Computing cryptographic properties of Boolean functions from the algebraic normal orm representation

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2013

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Çalık, Çağdaş

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Boolean functions play an important role in the design and analysis of symmetric-key cryptosystems, as well as having applications in other fields such as coding theory. Boolean functions acting on large number of inputs introduces the problem of computing the cryptographic properties. Traditional methods of computing these properties involve transformations which require computation and memory resources exponential in the number of input variables. When the number of inputs is large, Boolean functions are usually defined by the algebraic normal form (ANF) representation. In this thesis, methods for computing the weight and nonlinearity of Boolean functions from the ANF representation are investigated. The relation between the ANF coefficients and the weight of a Boolean function was introduced by Carlet and Guillot. This expression allows the weight to be computed in $\mathcal{O}(2^p)$ operations for a Boolean function containing $p$ monomials in its ANF. In this work, a more efficient algorithm for computing the weight is proposed, which eliminates the unnecessary calculations in the weight expression. By generalizing the weight expression, a formulation of the distances to the set of linear functions is obtained. Using this formulation, the problem of computing the nonlinearity of a Boolean function from its ANF is reduced to an associated binary integer programming problem. This approach allows the computation of nonlinearity for Boolean functions with high number of input variables and consisting of small number of monomials in a reasonable time.

Subject Keywords

Coding theory, Algebra, Boolean, Binary system (Mathematics)., Integer programming., Cryptography.

URI

http://etd.lib.metu.edu.tr/upload/12615762/index.pdf
https://hdl.handle.net/11511/22563

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Graduate School of Applied Mathematics, Thesis

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Citation Formats

Ç. Çalık, “Computing cryptographic properties of Boolean functions from the algebraic normal orm representation,” Ph.D. - Doctoral Program, Middle East Technical University, 2013.