Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 703–717 | Cite as

On the linear complexity of Legendre–Sidelnikov sequences



Linear complexity is an important cryptographic index of sequences. We study the linear complexity of \(p(q-1)\)-periodic Legendre–Sidelnikov sequences, which combine the concepts of Legendre sequences and Sidelnikov sequences. We get lower and upper bounds on the linear complexity in different cases, and experiments show that the upper bounds can be attained. Remarkably, we associate the linear complexity of Legendre–Sidelnikov sequences with some famous primes including safe prime and Fermat prime. If \(2\) is a primitive root modulo \(\frac{q-1}{2}\), and \(q\) is a safe prime greater than 7, the linear complexity is the period if \(p\equiv 3 \pmod 8\); \(p(q-1)-p+1\) if \(p\equiv q \equiv 7 \pmod 8\), and \(p(q-1)-\frac{p-1}{2}\) if \(p \equiv 7 \pmod 8, q \equiv 3 \pmod 8\). If \(q\) is a Fermat prime, the linear complexity is the period if \(p \equiv 3 \pmod 8\), and \(p(q-1)-q+2\) if \(p \equiv 5 \pmod 8\). It is very interesting that the Legendre–Sidelnikov sequence has maximal linear complexity and is balanced if we choose \(p=q\) to be some safe prime.


Linear complexity Legendre–Sidelnikov sequence Legendre sequence Sidelnikov sequence Cryptography 

Mathematics Subject Classification

94A60 94A55 65C10 68P25 



The author would like to thank anonymous referees for valuable comments, and express his sincere thanks for hospitality during his visit to RICAM, Austrian Academy of Science, and some useful discussions with Arne Winterhof that led to this work. He is supported by National Natural Science Foundation of China (No. 61003070), and partially by NSFC (No. 61070014, 61373018).


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceNankai UniversityTianjinChina

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