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Paper: Minimum Distance between Bent and 1-resilient Boolean Functions

Authors:
Soumen Maity
Subhamoy Maitra
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URL: http://eprint.iacr.org/2003/133
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Abstract: In this paper we study the minimum distance between the set of bent functions and the set of 1-resilient Boolean functions and present a lower bound on that. The bound is proved to be tight for functions up to $10$ input variables. As a consequence, we present a strategy to modify the bent functions, by toggling some of its outputs, in getting a large class of $1$-resilient functions with very good nonlinearity and autocorrelation. In particular, the technique is applied upto $12$-variable functions and we show that the construction provides a large class of $1$-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation values which were not known earlier. The technique is sound enough to theoretically solve some of the mysteries of $8$-variable, $1$-resilient functions with maximum possible nonlinearity. However, the situation becomes complicated from $10$ variables and above, where we need to go for complicated combinatorial analysis with trial and error using computational facility.
BibTeX
@misc{eprint-2003-11848,
  title={Minimum Distance between Bent and 1-resilient Boolean Functions},
  booktitle={IACR Eprint archive},
  keywords={secret-key cryptography / Boolean Functions},
  url={http://eprint.iacr.org/2003/133},
  note={Accepted in FSE 2004 subho@isical.ac.in 12441 received 12 Jul 2003, last revised 24 Jan 2004},
  author={Soumen Maity and Subhamoy Maitra},
  year=2003
}