The "birthday paradox" places an upper bound on collision resistance: if a hash function produces N bits of output, an attacker who computes only 2N/2 (or
) hash operations on random input is likely to find two matching outputs.
If there is an easier method to do this than brute-force attack, it is typically considered a flaw in the hash function.
[3][4] However, some hash functions have a proof that finding collisions is at least as difficult as some hard mathematical problem (such as integer factorization or discrete logarithm).
[2] A family of functions {hk : {0, 1}m(k) → {0, 1}l(k)} generated by some algorithm G is a family of collision-resistant hash functions, if |m(k)| > |l(k)| for any k, i.e., hk compresses the input string, and every hk can be computed within polynomial time given k, but for any probabilistic polynomial algorithm A, we have where negl(·) denotes some negligible function, and n is the security parameter.