Recent attacks on standardised hash functions such as SHA1 have reawakened interest in design strategies based on techniques common in provable security. In presenting the VSH hash function, a design based on RSA-like modular exponentiation, the authors introduce VSH-DL, a design based on exponentiation in DLP-based groups. In this article we explore a variant of VSH-DL that is based on cyclotomic subgroups of finite fields; we show that one can trade-off performance against bandwidth by using known techniques in such groups. Further, we investigate a variant of VSH-DL based on elliptic curves and extract a tighter reduction to the underlying DLP in comparison to the original VSH-DL proposal