On the linear complexity of binary threshold sequences derived from Fermat quotients |
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Authors: | Zhixiong Chen Xiaoni Du |
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Institution: | 1. Department of Mathematics, Putian University, Putian, 351100, Fujian, People’s Republic of China 2. State Key Laboratory of Information Security, Graduate School of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China 3. College of Mathematics and Information Science, Northwest Normal University, Lanzhou, 730070, Gansu, People’s Republic of China
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Abstract: | We determine the linear complexity of a family of p 2-periodic binary threshold sequences derived from Fermat quotients modulo an odd prime p, where p satisfies ${2^{p-1} \not\equiv 1 ({\rm mod}\, {p^2})}$ . The linear complexity equals p 2 ? p or p 2 ? 1, depending whether ${p \equiv 1}$ or 3 (mod 4). Our research extends the results from previous work on the linear complexity of the corresponding binary threshold sequences when 2 is a primitive root modulo p 2. Moreover, we present a partial result on their linear complexities for primes p with ${2^{p-1} \equiv 1 ({\rm mod} \,{p^2})}$ . However such so called Wieferich primes are very rare. |
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