pycryptodome/lib/Crypto/Math/_Numbers_int.py

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# Copyright (c) 2014, Legrandin <helderijs@gmail.com>
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from Crypto.Util.number import long_to_bytes, bytes_to_long
from Crypto.Util.py3compat import maxint

class Integer(object):
    """A class to model a natural integer (including zero)"""

    def __init__(self, value):
        if isinstance(value, float):
            raise ValueError("A floating point type is not a natural number")
        try:
            self._value = value._value
        except AttributeError:
            self._value = value

    # Conversions
    def __int__(self):
        return self._value

    def __str__(self):
        return str(int(self))

    def __repr__(self):
        return "Integer(%s)" % str(self)

    def to_bytes(self, block_size=0):
        if self._value < 0:
            raise ValueError("Conversion only valid for non-negative numbers")
        result = long_to_bytes(self._value, block_size)
        if len(result) > block_size > 0:
            raise ValueError("Value too large to encode")
        return result

    @staticmethod
    def from_bytes(byte_string):
        return Integer(bytes_to_long(byte_string))

    # Relations
    def __eq__(self, term):
        if term is None:
            return False
        return self._value == int(term)

    def __ne__(self, term):
        return not self.__eq__(term)

    def __lt__(self, term):
        return self._value < int(term)

    def __le__(self, term):
        return self.__lt__(term) or self.__eq__(term)

    def __gt__(self, term):
        return not self.__le__(term)

    def __ge__(self, term):
        return not self.__lt__(term)

    def __nonzero__(self):
        return self._value != 0

    def is_negative(self):
        return self._value < 0

    # Arithmetic operations
    def __add__(self, term):
        return self.__class__(self._value + int(term))

    def __sub__(self, term):
        return self.__class__(self._value - int(term))

    def __mul__(self, factor):
        return self.__class__(self._value * int(factor))

    def __floordiv__(self, divisor):
        return self.__class__(self._value // int(divisor))

    def __mod__(self, divisor):
        divisor_value = int(divisor)
        if divisor_value < 0:
            raise ValueError("Modulus must be positive")
        return self.__class__(self._value % divisor_value)

    def inplace_pow(self, exponent, modulus=None):
        exp_value = int(exponent)
        if exp_value < 0:
            raise ValueError("Exponent must not be negative")

        if modulus is not None:
            mod_value = int(modulus)
            if mod_value < 0:
                raise ValueError("Modulus must be positive")
            if mod_value == 0:
                raise ZeroDivisionError("Modulus cannot be zero")
        else:
            mod_value = None
        self._value = pow(self._value, exp_value, mod_value)
        return self

    def __pow__(self, exponent, modulus=None):
        result = self.__class__(self)
        return result.inplace_pow(exponent, modulus)

    def __abs__(self):
        return abs(self._value)

    def sqrt(self, modulus=None):

        value = self._value
        if modulus is None:
            if value < 0:
                raise ValueError("Square root of negative value")
            # http://stackoverflow.com/questions/15390807/integer-square-root-in-python

            x = value
            y = (x + 1) // 2
            while y < x:
                x = y
                y = (x + value // x) // 2
            result = x
        else:
            if modulus <= 0:
                raise ValueError("Modulus must be positive")
            result = self._tonelli_shanks(self.__class__(value % modulus), modulus)

        return self.__class__(result)

    @staticmethod
    def _tonelli_shanks(n, p):
        """Tonelli-shanks algorithm for computing the square root
        of n modulo a prime p.

        n must be in the range [0..p-1].
        p must be at least even.

        The return value r is the square root of modulo p. If non-zero,
        another solution will also exist (p-r).

        Note we cannot assume that p is really a prime: if it's not,
        we can either raise an exception or return the correct value.
        """

        # See https://rosettacode.org/wiki/Tonelli-Shanks_algorithm

        if n in (0, 1):
            return n

        if p % 4 == 3:
            root = pow(n, (p + 1)/4, p)
            if pow(root, 2, p) != n:
                raise ValueError("Cannot compute square root")
            return root

        s = 1
        q = (p - 1) / 2
        while not (q & 1):
            s += 1
            q >>= 1

        z = n.__class__(2)
        while True:
            euler = pow(z, (p - 1) / 2, p)
            if euler == 1:
                z += 1
                continue
            if euler == p - 1:
                break
            # Most probably p is not a prime
            raise ValueError("Cannot compute square root")
        
        m = s
        c = pow(z, q, p)
        t = pow(n, q, p)
        r = pow(n, (q + 1) / 2, p)
        
        while t != 1:
            for i in xrange(0, m):
                if pow(t, 2**i, p) == 1:
                    break
            if i == m:
                raise ValueError("Cannot compute square root of %d mod %d" % (n, p))
            b = pow(c, 2**(m - i - 1), p)
            m = i
            c = b**2 % p
            t = (t * b**2) % p
            r = (r * b) % p
        
        if pow(r, 2, p) != n:
            raise ValueError("Cannot compute square root")

        return r

    def __iadd__(self, term):
        self._value += int(term)
        return self

    def __isub__(self, term):
        self._value -= int(term)
        return self

    def __imul__(self, term):
        self._value *= int(term)
        return self

    def __imod__(self, term):
        modulus = int(term)
        if modulus == 0:
            raise ZeroDivisionError("Division by zero")
        if modulus < 0:
            raise ValueError("Modulus must be positive")
        self._value %= modulus
        return self

    # Boolean/bit operations
    def __and__(self, term):
        return self.__class__(self._value & int(term))

    def __or__(self, term):
        return self.__class__(self._value | int(term))

    def __rshift__(self, pos):
        try:
            return self.__class__(self._value >> int(pos))
        except OverflowError:
            if self._value >= 0:
                return 0
            else:
                return -1

    def __irshift__(self, pos):
        try:
            self._value >>= int(pos)
        except OverflowError:
            if self._value >= 0:
                return 0
            else:
                return -1
        return self

    def __lshift__(self, pos):
        try:
            return self.__class__(self._value << int(pos))
        except OverflowError:
            raise ValueError("Incorrect shift count")

    def __ilshift__(self, pos):
        try:
            self._value <<= int(pos)
        except OverflowError:
            raise ValueError("Incorrect shift count")
        return self


    def get_bit(self, n):
        if self._value < 0:
            raise ValueError("no bit representation for negative values")
        try:
            try:
                result = (self._value >> n._value) & 1
                if n._value < 0:
                    raise ValueError("negative bit count")
            except AttributeError:
                result = (self._value >> n) & 1
                if n < 0:
                    raise ValueError("negative bit count")
        except OverflowError:
            result = 0
        return result

    # Extra
    def is_odd(self):
        return (self._value & 1) == 1

    def is_even(self):
        return (self._value & 1) == 0

    def size_in_bits(self):

        if self._value < 0:
            raise ValueError("Conversion only valid for non-negative numbers")

        if self._value == 0:
            return 1

        bit_size = 0
        tmp = self._value
        while tmp:
            tmp >>= 1
            bit_size += 1

        return bit_size

    def size_in_bytes(self):
        return (self.size_in_bits() - 1) // 8 + 1

    def is_perfect_square(self):
        if self._value < 0:
            return False
        if self._value in (0, 1):
            return True

        x = self._value // 2
        square_x = x ** 2

        while square_x > self._value:
            x = (square_x + self._value) // (2 * x)
            square_x = x ** 2

        return self._value == x ** 2

    def fail_if_divisible_by(self, small_prime):
        if (self._value % int(small_prime)) == 0:
            raise ValueError("Value is composite")

    def multiply_accumulate(self, a, b):
        self._value += int(a) * int(b)
        return self

    def set(self, source):
        self._value = int(source)

    def inplace_inverse(self, modulus):
        modulus = int(modulus)
        if modulus == 0:
            raise ZeroDivisionError("Modulus cannot be zero")
        if modulus < 0:
            raise ValueError("Modulus cannot be negative")
        r_p, r_n = self._value, modulus
        s_p, s_n = 1, 0
        while r_n > 0:
            q = r_p // r_n
            r_p, r_n = r_n, r_p - q * r_n
            s_p, s_n = s_n, s_p - q * s_n
        if r_p != 1:
            raise ValueError("No inverse value can be computed" + str(r_p))
        while s_p < 0:
            s_p += modulus
        self._value = s_p
        return self

    def inverse(self, modulus):
        result = self.__class__(self)
        result.inplace_inverse(modulus)
        return result

    def gcd(self, term):
        r_p, r_n = abs(self._value), abs(int(term))
        while r_n > 0:
            q = r_p // r_n
            r_p, r_n = r_n, r_p - q * r_n
        return self.__class__(r_p)

    def lcm(self, term):
        term = int(term)
        if self._value == 0 or term == 0:
            return self.__class__(0)
        return self.__class__(abs((self._value * term) // self.gcd(term)._value))

    @staticmethod
    def jacobi_symbol(a, n):
        a = int(a)
        n = int(n)

        if (n & 1) == 0:
            raise ValueError("n must be even for the Jacobi symbol")

        # Step 1
        a = a % n
        # Step 2
        if a == 1 or n == 1:
            return 1
        # Step 3
        if a == 0:
            return 0
        # Step 4
        e = 0
        a1 = a
        while (a1 & 1) == 0:
            a1 >>= 1
            e += 1
        # Step 5
        if (e & 1) == 0:
            s = 1
        elif n % 8 in (1, 7):
            s = 1
        else:
            s = -1
        # Step 6
        if n % 4 == 3 and a1 % 4 == 3:
            s = -s
        # Step 7
        n1 = n % a1
        # Step 8
        return s * Integer.jacobi_symbol(n1, a1)