**A method of construction of balanced functions with optimum algebraic immunity**

*C. Carlet*

**Abstract: **Because of the recent algebraic attacks, a high algebraic immunity is now an absolutely necessary (but not sufficient) property for Boolean functions used in stream ciphers. A difference of only 1 between the algebraic immunities of two functions can make a crucial difference with respect to algebraic attacks. Very few examples of (balanced) functions with high algebraic immunity have been found so far. These examples seem to be isolated and no method for obtaining such functions is known. In this paper, we introduce a general method for proving that a given function, in any number of variables, has a prescribed algebraic immunity. We deduce an algorithm for generating balanced functions in any odd number of variables, with optimum algebraic immunity. We also give an algorithm, valid for any even number of variables, for constructing (possibly) balanced functions with optimum (or, if this can be useful, with high but not optimal) algebraic immunity. We also give a new example of an infinite class of such functions. We study their Walsh transforms. To this aim, we completely characterize the Walsh transform of the majority function.

**Category / Keywords: **secret-key cryptography / Boolean functions

**Date: **received 16 Apr 2006, last revised 8 Jun 2007

**Contact author: **claude carlet at inria fr

**Available format(s): **PDF | BibTeX Citation

**Note: **We completed the paper with a method of construction of balanced functions with optimum algebraic immunity in odd numbers of variables. This method is very general and will most probably allow obtaining many functions also achieving good nonlinearity.

**Version: **20070608:170742 (All versions of this report)

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