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#!/usr/bin/env python3
######## ENCODING and DECODING ########
FIELD_BITS = 32
FIELD_MODULUS = (1 << FIELD_BITS) + 0b10001101
def mul2(x):
"""Compute 2*x in GF(2^FIELD_BITS)"""
return (x << 1) ^ (FIELD_MODULUS if x.bit_length() >= FIELD_BITS else 0)
def mul(x, y):
"""Compute x*y in GF(2^FIELD_BITS)"""
ret = 0
for bit in [(x >> i) & 1 for i in range(x.bit_length())]:
ret ^= bit * y
y = mul2(y)
return ret
######## ENCODING only ########
def sketch(shortids, capacity):
"""Compute the bytes of a sketch for given shortids and given capacity."""
odd_sums = [0 for _ in range(capacity)]
for shortid in shortids:
squared = mul(shortid, shortid)
for i in range(capacity):
odd_sums[i] ^= shortid
shortid = mul(shortid, squared)
return b''.join(elem.to_bytes(4, 'little') for elem in odd_sums)
######## DECODING only ########
import random
def inv(x):
"""Compute 1/x in GF(2^FIELD_BITS)"""
t = x
for i in range(FIELD_BITS - 2):
t = mul(mul(t, t), x)
return mul(t, t)
def berlekamp_massey(s):
"""Given a sequence of LFSR outputs, find the coefficients of the LFSR."""
C, B, L, m, b = [1], [1], 0, 1, 1
for n in range(len(s)):
d = s[n]
for i in range(1, L + 1):
d ^= mul(C[i], s[n - i])
if d == 0:
m += 1
else:
T = list(C)
while len(C) <= len(B) + m:
C += [0]
t = mul(d, inv(b))
for i in range(len(B)):
C[i + m] ^= mul(t, B[i])
if 2 * L <= n:
L, B, b, m = n + 1 - L, T, d, 1
else:
m += 1
return C[0:L + 1]
def poly_monic(p):
"""Return the monic multiple of p, or 0 if the input is 0."""
if len(p) == 0:
return []
i = inv(p[-1])
return [mul(v, i) for v in p]
def poly_divmod(m, p):
"""Compute the polynomial quotient p/m, and replace p with p mod m."""
assert(len(m) > 0 and m[-1] == 1)
div = [0 for _ in range(len(p) - len(m) + 1)]
while len(p) >= len(m):
div[len(p) - len(m)] = p[-1]
for i in range(len(m)):
p[len(p) - len(m) + i] ^= mul(p[-1], m[i])
assert(p[-1] == 0)
p.pop()
while (len(p) > 0 and p[-1] == 0):
p.pop()
return div
def poly_gcd(a, b):
"""Compute the GCD of a and b (destroys the inputs)."""
if len(a) < len(b):
a, b = b, a
while len(b):
if len(b) == 1:
return [1]
b = poly_monic(b)
poly_divmod(b, a)
a, b = b, a
return a
def poly_sqr(p):
"""Compute the coefficients of the square of polynomial with coefficients p."""
return [0 if i & 1 else mul(p[i // 2], p[i // 2]) for i in range(2 * len(p))]
def poly_trace(m, a):
"""Compute the coefficients of the trace polynomial of (a*x) mod m."""
out = [0, a]
for i in range(FIELD_BITS - 1):
out = poly_sqr(out)
while len(out) < 2:
out += [0]
out[1] = a
poly_divmod(m, out)
return out
def find_roots_inner(p, a):
"""Recursive helper function for find_roots (destroys p). a is randomizer."""
# p must be monic
assert(len(p) > 0 and p[-1] == 1)
# Deal with degree 0 and degree 1 inputs
if len(p) == 1:
return []
elif len(p) == 2:
return [p[0]]
# Otherwise, split p in left*right using paramater a_vals[0].
t = poly_monic(poly_trace(p, a))
left = poly_gcd(list(p), t)
right = poly_divmod(list(left), p)
# Invoke recursion with the remaining a_vals.
ret_right = find_roots_inner(right, mul2(a))
ret_left = find_roots_inner(left, mul2(a))
# Concatenate roots
return ret_left + ret_right
def find_roots(p):
"""Find the roots of polynomial with coefficients p."""
# Compute x^(2^FIELD_BITS)+x mod p in a roundabout way.
t = poly_trace(p, 1)
t2 = poly_sqr(t)
for i in range(len(t)):
t2[i] ^= t[i]
poly_divmod(p, t2)
# If distinct from 0, p is not fully factorizable into non-repeating roots.
if len(t2):
return None
# Invoke the recursive splitting algorithm
return find_roots_inner(list(p), random.randrange(1, 2**32-1))
def decode(sketch):
"""Recover the shortids from a sketch."""
odd_sums = [int.from_bytes(sketch[i*4:(i+1)*4], 'little') for i in range(len(sketch) // 4)]
sums = []
for i in range(len(odd_sums) * 2):
if i & 1:
sums.append(mul(sums[(i-1)//2], sums[(i-1)//2]))
else:
sums.append(odd_sums[(i+1)//2])
return find_roots(list(reversed(berlekamp_massey(sums))))
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