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

*Enes Pasalic and Thomas Johansson and Subhamoy Maitra and Palash Sarkar*

**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.

**Category / Keywords: **secret-key cryptography / boolean function

**Date: **received 26 Sep 2000

**Contact author: **subho at isical ac in

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | BibTeX Citation

**Version: **20000926:205428 (All versions of this report)

**Short URL: **ia.cr/2000/048

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