Constructing balanced functions with optimum algebraic immunity

Because of the 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, valid for any even number of variables, for constructing functions with optimum (or, if this can be useful, with high but not optimal) algebraic immunity and which can be balanced if we wish. We also give a new example of an infinite class of such functions. We study their Walsh transforms.