Revisiting LFSMs

Abstract—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) stu died and implemented in many cryptographic applications such as design of stream ciphers or pseudo-random generation. LFSR s could be seen as particular LFSMs without inputs. In this paper, we give first a general representation of LFSMs using traditional matrices representation linking this definition together with a new polynomial representation leading to sp arse representations and implementations. As direct applicati ons, we focus our work on the LFSRs case and show how the new LFSMs representation leads to a powerful design for LFSRs called R ing LFSRs efficient in both hardware and software. We also study a particular LFSRs subcase called windmill LFSRs used for example in the E0 stream cipher and we generalize their representation leading to better hardware performances.

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