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Correlation Immunity, Avalanche Features, and Other Cryptographic Properties of Generalized Boolean Functions

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Naval Postgraduate School Monterey United States

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This dissertation investigates correlation immunity, avalanche features, and the bent cryptographic properties for generalized Boolean functions defined on V sub n with values in Z sub q. 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 construction 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 components 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 transform 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 Z sub 8 and establish necessary conditions for generalized bent Boolean functions.

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Technical Report



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Approved For Public Release;

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