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 article, 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≥2 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.
Highlights
As important components of cipher systems, pseudo-random sequences have widely applications in cryptography
We present a figure to explain the significance of the lower bound on the 2-adic complexity of each LSB sequence in this article
In this article, we first turned the problem of determining the autocorrelation distribution of the LSB sequence of a p-ary m-sequence with period pn − 1 for any order n ≥ 2 into the problem of calculating the autocorrelation distribution of a corresponding Costas sequence with period p − 1 directly by computer
Summary
As important components of cipher systems, pseudo-random sequences have widely applications in cryptography. Autocorrelation distributions, linear complexity and 2-adic complexity of sequences become three important indexes to measure a cipher system, i.e., sequences used as a key stream should have low autocorrelation, high linear complexity and large 2-adic complexity Due to their ideal correlation property and other good performance measures such as highly efficient implementation, maximal length LFSR sequences (i.e., m-sequences) have. Similar to BMA of LFSRs, Klapper and Goresky proposed an algorithm, called Rational Approximation Algorithm (RAA), to determine the 2-adic complexity of s Zhang et al introduced a new method to determine the 2-adic complexity of a binary sequence by ‘‘Gauss periods’’ and ‘‘Gauss sum’’ over a ring ZN of residue classes modulo an integer N [29].
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