Abstract
In this paper, a survey of Legendre sequences and modified Jacobi sequences is presented, firstly. We introduce the construction and periodic autocorrelation functions of these two sequences (binary and polyphase). Then we determine the linear complexity of all modified polyphase Jacobi sequences and the corresponding feedback polynomials of the shortest linear feedback shift register that generates such a sequence. Making use of these results, at the same time, we prove the conjectures on the linear complexity and feedback polynomials of modified Jacobi sequences brought forward by D.H. Green and J. Choi.
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References
D. Everett. Periodic digit sequences with pseudoradom properties. GEC J., 33, 1966.
P. Fan and M. Darnell. Sequence Design for Communications Applications. John Wiley, Research Studies Press, Taunton, 1996.
I. Damgaard. On the randomness of Legendre and Jacobi sequences. Advances in Cryptology: Crypto’88, Berlin, Germany: Springer-Verlage, LNCS 403, 1990.
J.H. Kim and H.Y. Song. Trace representation of Legendre sequences. Designs, Codes Cryptogr., 24(3), 2001.
C. Ding, T. Helleseth, and W. Shan. On the linear complexity of Legendre sequences. IEEE Trans. Inf. Theory, 44(3), 1998.
D.H. Green and P.R. Green. Modified Jocobi sequences. IEE Proc. Comput. Digit. Tech., 147(4), 2000.
D.H. Green and J. Choi. Linear complexity of modified Jacobi sequences. In IEE Proc. Comput. Digit. Tech., 149(3), 2002.
D.H. Green, M.D. Smith, and N. Martzoukos. Linear complexity of polyphase power residue sequence. IEE Proc. Commun., 149(4), 2002.
D.H. Green and P.R. Green. Polyphase related-prime sequences. IEE Proc. Comput. Digit. Tech., 148(2), 2001. MIT Press, Cambridge, MA, 1994.
D.H. Green. Structural properties of pseudorandom arrays and volumes and their related sequences. IEE Proc. Comput. Digit. Tech., 132(3), 1985.
R. Lidl and H. Niederreiter. Finite Fields. in Encyclopedia of Mathematics and Its Applications, vol.20, Reading, MA: Assison-Wesley, 1983.
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© 2004 Kluwer Academic Publishers
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Bai, E., Zhang, B. (2004). Legendre Sequences and Modified Jacobi Sequences. In: Chen, K. (eds) Progress on Cryptography. The International Series in Engineering and Computer Science, vol 769. Springer, Boston, MA. https://doi.org/10.1007/1-4020-7987-7_2
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DOI: https://doi.org/10.1007/1-4020-7987-7_2
Publisher Name: Springer, Boston, MA
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