Sequences, DFT and Resistance against Fast Algebraic Attacks

Abstract

The discrete Fourier transform (DFT) of a boolean function yields a trace representation or equivalently, a polynomial representation, of the boolean function, which is identical to the DFT of the sequence associated with the boolean function. Using this tool, we investigate characterizations of boolean functions for which the fast algebraic attack is applicable. In order to apply the fast algebraic attack, the question that needs to be answered is that: for a given boolean function f in n variables and a pair of positive integers (d, e), when there exists a function g with degree at most d such that $h=fg\ne 0$ where h’s degree is at most e. We give a sufficient and necessary condition for the existence of those multipliers of f. An algorithm for finding those multipliers is given in terms of a polynomial basis of 2ⁿ dimensional space over $\mathbb{F}_2$ which is established by an arbitrary m-sequence of period 2ⁿ − 1 together with all its decimations and certain shifts. We then provide analysis for degenerated cases and introduce a new concept of resistance against the fast algebraic attack in terms of the DFT of sequences or boolean functions. Some functions which made the fast algebraic attack inefficient are identified.