© 2015 by De Gruyter 2015.T. Harayama and D. K. Friesen  proposed the linearized binomial attack for multivariate quadratic cryptosystems and introduced weak Dembowski-Ostrom (DO) polynomials in this framework over the finite field F2. We extend the linearized binomial attack to multivariate quadratic cryptosystems over Fp for any prime p and redefine the weak DO polynomials for general case. We identify infinite classes of weak DO polynomials for these systems by considering highly degenerate quadratic forms over algebraic function fields and Artin-Schreier type curves to achieve our results. This gives a general answer to the conjecture stated by Harayama and Friesen and also a partial enumeration of weak DO polynomials over finite fields.