A method of construction of balanced functions with optimum algebraic immunity

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

New construction of Boolean functions with maximun algebraic immunity

Because of the algebraic attacks, a high algebraic immunity is now an important criteria for Boolean functions used in stream ciphers. In this paper, by using the relationship between some flats and support of a n variables Boolean function f, we introduce a general method to determine the algebraic immunity of a Boolean function and finally construct some balanced functions with optimum algebraic immunity