On Rueppel's Linear Complexity Conjecture

Abstract

Rueppel's conjecture on the linear complexity of the first $n$ terms of the sequence $(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots)$ was first proved by Dai using the Euclidean algorithm. We have previously shown that we can attach a homogeneous (annihilator) ideal of $F[x,z]$ to the first $n$ terms of a sequence over a field $F$ and construct a pair of generating forms for it. This approach gives another proof of Rueppel's conjecture. We also prove additional properties of these forms and deduce the outputs of the LFSR synthesis algorithm applied to the first $n$ terms. Further, dehomogenising the leading generators yields the minimal polynomials of Dai.