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numTh.py
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numTh.py
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############################################################
# #
# File Name: numTh.py #
# #
# Author: Jyun-Neng Ji (jyunnengji@gmail.com) #
# #
# Creation Date: 2018/02/15 #
# #
# Last Modified: 2018/04/12 #
# #
# Description: Number theory library. #
# #
############################################################
import math
import random
import numpy as np
from sympy import isprime, nextprime
from sympy.ntheory.residue_ntheory import nthroot_mod
def findPrimes(prime_bit, N, num):
"""
Give prime bits and number of primes,
then generate primes, which congruence to 1 modular 2N.
Examples
========
>>> from numTh import findPrimes
>>> findPrimes(12, 4, 5)
[2081, 2089, 2113, 2129, 2137], 60
"""
primes = []
total_bits = 0
prime = pow(2, prime_bit - 1)
while len(primes) != num:
prime = nextprime(prime)
if prime % (2 * N) == 1:
primes.append(prime)
total_bits += prime.bit_length()
return primes, total_bits
def findPrimitiveNthRoot(M, N):
"""
Generate the smallest primitive Nth root of unity (expect 1).
Find w s.t w^N = 1 mod M and there are not other numbers k (k < N)
s.t w^k = 1 mod M
"""
roots = nthroot_mod(1, N, M, True)[1:] # find Nth root of unity
for root in roots: # find primitive Nth root of unity
is_primitive = True
for k in range(1, N):
if pow(root, k, M) == 1:
is_primitive = False
if is_primitive:
return root
return None
def isPrimitiveNthRoot(M, N, beta):
"""
verify B^N = 1 (mod M)
"""
return pow(beta, N, M) == 1 # modular(M).modExponent(beta, N) == 1
def uniform_sample(upper, num):
"""
Sample num values uniformly between [0,upper).
"""
sample = []
for i in range(num):
value = random.randint(0, upper - 1)
sample.append(value)
return sample
def gauss_sample(num, stdev):
"""
Sample num values from gaussian distribution
mean = 0, standard deviation = stdev
"""
sample = np.random.normal(0, stdev, num)
sample = sample.round().astype(int)
return sample
def hamming_sample(num, hwt):
"""
Sample a vector uniformly at random from -1, 0, +1,
subject to the condition that it has exactly hwt nonzero entries.
"""
i = 0
sample = [0] * num
while i < hwt:
degree = random.randint(0, num - 1)
if sample[degree] == 0:
coeff = random.randint(0, 1)
if coeff == 0:
coeff = -1
sample[degree] = coeff
i += 1
return sample
def small_sample(num):
"""
Sample vectors with entires -1, 0, +1.
Each element is 0 with probabilty 0.5 and +-1 with probabilty 0.25.
"""
sample = [0] * num
for i in range(num):
u = random.randint(0, 3)
if u == 3:
sample[i] = -1
if u == 2:
sample[i] = 1
return sample
# Not used
class modular:
def __init__(self, M):
self.mod = M
self.M_bit = M.bit_length()
self.u = (1 << (2 * self.M_bit)) // M
def modReduce(self, x):
"""
Barrett modular reduction algorithm.
Compute x mod M. M is initialized by modular class.
Examples
========
>>> from numTh import numTh
>>> modular(11).modReduce(12)
1
"""
assert 0 <= x < pow(self.mod, 2), 'out of range.'
q = (x * self.u) >> (2 * self.M_bit)
r = x - q * self.mod
while r >= self.mod:
r -= self.mod
return r
def modReducem(self, x, M):
"""
Barrett modular reduction algorithm.
Compute x mod M. M can be redefined.
Examples
========
>>> from numTh import numTh
>>> modular(5).modReducem(12, 11)
1
"""
tmp_mod, tmp_M_bit, tmp_u = self.mod, self.M_bit, self.u
self.mod = M
self.M_bit = M.bit_length()
self.u = (1 << (2 * self.M_bit)) // M
r = self.modReduce(x)
# return initial modular, bit size of modular and precompute u
self.mod, self.M_bit, self.u = tmp_mod, tmp_M_bit, tmp_u
return r
def modInv(self, x):
"""
Calculate modular inverse.
Examples
========
>>> from numTh import modular
>>> modular(5).modInv(3)
2
"""
t, new_t, r, new_r = 0, 1, self.mod, x
while new_r != 0:
q = r // new_r
r, new_r = new_r, (r % new_r)
t, new_t = new_t, (t - q * new_t)
assert r <= 1, 'x is not invertible'
return t if t > 0 else t + self.mod
# Slower than pow(base, power, modulus)
def modExponent(self, base, power):
"""
Modular exponentiation algorithm.
It's a fast method to compute a^b mod p
Examples
========
>>> from numTh import modular
>>> modular(2013265921).modExponent(1003203377, 2048)
1
"""
result = 1
power = int(power)
base = base % self.mod
while power > 0:
if power & 1:
# self.modReduce(result * base)
result = result * base % self.mod
base = base * base % self.mod # self.modReduce(base * base)
power = power >> 1
return result