Autocorrelation and Lower Bound on the 2-Adic Complexity of LSB Sequence of $p$-ary $m$-Sequence

Abstract

LSB (Least Significant Bit) sequences are widely used as the initial inputs in some modern stream ciphers, such as the ZUC algorithm-the core of the 3GPP LTE International Encryption Standard. Therefore, analyzing the statistical properties (for example, autocorrelation, linear complexity and 2-adic complexity) of these sequences becomes an important research topic. In this paper, we first reduce the autocorrelation distribution of the LSB sequence of a $p$-ary $m$-sequence with period $p^n-1$ for any order $n\geq2$ to the autocorrelation distribution of a corresponding Costas sequence with period $p-1$, and from the computing of which by computer, we obtain the explicit autocorrelation distribution of the LSB sequence for each prime $p<100$. In addition, we give a lower bound on the 2-adic complexity of each of these LSB sequences for all primes $p < 20$, which proves to be large enough to resist the analysis of RAA (Rational Approximation Algorithm) for FCSRs (Feedback with Carry Shift Registers). In particular, for a Mersenne prime $p=2^k-1$ (i.e., $k$ is a prime such that $p$ is also a prime), our results hold for all its bit-component sequences since they are shift equivalent to the LSB sequence.