The Linear Vector Space Spanned by the Nonlinear Filter Generator

Abstract

The filter generator is an important building block in many stream ciphers. The generator consists of a linear feedback shift register (LFSR) of length n and a Boolean filtering function of degree d that combines bits from the shift register and creates an output bit z _t at any time t. A new attack on stream ciphers based on linear shift registers has recently been described by the authors in [3]. This attack is modified to stream ciphers based on any linear shift register and not only for LFSRs. The focal point of this paper is to present a linear description of the filter generator in terms of matrices. The filter generator is viewed entirely in terms of powers of a unique linear operator T together with a vector representing the filtering function. It is proved that T embodies the coefficient sequences described in [3]. Thus, interesting properties of the vector space (e.g. the dimension of the equation systems) generated by the filter generator can be analysed using theory of cyclic vector spaces, which very elegantly complements analysis in terms of the roots of the LFSR.