Parameter determination for post-quantum hash functions

In the design of the hash function one has control over two parameters; the dimension of the special linear group considered, and the characteristic of the underlying field. We would like to determine appropriate choices for both via two branches of cryptanalysis: first, by analysis of the multivariate system implied by long matrix products; and second by direct analysis of group-theoretic properties. We should also study the impact of the partial homomorphicity exhibited by the scheme.