Network-driven Boolean Normal Forms

Abstract

We apply the PolyBoRi framework for Groebner 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.