The degree of regularity of a quadratic polynomial

Let F be a finite field of odd characteristic q. We calculate the degree of regularity for a quadratic element of the algebra , the first degree at which non-trivial annihilation occurs. We also give an explicit formula for the dimension of the space of non-trivial annihilators of a given degree. These questions arise in multivariate cryptography, particularly in the study of the Hidden Field Equation cryptosystems.