We explore the idea of building a secure hash using quadratic or higher degree multivariate polynomials over a finite field as the Merkle-Damg aard compression function. We analyze the security properties and potential feasibility, there the compression functions are randomly chosen quadratic polynomials. Next, we propose to improve on the efficiency of the system by using some specially designed quadrctic polynomials with certain sparsity property, and the security of the system relies on stronger assumptions.