Correlation immunity, avalanche features, and other cryptographic properties of generalized Boolean functions
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This dissertation investigates correlation immunity, avalanche features, and the bent cryptographic properties for generalized Boolean functions defined on Vn with values in Zԛ. We extend the concept of correlation immunity from the Boolean case to the generalized setting, and provide multiple construction methods for order 1 and higher correlation immune generalized Boolean functions. We establish necessary and sufficient conditions for generalized Boolean functions. Additionally, we discuss correlation immune and rotation symmetric generalized Boolean functions, introducing a construction method along the way. Using a graph-theoretic and probabilistic frame of reference, we subsequently establish several, increasingly stringent, strict avalanche criteria along with a con-struction method for generalized Boolean functions. We introduce the notion of a uniform avalanche criterion and demonstrate that generalized Boolean functions that satisfy this criterion are also order 1 correlation immune and always have Boolean function compo-nents that are both order 1 correlation immune and satisfy the strict avalanche criterion.We subsequently investigate linear structures, directional derivatives and define a unit vector gradient for generalized Boolean function. We introduce the Walsh-Hadamard trans-form of a generalized Boolean function along with the notion of generalized bent Boolean functions. We provide a construction of generalized bent Boolean functions with outputs in Z8 and establish necessary conditions for generalized bent Boolean functions.
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