
"""Raise an exception if the small prime is a divisor."""
if isinstance(small_prime, int):
if 0 < small_prime < 65536:
if _gmp.mpz_divisible_ui_p(self._mpz_p, c_ulong(small_prime)):
raise ValueError("The value is composite")
return
small_prime = Integer(small_prime)
if _gmp.mpz_divisible_p(self._mpz_p, small_prime._mpz_p):
raise ValueError("The value is composite")
def multiply_accumulate(self, a, b):
"""Increment the number by the product of a and b."""
if not isinstance(a, Integer):
a = Integer(a)
if isinstance(b, int):
if 0 < b < 65536:
_gmp.mpz_addmul_ui(self._mpz_p, a._mpz_p, c_ulong(b))
return self
if -65535 < b < 0:
_gmp.mpz_submul_ui(self._mpz_p, a._mpz_p, c_ulong(-b))
return self
b = Integer(b)
_gmp.mpz_addmul(self._mpz_p, a._mpz_p, b._mpz_p)
return self
def set(self, source):
"""Set the Integer to have the given value"""
if not isinstance(source, Integer):
source = Integer(source)
_gmp.mpz_set(self._mpz_p, source._mpz_p)
return self
def inplace_inverse(self, modulus):
"""Compute the inverse of this number in the ring of
modulo integers.
Raise an exception if no inverse exists.
"""
if not isinstance(modulus, Integer):
modulus = Integer(modulus)
comp = _gmp.mpz_cmp(modulus._mpz_p, self._zero_mpz_p)
if comp == 0:
raise ZeroDivisionError("Modulus cannot be zero")
if comp < 0:
raise ValueError("Modulus must be positive")
result = _gmp.mpz_invert(self._mpz_p, self._mpz_p, modulus._mpz_p)
if not result:
raise ValueError("No inverse value can be computed")
return self
def inverse(self, modulus):
result = Integer(self)
result.inplace_inverse(modulus)
return result
def gcd(self, term):
"""Compute the greatest common denominator between this
number and another term."""
result = Integer(0)
if isinstance(term, int):
if 0 < term < 65535:
_gmp.mpz_gcd_ui(result._mpz_p, self._mpz_p, c_ulong(term))
return result
term = Integer(term)
_gmp.mpz_gcd(result._mpz_p, self._mpz_p, term._mpz_p)
return result
def lcm(self, term):
"""Compute the least common multiplier between this
number and another term."""
result = Integer(0)
if not isinstance(term, Integer):
term = Integer(term)
_gmp.mpz_lcm(result._mpz_p, self._mpz_p, term._mpz_p)
return result
@staticmethod
def jacobi_symbol(a, n):
"""Compute the Jacobi symbol"""
if not isinstance(a, Integer):
a = Integer(a)
if not isinstance(n, Integer):
n = Integer(n)
if n <= 0 or n.is_even():
raise ValueError("n must be positive even for the Jacobi symbol")
return _gmp.mpz_jacobi(a._mpz_p, n._mpz_p)
# Clean-up
def __del__(self):
try:
if self._mpz_p is not None:
if self._initialized:
_gmp.mpz_clear(self._mpz_p)
self._mpz_p = None
except AttributeError:
pass