Sequence Design and Construction of Cryptographic Boolean Functions

In this thesis, we study the application of well-known sequences from communications theory, in the construction of cryptographic Boolean functions. First, we explain the basic connection between binary sequences, polynomials over the finite field GF (2), and Boolean functions. Second, we classify highly nonlinear quadratic polynomials which may have useful applications in the design and cryptanalysis of finite field based crypto-systems. Third, we construct Boolean functions to be used in stream cipher systems. They have good cryptographic properties like balance, resiliency, high nonlinearity and low additive auto-correlation for protection against various statistical attacks. Fourth, we explore using an S-box instead of a Boolean function in stream cipher systems for higher communication speed. However in that case, we require the S-box to have an additional requirement: low maximum correlation. We construct S-boxes with good cryptographic properties and having low maximum correlation which improves on currently known bounds. These Boolean function and S-box constructions are based on the m-sequences, GMW sequences and ideal 2-level auto-correlation sequences from communications theory. Finally, we consider efficient methods to compute the Hadamard transform of polynomials, which is useful in the design of Boolean functions from sequences. We also present some experimental results to demonstrate the application of these methods.

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