International Association for Cryptologic Research

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for Cryptologic Research

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Paper: New Constructions of Resilent and Correlation Immune Boolean Functions achieving Upper Bounds on Nonlinearity

Authors:
Enes Pasalic
Thomas Johansson
Subhamoy Maitra
Palash Sarkar
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URL: http://eprint.iacr.org/2000/048
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Abstract: Recently weight divisibility results on resilient and correlation immune Boolean functions have received a lot of attention. These results have direct consequences towards the upper bound on nonlinearity of resilient and correlation immune Boolean functions of certain order. Now the clear benchmark in the design of resilient Boolean functions (which optimizes Sigenthaler's inequality) is to provide results which attain the upper bound on nonlinearity. Here we construct a 7-variable, 2-resilient Boolean function with nonlinearity 56. This solves the maximum nonlinearity issue for 7-variable functions with any order of resiliency. Using this 7-variable function, we also construct a 10-variable, 4-resilient Boolean function with nonlinearity 480. Construction of these two functions were justified as important open questions in Crypto 2000. Also we provide methods to generate an infinite sequence of Boolean functions on $n = 7 + 3i$ variables $(i \geq 0)$ with order of resiliency $m = 2 + 2i$, algebraic degree $4 + i$ and nonlinearity $2^{n-1} - 2^{m+1}$, which were not known earlier. We conclude with a few interesting construction results on unbalanced correlation immune functions of 5 and 6 variables.
BibTeX
@misc{eprint-2000-11392,
  title={New Constructions of Resilent and Correlation Immune Boolean Functions achieving Upper Bounds on Nonlinearity},
  booktitle={IACR Eprint archive},
  keywords={secret-key cryptography / boolean function},
  url={http://eprint.iacr.org/2000/048},
  note={ subho@isical.ac.in 11226 received 26 Sep 2000},
  author={Enes Pasalic and Thomas Johansson and Subhamoy Maitra and Palash Sarkar},
  year=2000
}