On obtaining long m-sequences from low-degree primitive polynomials

Abstract

Maximum-length sequences of length $2^n-1$ (m-sequences) are typically obtained by starting from a primitive polynomial of degree $n$ over $GF\left(2\right)$ and configuring a Linear Feedback Shift Register (LFSR) based on that polynomial. In this study, we investigate the generation of long m-sequences based on a primitive polynomial of low degree. Specifically, we investigate a very simple form of an LFSR structure, referred to as Two-Multiplier Split LFSR (2M-SLFSR), that consists of $m$ $\delta$-bit cells and is based on a single low-degree primitive polynomial of degree $\delta \ge 2$ over $GF\left(2\right)$ and which can generate, with proper configuration, an m-sequence of length $2^{m\delta }-1$. For example, we show that starting from the primitive polynomial $x^2+x+1$ over $GF\left(2\right)$, a 2M-SLFSR with $m=599$ 2-bit cells can be constructed that yields an m-sequence of length $2^{1198}-1$. M-sequences of large length such as $2^{512}-1$ obtained from low degree primitive polynomials via LFSR structures akin to 2M-SLFSR find current applications in stream ciphers like those used in SNOW-V and SNOW-Vi.