On the Algebraic Immunity of Symmetric Boolean Functions

Abstract

In this paper, we analyze the algebraic immunity of symmetric Boolean functions. The algebraic immunity is a property which measures the resistance against the algebraic attacks on symmetric ciphers. We identify a set of lowest degree annihilators for symmetric functions and propose an efficient algorithm for computing the algebraic immunity of a symmetric function. The existence of several symmetric functions with maximum algebraic immunity is proven. In this way, we have found a new class of functions which have good implementation properties and maximum algebraic immunity.

An Infinite Class of Balanced Functions with Optimal Algebraic Immunity, Good Immunity to Fast Algebraic Attacks and Good Nonlinearity

“…Moreover, they are not balanced (but it is possible to build balanced functions from these ones) and are weak against fast algebraic attacks [2,18]. The second class contains symmetric functions (whose values depend only on the Hamming weight of the input vectors) [3,18] or functions whose values depend on the Hamming weight of the input vectors except for a few inputs [7]. The nonlinearities of these functions are often not exceeding 2…”

A Proof of Constructions for Balanced Boolean Function with Optimum Algebraic Immunity

“…However, further study [5] showed that the functions are not balanced. Another class of constructions [7][8] contains symmetric functions. Being symmetric, they present a risk if attacks using this peculiarity can be found in the future.…”

Further properties of several classes of Boolean functions with optimum algebraic immunity

“…Moreover, they are not balanced. The second way is based on modifying symmetric functions [13,2]. Speaking concretely, up to affine equivalence, the obtained functions of n-variable are symmetric on the set consisting of all elements with weight not equal to are not exceeding 2 n−1 − n−1 n 2 .…”