I recently had some time to think about the following cryptography problem:

Given two functions, function f and function g find corresponding variables such that a collision between the functions occurs. Logically this means you need to find the correct arguments such that f equals g.

apply definitions of functions

step 1
xor both sides by k2

step 2
apply AES decryption to both sides with key k2

step 3

Now we have solved for message 2; the other values we can simply choose at random.

I've written up a short example in Python3 to show this in action (in case you doubt me!)

from Crypto.Cipher import AES
from Crypto.Util.strxor import strxor as xor
from Crypto import Random
from binascii import hexlify as hexify

# find a, b, x, and y such that
# AES(y, x) XOR y == AES(b, a) XOR b
# Note that for clarity, these names are not used
# each variable is referred to as a m (message) or a k(key)
# and AES takes arguments (key, message)
def collision():
  random = Random.new()
  k1 = random.read(16)
  m1 = random.read(16)
  k2 = random.read(16)

  key1 = AES.new(k1, AES.MODE_ECB)
  key2 = AES.new(k2, AES.MODE_ECB)

  m2 = key2.decrypt(xor(xor(key1.encrypt(m1), k1), k2))

  xor1 = xor(key1.encrypt(m1), k1)
  xor2 = xor(key2.encrypt(m2), k2)

  print("key 1: %s\nkey 2: %s" %(hexify(k1), hexify(k2)))
  print("msg 1: %s\nmsg 2: %s" %(hexify(m1), hexify(m2)))
  print("AES(key1, msg1) xor key1: %s\nAES(key2, msg2) xor key2: %s" %(hexify(xor1), hexify(xor2)))

if __name__ == '__main__':

For more information: Davies-Meyer.