Divisibility properties on boolean functions using the numerical normal form

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2004
Göloğlu, Faruk
A Boolean function can be represented in several different forms. These different representation have advantages and disadvantages of their own. The Algebraic Normal Form, truth table, and Walsh spectrum representations are widely studied in literature. In 1999, Claude Carlet and Phillippe Guillot introduced the Numerical Normal Form. NumericalNormal Form(NNF) of a Boolean function is similar to Algebraic Normal Form, with integer coefficients instead of coefficients from the two element field. Using NNF representation, just like the Walsh spectrum, characterization of several cryptographically important functions, such as resilient and bent functions, is possible. In 2002, Carlet had shown several divisibility results concerning resilient and correlation-immune functions using NNF. With these divisibility results, Carlet is able to give bounds concerning nonlinearity of resilient and correlation immune functions. In this thesis, following Carlet and Guillot, we introduce the Numerical Normal Form and derive the pairwise relations between the mentioned representations. Characterization of Boolean, resilient and bent functions using NNF is also given. We then review the divisibility results of Carlet, which will be linked to some results on the nonlinearity of resilient and correlation immune functions. We show the Möbius inversion properties of NNF of a Boolean function, using Gian-Carlo Rota̕s work as a guide. Finally, using a lot of the mentioned results, we prove a necessary condition on theWalsh spectrum of Boolean functions with given degree.

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Citation Formats
F. Göloğlu, “Divisibility properties on boolean functions using the numerical normal form,” M.S. - Master of Science, Middle East Technical University, 2004.