Efficient Computation of Algebraic Immunity for Algebraic and Fast Algebraic Attacks

Abstract

Abstract. In this paper we propose several efficient algorithms for assessing the resistance of Boolean functions against algebraic and fast algebraic attacks when implemented in LFSR-based stream ciphers. An algorithm is described which permits to compute the algebraic immu-operations necessary in all previous algorithms. Our algorithm is based on multivariate polynomial interpolation. For assessing the vulnerability of arbitrary Boolean functions with respect to fast algebraic attacks, an efficient generic algorithm is presented that is not based on interpolation. This algorithm is demonstrated to be particularly efficient for symmetric Boolean functions. As an application it is shown that large classes of symmetric functions are very vulnerable to fast algebraic attacks despite their proven resistance against conventional algebraic attacks.

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

“…The research of Boolean functions that can resist algebraic attacks, the Berlekamp-Massey attack and the fast correlation attacks has not given fully satisfactory results: we know that functions achieving optimal or suboptimal algebraic immunity and in the same time balancedness, high algebraic degree and high nonlinearity must exist thanks to the results of [19,37]. Such functions have been found with sufficient numbers of variables thanks to Algorithm 1 of [2] (others can be found by using the algorithm of [20]). But the functions given in [2] belong to classes which have not, potentially, a good asymptotic algebraic immunity (see [35]), and there remains to see whether these functions behave well against fast algebraic attacks.…”

“…Such functions have been found with sufficient numbers of variables thanks to Algorithm 1 of [2] (others can be found by using the algorithm of [20]). But the functions given in [2] belong to classes which have not, potentially, a good asymptotic algebraic immunity (see [35]), and there remains to see whether these functions behave well against fast algebraic attacks. No infinite class of functions with good algebraic immunity and good nonlinearity has been exhibited so far.…”

“…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].…”

On the Higher Order Nonlinearities of Algebraic Immune Functions

“…Note that a high algebraic immunity does not necessarily prevent the system from fast algebraic attacks, see e.g. [2], but these attacks were not the subject of the present paper. The requirements concerning the Boolean functions used in symmetric ciphers are going towards a simplification.…”

“…These functions have some drawbacks: all of them have insufficient nonlinearities and all but one are non-balanced. Moreover, the functions studied in [19,3], and to a slightly smaller extent the functions introduced in [18], have not a good behavior against fast algebraic attacks, see [2,20]. But the research in this domain is very active and it is probable that better examples of functions will be found in the future.…”