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Paper: There exist Boolean functions on $n$ (odd) variables having nonlinearity $> 2^{n-1} - 2^{\frac{n-1}{2}}$ if and only if $n > 7$

Authors:
Selçuk Kavut
Melek D. Yücel
Subhamoy Maitra
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URL: http://eprint.iacr.org/2006/181
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Abstract: For the first time we find Boolean functions on 9 variables having nonlinearity 241, that remained as an open question in literature for almost three decades. Such functions are discovered using a suitably modified steepest-descent based iterative heuristic search in the class of rotation symmetric Boolean functions (RSBFs). This shows that there exist Boolean functions on $n$ (odd) variables having nonlinearity $> 2^{n-1} - 2^{\frac{n-1}{2}}$ if and only if $n > 7$. Using the same search method, we also find several other important functions and we study the autocorrelation, propagation characteristics and resiliency of the RSBFs (using proper affine transformations, if required). The results show that it is possible to get balanced Boolean functions on $n=10$ variables having autocorrelation spectra with maximum absolute value $< 2^{\frac{n}{2}}$, which was not known earlier. In certain cases the functions can be affinely transformed to get first order propagation characteristics. We also obtain 10-variable functions having first order resiliency and nonlinearity 492 which was posed as an open question in Crypto 2000.
BibTeX
@misc{eprint-2006-21674,
  title={There exist Boolean functions on $n$ (odd) variables having nonlinearity $> 2^{n-1} - 2^{\frac{n-1}{2}}$ if and only if $n > 7$},
  booktitle={IACR Eprint archive},
  keywords={foundations / Boolean Functions, Nonlinearity},
  url={http://eprint.iacr.org/2006/181},
  note={ subho@isical.ac.in 13348 received 28 May 2006, last revised 19 Jul 2006},
  author={Selçuk Kavut and Melek D. Yücel and Subhamoy Maitra},
  year=2006
}