Part VIII of a deep-dive series on Linear Feedback Shift Registers (LFSRs) explores companion matrix representations and their relationship to finite field polynomial methods. The post derives lookahead bitmasks for time-shifted LFSR output using powers of the companion matrix, then shows equivalent bitwise/finite-field computations that avoid matrix math entirely. It then tackles state recovery — reconstructing the LFSR internal state from observed output bits — presenting four methods: two matrix-based (scipy triangular solver and manual Toeplitz back-substitution) and two pure LFSR-shifting methods (forward and backward). Working Python examples using the libgf2 library are provided throughout, and the post concludes that while matrices aid understanding, efficient implementations can almost always avoid them.
