**On generalized semi-bent (and partially bent) Boolean functions**

*Brajesh Kumar Singh*

**Abstract: **In this paper, we obtain a characterization of generalized Boolean functions based on spectral analysis. We investigate a relationship between the Walsh-Hadamard spectrum and $\sigma_f$, the sum-of-squares-modulus indicator (SSMI) of the generalized Boolean function. It is demonstrated that $\sigma_f = 2^{2n + s}$ for every $s$-plateaued generalized Boolean function in $n$ variables. Two classes of generalized semi-bent Boolean functions are constructed.% and it is demonstrated that their SSMI is over generalized $s$-plateaued Boolean functions is $2^{2n + s}$. We have constructed a class of generalized semi-bent functions in $(n+1)$ variables from generalized semi-bent functions in $n$ variables and identify a subclass of it for which $\sigma_f$ and $\triangle_{f}$ both have optimal value. Finally, some construction on generalized partially bent Boolean functions are given.

**Category / Keywords: **secret-key cryptography / Boolean functions, generalized functions; Walsh-Hadamard spectrum; generalized bent Boolean functions; generalized semi-bent functions; sum-of-squares-modulus indicator; modulus indicator

**Date: **received 21 Mar 2013

**Contact author: **bksingh0584 at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20130328:124336 (All versions of this report)

**Short URL: **ia.cr/2013/166

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