$p$ -元$m$ -序列的LSB序列的2-Adic复杂度的自相关和下界
Autocorrelation and Lower Bound on the 2-Adic Complexity of LSB Sequence of $p$-ary $m$-Sequence
摘要：
LSB（Least Significant Bit，最低有效位）序列被广泛地用作一些现代流密码算法的初始输入，如3GPP LTE国际加密标准的核心--ZUC算法。因此，分析这些序列的统计特性（例如，自相关性，线性复杂度和2-adic复杂度）成为一个重要的研究课题。本文首先将周期为PDMATHABM 0的$p$元$m$ -序列的LSB序列的自相关分布约化为相应的周期为$p-1$的Costas序列的自相关分布，并通过计算机计算得到了每个素数$p<100$的LSB序列的显式自相关分布.此外，我们给出了所有素数$p < 20$的LSB序列的2-adic复杂度的下界，证明了该下界足够大以抵抗FCSR（带进位移位寄存器的反馈）的RAA（有理逼近算法）的分析。特别地，对于梅森素数$p=2^k-1$（即，$k$是一个素数，使得$p$也是一个素数），我们的结果适用于所有它的位分量序列，因为它们与LSB序列移位等价。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
