A new measure to estimate pseudo-randomness of Boolean functions and relations with Gröbner bases

Abstract

In this short note we will introduce a generic measure of the algebraic complexity of vector valued Boolean functions: Normalized Average Number of Terms (NANT). NANT can be considered as a tool that extracts those vector valued Boolean functions that are suitable for effective application of Gröbner bases. As an example, we use NANT to show clear differences between two popular cryptographic hash functions: SHA-1 and SHA-2. The obtained results show that SHA-1 is susceptible to attacks based on Gröbner bases, which lead us to believe that SHA-1 is much weaker than SHA-2 from a design point of view. © 2009 Springer-Verlag Berlin Heidelberg.