The relation between the Walsh-Hadamard transform and the auto-correlation function of Boolean functions is used to study propagation characteristics of these functions. The Strict Avalanche Criterion and the Perfect Nonlinearity Criterion are generalized in a Propagation Criterion of degree k. New properties and constructions for Boolean bent functions are given and also the extension of the definition to odd values of n is discussed. New properties of functions satisfying higher order SAC are derived. Finally a general framework is established to classify functions according to their propagation characteristics if a number of bits is kept constant.

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Propagation characteristics of Boolean functions

Two general classes of binary bent sequences, bent-based and linear-based, are introduced. Algorithms that allow easy generation of bent sequences from either class are given. Based on some simple computation and a computer search, the authors conjecture a lower bound on the total number of binary bent sequences of a given order. This lower bound is exact for bent sequences of order 16; a list is included from which all such sequences can be derived. >

Generating and counting binary bent sequences

In this paper, the eigenvalues and eigenvectors of the generalized discrete Fourier transform (GDFT), the generalized discrete Hartley transform (GDHT), the type-IV discrete cosine transform (DCT-IV), and the type-IV discrete sine transform (DST-IV) matrices are investigated in a unified framework. First, the eigenvalues and their multiplicities of the GDFT matrix are determined, and the theory of commuting matrices is applied to find the real, symmetric, orthogonal eigenvectors set that constitutes the discrete counterpart of Hermite Gaussian function. Then, the results of the GDFT matrix and the relationships among these four unitary transforms are used to find the eigenproperties of the GDHT, DCT-IV, and DST-IV matrices. Finally, the fractional versions of these four transforms are defined, and an image watermarking scheme is proposed to demonstrate the effectiveness of fractional transforms.

Eigenvalues and eigenvectors of generalized DFT, generalized DHT, DCT-IV and DST-IV matrices

This paper presents an analysis and synthesis of bent sequences in the Hadamard domain. It is shown that the basic bent sequence of length 4k is threshold realizable using 3k nonzero weights and the realisation of arbitrary bent sequences is also discussed.

Analysis and synthesis of bent sequences

Linear functions are a source of weakness in a cryptographic algorithm because such functions are easy to invert and have very predictable outputs; hence the essential cryptographic property of randomness is made more difficult to achieve if any part of the algorithm is linear, or close to linear. Thus it is desirable for the functions used in the algorithm to have high nonlinearity. Bent Boolean functions, which are only defined for an even number n of variables, are functions which have the maximum Hamming distance from the set of all linear functions, and so have the maximum possible nonlinearity. This chapter gives a full discussion of all aspects of the theory of bent functions, including equivalent definitions of bent, ways to construct bent functions, properties of bent functions and counts of bent functions as n varies. The chapter concludes with discussions of the related partially bent functions and semi-bent functions.

R. Yarlagadda

Papers

A note on the eigenvectors of Hadamard matrices of order 2n

Stabilization of matrices