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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.
Senior Honors Thesis. Advisor: Amit Chakrabarti.
Bibliographic citation for this report: [plain text] [BIB] [BibTeX] [Refer]
Or copy and paste:
Matthew J. Harding, "Polynomial and Query Complexity of Minterm-Cyclic Functions." Dartmouth Computer Science Technical Report TR2013-740, May 2013.
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