Growth of the Ideal Generated by a Quadratic Multivariate Function Over Gf(3)

Abstract

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.

The degree of regularity of a quadratic polynomial

“…The result is proved here for q an odd prime. When q = 2 the same result is proved implicitly in [4]. There are a number of directions in which one might want to extend this result and we comment briefly now on each of these.…”

“…In addition, it was shown that non-trivial annihilators of λ occur only in degrees n/2 and n/2 + 1 and have dimension n/2 in both cases. Analogous results were proved for the case q = 3 in [4] and conjectures were made concerning the general case. We prove these conjectures in the case where q is an arbitrary odd prime.…”

“…We may now state our main theorem which proves the conjectures stated in [4] for the case when q is prime.…”

“…In [4] it was conjectured that Theorem 4.7 holds also in the case when q is a prime power, F is a finite field of order q and B = F[X 1 , . .…”

“…From this result one can deduce dim H(λ r , B k ) for a quadratic λ of arbitrary rank using the same approach as used in the case when q = 3 [4]. We will not go into these details here.…”