Associative noncommutative hash function

Question

Is there a hash function with following properties?

• is associative
• is not commutative
• easily implementable on 32 bit integers: int32 hash(int32, int32)

If I am correct, such function allows achieving following goals

• calculate hash of concatenated string from hashes of substrings
• calculate hash concurrently
• calculate hash of list implemented on binary tree - including order, but excluding how tree is balanced

The best I found so far is multiplication of 4x4 matrix of bits, but thats awkward to implement and reduces space to 16bits.

I am grateful for any help.

Answer 1

Polynomial rolling hash could help:

• H(A1,...,An) = (H(A1,...,An-1) * Base + An) Mod P

It's easy to concat two results or substract prefix/suffix from result, as long as the length is known.

Answer 2

Matrix multiplication is associative and non-commutative.

You could try representing your hashes as matrices but this will result in a loss of information if they have 0 determinant (which is likely!).

So instead you should generate a triangle matrix with a diagonal of 1's to ensure that you have a determinant of 1 (this guarantees that composition does not loose information).

Furthermore the composition of triangle matrices produces a new triangle matrix, making reading the composition the same as generation.

Note: to use this method the length of your hash must be a triangle number!