Formal verification techniques based on computer algebra have proven highly effective for circuit verification. The circuit, given as an and-inverter graph, is encoded using polynomials that automatically generate a Gröbner basis with respect to a lexicographic term ordering. Correctness of the circuit is derived by computing the polynomial remainder of the specification. However, the main obstacle is the monomial blow-up during the reduction, as the degree can increase. In this paper, we investigate an orthogonal approach and focus the computational effort on rewriting the Gröbner basis itself. Our goal is to ensure the basis contains linear polynomials that can be effectively used to rewrite the linearized specification. We first prove the soundness and completeness of this technique and then demonstrate its practical application. Our implementation of this method shows promising results on benchmarks related to multiplier verification.