Revisiting LFSMs

Linear Finite State Machines (LFSMs) are particular primitives widely used in information theory, coding theory and cryptography. Among those linear automata, a particular case of study is Linear Feedback Shift Registers (LFSRs) used in many cryptographic applications such as design of stream ciphers or pseudo-random generation. LFSRs could be seen as particular LFSMs without inputs. In this paper, we first recall the description of LFSMs using traditional matrices representation. Then, we introduce a new matrices representation with polynomial fractional coefficients. This new representation leads to sparse representations and implementations. As direct applications, we focus our work on the Windmill LFSRs case, used for example in the E0 stream cipher and on other general applications that use this new representation. In a second part, a new design criterion called diffusion delay for LFSRs is introduced and well compared with existing related notions. This criterion represents the diffusion capacity of an LFSR. Thus, using the matrices representation, we present a new algorithm to randomly pick LFSRs with good properties (including the new one) and sparse descriptions dedicated to hardware and software designs. We present some examples of LFSRs generated using our algorithm to show the relevance of our approach.

💡 Research Summary

The paper revisits Linear Finite State Machines (LFSMs) and, in particular, Linear Feedback Shift Registers (LFSRs), which are the cornerstone of many stream ciphers and pseudo‑random generators. It begins by recalling the classical representation of LFSMs using binary (0‑1) matrices that describe state transition, input, and output. While mathematically convenient, these dense matrices lead to costly hardware implementations (many registers and XOR gates) and to inefficient software due to numerous bit‑wise operations.

To overcome these limitations, the authors propose a novel matrix formalism where each matrix entry is a rational function over the polynomial ring 𝔽₂