Arnau Gàmez-Montolio (City, University of London; Activision Research), Enric Florit (Universitat de Barcelona), Martin Brain (City, University of London), Jacob M. Howe (City, University of London)

Polynomials over fixed-width binary numbers (bytes, Z/2 wZ, bit-vectors, etc.) appear widely in computer science including obfuscation and reverse engineering, program analysis, automated theorem proving, verification, errorcorrecting codes and cryptography. As some fixed-width binary numbers do not have reciprocals, these polynomials behave differently to those normally studied in mathematics. In particular, polynomial equality is harder to determine; polynomials having different coefficients is not sufficient to show they always compute different values. Determining polynomial equality is a fundamental building block for most symbolic algorithms. For larger widths or multivariate polynomials, checking all inputs is computationally infeasible. This paper presents a study of the mathematical structure of null polynomials (those that evaluate to 0 for all inputs) and uses this to develop efficient algorithms to reduce polynomials to a normalized form. Polynomials in such normalized form are equal if and only if their coefficients are equal. This is a key building block for more mathematically sophisticated approaches to a wide range of fundamental problems.