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Paper: Covering Radius of the $(n-3)$-rd Order Reed-Muller Code in the Set of Resilient Functions

Authors:
Yuri Borissov
An Braeken
Svetla Nikova
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URL: http://eprint.iacr.org/2004/202
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Abstract: In this paper, we continue the study of the covering radius in the set of resilient functions, which has been defined by Kurosawa. This new concept is meaningful to cryptography especially in the context of the new class of algebraic attacks on stream ciphers proposed by Courtois and Meier at Eurocrypt 2003 and Courtois at Crypto 2003. In order to resist such attacks the combining Boolean function should be at high distance from lower degree functions. Using a result from coding theory on the covering radius of $(n-3)$-rd Reed-Muller codes, we establish exact values of the the covering radius of $RM(n-3,n)$ in the set of $1$-resilient Boolean functions of $n$ variables, when $\lfloor n/2 \rfloor = 1 \mod\;2$. We also improve the lower bounds for covering radius of the Reed-Muller codes $RM(r,n)$ in the set of $t$-resilient functions, where $\lceil r/2 \rceil = 0 \mod\;2$, $t \leq n-r-2$ and $n\geq r+3$.
BibTeX
@misc{eprint-2004-12174,
  title={Covering Radius of the $(n-3)$-rd Order Reed-Muller Code in the Set of Resilient Functions},
  booktitle={IACR Eprint archive},
  keywords={covering radius, resilient functions},
  url={http://eprint.iacr.org/2004/202},
  note={published at the the 25th Symposium on Information Theory in the Benelux svetla.nikova@esat.kuleuven.ac.be 12647 received 17 Aug 2004},
  author={Yuri Borissov and An Braeken and Svetla Nikova},
  year=2004
}