Let K be the field GF(3). We calculate the growth of the ideal Aλ where A is the algebra of functions from Kn → Kn and λ is a quadratic function. Specifically we calculate dim Akλ where Ak is the space of polynomials of degree less than or equal to k. This question arises in the analysis of the complexity of Gröbner basis attacks on multivariate quadratic cryptosystems such as the Hidden Field Equation systems. We also prove analogous results over the associated graded ring [Formula: see text] and state conjectures for the case of a general finite field of odd order.

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