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# Copyright 2011 Sybren A. Stüvel <sybren@stuvel.eu>
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""Common functionality shared by several modules."""
import typing
class NotRelativePrimeError(ValueError):
def __init__(self, a: int, b: int, d: int, msg: str = "") -> None:
super().__init__(msg or "%d and %d are not relatively prime, divider=%i" % (a, b, d))
self.a = a
self.b = b
self.d = d
def bit_size(num: int) -> int:
"""
Number of bits needed to represent a integer excluding any prefix
0 bits.
Usage::
>>> bit_size(1023)
10
>>> bit_size(1024)
11
>>> bit_size(1025)
11
:param num:
Integer value. If num is 0, returns 0. Only the absolute value of the
number is considered. Therefore, signed integers will be abs(num)
before the number's bit length is determined.
:returns:
Returns the number of bits in the integer.
"""
try:
return num.bit_length()
except AttributeError as ex:
raise TypeError("bit_size(num) only supports integers, not %r" % type(num)) from ex
def byte_size(number: int) -> int:
"""
Returns the number of bytes required to hold a specific long number.
The number of bytes is rounded up.
Usage::
>>> byte_size(1 << 1023)
128
>>> byte_size((1 << 1024) - 1)
128
>>> byte_size(1 << 1024)
129
:param number:
An unsigned integer
:returns:
The number of bytes required to hold a specific long number.
"""
if number == 0:
return 1
return ceil_div(bit_size(number), 8)
def ceil_div(num: int, div: int) -> int:
"""
Returns the ceiling function of a division between `num` and `div`.
Usage::
>>> ceil_div(100, 7)
15
>>> ceil_div(100, 10)
10
>>> ceil_div(1, 4)
1
:param num: Division's numerator, a number
:param div: Division's divisor, a number
:return: Rounded up result of the division between the parameters.
"""
quanta, mod = divmod(num, div)
if mod:
quanta += 1
return quanta
def extended_gcd(a: int, b: int) -> typing.Tuple[int, int, int]:
"""Returns a tuple (r, i, j) such that r = gcd(a, b) = ia + jb"""
# r = gcd(a,b) i = multiplicitive inverse of a mod b
# or j = multiplicitive inverse of b mod a
# Neg return values for i or j are made positive mod b or a respectively
# Iterateive Version is faster and uses much less stack space
x = 0
y = 1
lx = 1
ly = 0
oa = a # Remember original a/b to remove
ob = b # negative values from return results
while b != 0:
q = a // b
(a, b) = (b, a % b)
(x, lx) = ((lx - (q * x)), x)
(y, ly) = ((ly - (q * y)), y)
if lx < 0:
lx += ob # If neg wrap modulo original b
if ly < 0:
ly += oa # If neg wrap modulo original a
return a, lx, ly # Return only positive values
def inverse(x: int, n: int) -> int:
"""Returns the inverse of x % n under multiplication, a.k.a x^-1 (mod n)
>>> inverse(7, 4)
3
>>> (inverse(143, 4) * 143) % 4
1
"""
(divider, inv, _) = extended_gcd(x, n)
if divider != 1:
raise NotRelativePrimeError(x, n, divider)
return inv
def crt(a_values: typing.Iterable[int], modulo_values: typing.Iterable[int]) -> int:
"""Chinese Remainder Theorem.
Calculates x such that x = a[i] (mod m[i]) for each i.
:param a_values: the a-values of the above equation
:param modulo_values: the m-values of the above equation
:returns: x such that x = a[i] (mod m[i]) for each i
>>> crt([2, 3], [3, 5])
8
>>> crt([2, 3, 2], [3, 5, 7])
23
>>> crt([2, 3, 0], [7, 11, 15])
135
"""
m = 1
x = 0
for modulo in modulo_values:
m *= modulo
for (m_i, a_i) in zip(modulo_values, a_values):
M_i = m // m_i
inv = inverse(M_i, m_i)
x = (x + a_i * M_i * inv) % m
return x
if __name__ == "__main__":
import doctest
doctest.testmod()
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