Chinese Remainder Theorem Applications
Solve systems of congruences
Formula & Implementation
Formula
x ≡ aᵢ (mod nᵢ) for i = 1, 2, ..., k
Find
x
that satisfies allcongruences
Implementation
def chinese_remainder_theorem(remainders, moduli):
"""
Solve x ≡ rᵢ (mod mᵢ) for all i
Returns: x that satisfies all congruences
"""
total = 0
product = 1
for modulus in moduli:
product *= modulus
for remainder, modulus in zip(remainders, moduli):
p = product // modulus
total += remainder * p * pow(p, -1, modulus)
return total % product
Hash Function Collision Attacks
POC
def hash_collision_via_crt(hash_function, target_bits):
"""
Use CRT to construct messages with specific hash properties
Applies to MD5, SHA-1, custom hash functions
"""
# Construct messages that collide on specific bit positions
# Using CRT to combine constraints
pass
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