# Collision resistance

Collision resistance is a property of cryptographic hash functions: a hash function H is collision resistant if it is hard to find two inputs that hash to the same output; that is, two inputs a and b such that H(a) = H(b), and ab.:136

Every hash function with more inputs than outputs will necessarily have collisions.:136 Consider a hash function such as SHA-256 that produces 256 bits of output from a large input (≤ 264-1 bits). Since it must generate one of 2256 outputs for each member of a much larger set of inputs, the pigeonhole principle guarantees that some inputs will hash to the same output. Collision resistance does not mean that no collisions exist; simply that they are hard to find.:143

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 ${\sqrt {2^{N}}}$ ) hash operations on random input is likely to find two matching outputs. If there is an easier method than this brute-force attack, it is typically considered a flaw in the hash function.

Cryptographic hash functions are usually designed to be collision resistant. But many hash functions that were once thought to be collision resistant were later broken. MD5 and SHA-1 in particular both have published techniques more efficient than brute force for finding collisions. 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). Those functions are called provably secure.

## Definition

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

Pr [kG(1n), (x1, x2) ← A(k, 1n) s.t. x1x2 but hk(x1) = hk(x2)] < negl(n),

where negl(·) denotes some negligible function, and n is the security parameter.

## Rationale

Collision resistance is desirable for several reasons.

• In some digital signature systems, a party attests to a document by publishing a public key signature on a hash of the document. If it is possible to produce two documents with the same hash, an attacker could get a party to attest to one, and then claim that the party had attested to the other.
• In some proof-of-work systems, users provide hash collisions as proof that they have performed a certain amount of computation to find them. If there is an easier way to find collisions than brute force, users can cheat the system.
• In some distributed content systems, parties compare cryptographic hashes of files in order to make sure they have the same version. An attacker who could produce two files with the same hash could trick users into believing they had the same version of a file when they in fact did not.