This is a follow-up to a prior question Does matrix multiplication of hash digests admit manipulation of the result?; this formulation failed because it admitted singular matrices and therefore degenerates to the zero matrix after enough elements are multiplied. The answerer suggested using a field like $GF(256)$ instead of a ring and rejecting singular matrices, which is what this question explores.
This is cross-posted from the Mathematics SE.
Consider an ordered sequence of elements $a_n$, a function $h$ that derives an invertible matrix over finite field ₂₅₆ from a single element's cryptographic hash, and a function $H$ that finds the product of all such matrices from a sequence:
$H(a) = \prod_{i = 1}^{n} h(a_i)$
Define $H(a)$ to be the hash of the sequence $a_n$.
Note that due to associativity, given two sequences $a_n$, $b_m$, then $H(a)*H(b) = H(a ⧺ b)$ (where $⧺$ means concatenate the sequences).
Assuming that we can trust the invertible matrix derivation function has the properties of a cryptographic hash function, Is there an algorithm better than brute force that can find two different sequences that have the same hash?
I coded up an example of this definition in a Julia notebook that I published here.
