Polynomial and Query Complexity of Minterm-Cyclic Functions

Boolean functions are at the heart of all computations, and all Boolean functions can be reduced to a sum of pattern-matching functions, called minterm-cyclic functions. In this thesis, we examine properties of polynomials representing minterm-cyclic Boolean functions. We use the term "saturated" to represent a polynomial with degree equal to input size n for all n; this indicates the intuitive notion that such functions are in some way complex, or difficult to compute. We present three main results. Firstly, there exist an infinite number of monotone minterm-cyclic functions that are not saturated. Secondly, for a specific class of minterms called self-avoiding minterms, we prove that the associated pattern-matching functions are not saturated; specifically, they can only have non-zero degree-n coefficients for n a multiple of the size of the minterm. Thirdly, for self-avoiding minterms that contain some '*', the degree-n coefficients are always zero. These results may have implications in the fields of algebraic cryptographic attacks or efficiency of error-correcting codes.