Network-driven Boolean Normal Forms

We apply the PolyBoRi framework for Gröbner bases computations with Boolean polynomials to bit-valued problems from algebraic cryptanalysis and formal verification. First, we proposed zero-suppressed binary decision diagrams (ZDDs) as a suitable data structure for Boolean polynomials. Utilizing the advantages of ZDDs we develop new reduced normal form algorithms for linear lexicographical lead rewriting systems. The latter play an important role in modeling bit-valued components of digital systems. Next, we reorder the variables in Boolean polynomial rings with respect to the topology of digital components. This brings computational algebra to digital circuits and small scale crypto systems in the first place. We additionally propose an optimized topological ordering, which tends to keep the intermediate results small. Thus, we successfully applied the linear lexicographical lead techniques to non-trivial examples from formal verification of digital systems. Finally, we evaluate the performance using benchmark examples from formal verification and cryptanalysis including equivalence checking of a bit-level formulation of multiplier components. Before we introduced topological orderings in PolyBoRi, state of the art for the algebraic approach was a bit-width of 4 for each factor. By combining our techniques we raised this bound to 16, which is an important step towards real-world applications.