A Geometric Sequence Binarized with Legendre Symbol over Odd Characteristic Field and Its Properties
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Abstract
Let <i>p</i> be an odd characteristic and <i>m</i> be the degree of a primitive polynomial <i>f</i>(<i>x</i>) over the prime field F<i>p</i>. Let <i>ω</i> be its zero, that is a primitive element in F<sup>*</sup><i><sub>p<sup>m</sup></sub></i>, the sequence <i>S</i>={<i>s<sub>i</sub></i>}, <i>s<sub>i</sub></i>=Tr(<i>ω<sup>i</sup></i>) for <i>i</i>=0,1,2,… becomes a nonbinary maximum length sequence, where Tr(·) is the trace function over F<i>p</i>. On this fact, this paper proposes to binarize the sequence by using Legendre symbol. It will be a class of geometric sequences but its properties such as the period, autocorrelation, and linear complexity have not been discussed. Then, this paper shows that the generated binary sequence (geometric sequence by Legendre symbol) has the period <i>n</i>=2(<i>p<sup>m</sup></i>1)/(<i>p</i>1) and a typical periodic autocorrelation. Moreover, it is experimentally observed that its linear complexity becomes the maximum, that is the period <i>n</i>. Among such experimental observations, especially in the case of <i>m</i>=2, it is shown that the maximum linear complexity is theoretically proven. After that, this paper also demonstrates these properties with a small example.
Journal

 IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E97.A(12), 23362342, 2014
The Institute of Electronics, Information and Communication Engineers