Generating cryptographically strong Boolean functions using partial information

Boolean functions play an important role in cryptography. They are elementary building blocks for various cryptographic algorithms – stream ciphers, block ciphers, hash functions, etc. The most common usage for Boolean functions is the construction of larger blocks – substitution boxes [4, 5, 6]. Boolean functions used in these constructions ought to satisfy certain criteria in order to resist various attacks. Mutual fulfilment of several criteria makes finding cryptographically strong Boolean functions an interesting problem. There are methods (besides others, more “analytical” ones) that try to employ good cryptographic properties of random Boolean functions and improve them further [1, 2, 3]. Our paper is the first attempt to use partially defined Boolean functions (PDBFs) for generating cryptographically strong Boolean functions. A PDBF can be considered as a Boolean function with some undefined (unknown) values, i.e. its values are from the set {0, 1, ?}. We generalize certain cryptographic properties to PDBFs, such as balancedness, nonlinearity and propagation characteristics. It is also shown that usual relationships among properties hold for these generalizations as well. We apply the results in methods for generating strong Boolean functions. The methods (algorithms) form the central part of the paper. We focus on greedy approach and test it in various settings. The paper compares obtained results with other methods. Conclusion outlines few ideas for subsequent research. We believe that this alternative approach can be refined further. PDBFs can be an attractive model for constructing Boolean functions in cryptography.