wagner

Python implementation of Wagner's Algorithm for the Generalized Birthday Problem.

This algorithm is used to solve what is known as the generalized birthday problem. Given a modulus $n$, and $k$ lists of random numbers $\{L_1, L_2, ..., L_k\}$, how can we find $k$ elements $\{x_1, x_2, ..., x_k\} : x_i \in L_i$ from those lists, such that they all sum to some constant $c$ mod $n$?

Check out my full-length article on the subject for more detailed info. This repository is meant as a demonstration for practically minded and inquisitive readers.

Usage

The primary export of this library is the solve method.

>>> import wagner
>>> wagner.solve(2**16)
[50320, 16960, 11687, 52082, 17220, 47751, 11228, 54896]
>>> sum(_) % (2**16)
0

This method solves the generalized birthday problem for a given modulus $n$. The higher $n$ is, the more difficult it is to find a solution and the longer the algorithm will take.

At no cost, the caller can also choose a desired sum other than zero.

n = 2 ** 16
sum(wagner.solve(n, 885)) % n # -> 885

To change the number of elements returned by solve, specify the height $H$ of the tree used to solve the problem. The number of elements in the solution will be $2^H$.

len(wagner.solve(n, tree_height=2)) # -> 4
len(wagner.solve(n, tree_height=3)) # -> 8
len(wagner.solve(n, tree_height=4)) # -> 16
len(wagner.solve(n, tree_height=5)) # -> 32

To specify how the random elements are generated, provide a generator callback. By default, wagner uses random.randrange(n) to generate random values. A common use case for Wagner's Algorithm is to find inputs whose hashes sum to some desired number. To ensure solve returns the preimages and not the hash outputs, return Lineage instances from your generator callback. This class holds is basically an integer with pointers to the element(s) which created it.

import random
import hashlib
import wagner


def hashfunc(r, n, index):
  r_bytes = r.to_bytes((int.bit_length(n) + 7) // 8, 'big')
  preimage = r_bytes + index.to_bytes(16, 'big')
  h = hashlib.sha1(preimage).digest()
  return int.from_bytes(h, 'big') % n


def generator(n, index):
  r = random.randrange(0, n)
  return wagner.Lineage(hashfunc(r, n, index), r)


if __name__ == "__main__":
  n = 2 ** 128
  preimages = wagner.solve(n, generator=generator)
  print(sum(hashfunc(r, n, index) for index, r in enumerate(preimages)) % n) # -> 0