Journal of information and communication convergence engineering

The Korea Institute of Information and Commucation Engineering (한국정보통신학회)
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Relation between the Irreducible Polynomials that Generates the Same Binary Sequence Over Odd Characteristic Field

  • Ali, Md. Arshad (Department of Information and Communication Systems, Okayama University) ;
  • Kodera, Yuta (Department of Information and Communication Systems, Okayama University) ;
  • Park, Taehwan (Department of Electrical and Computer Engineering, Pusan National University) ;
  • Kusaka, Takuya (Department of Information and Communication Systems, Okayama University) ;
  • Nogmi, Yasuyuki (Department of Information and Communication Systems, Okayama University) ;
  • Kim, Howon (Department of Electrical and Computer Engineering, Pusan National University)
  • Received : 2018.06.19
  • Accepted : 2018.07.30
  • Published : 2018.09.30
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Abstract

A pseudo-random sequence generated by using a primitive polynomial, trace function, and Legendre symbol has been researched in our previous work. Our previous sequence has some interesting features such as period, autocorrelation, and linear complexity. A pseudo-random sequence widely used in cryptography. However, from the aspect of the practical use in cryptographic systems sequence needs to generate swiftly. Our previous sequence generated by utilizing a primitive polynomial, however, finding a primitive polynomial requires high calculating cost when the degree or the characteristic is large. It’s a shortcoming of our previous work. The main contribution of this work is to find some relation between the generated sequence and irreducible polynomials. The purpose of this relationship is to generate the same sequence without utilizing a primitive polynomial. From the experimental observation, it is found that there are (p - 1)/2 kinds of polynomial, which generates the same sequence. In addition, some of these polynomials are non-primitive polynomial. In this paper, these relationships between the sequence and the polynomials are shown by some examples. Furthermore, these relationships are proven theoretically also.

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References

  1. T. W. Cusick, C. Ding, and A. R. Renvall, Stream Ciphers and Number Theory. Amsterdam: Elsevier, 1998.
  2. S. W. Golomb, Shift Register Sequences. San Francisco, CA: Holden-Day, 1967.
  3. S. W. Golomb and G. Gong, Sequence Design for Good Correlation. Cambridge, MA: Cambridge University Press, 2005. DOI: 10.1017/CBO9780511546907.
  4. A. J. Menezes, P. C. Van Oorschot, and S. A. Vanstone, Handbook of Applied Cryptography. Boca Raton, FL: CRC Press, 1996.
  5. N. Zierler, "Linear recurring sequences," Journal of the Society for Industrial and Applied Mathematics, vol. 7, no. 1, pp. 31-48, 1959. DOI: 10.1137/0107003.
  6. C. Ding, T. Hesseseth, and W. Shan, "On the linear complexity of Legendre sequences," IEEE Transactions on Information Theory, vol. 44, no. 3, pp. 1276-1278, 1998. DOI: 10.1109/18.669398.
  7. N. Zierler, Legendre Sequences. Lexington, MA: MIT Lincoln Laboratory, 1958.
  8. J. S. No, H. K. Lee, H. Chung, H. Y. Song, and K. Yang, "Trace representation of Legendre sequences of Mersenne prime period," IEEE Transactions on Information Theory, vol. 42, no. 6, pp. 2254-2255, 1996. DOI: 10.1109/18.556617.
  9. Y. Nogami, K. Tada, and S. Uehara, "A geometric sequence binarized with Legendre symbol over odd characteristic field and its properties," IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 97, no. 12, pp. 2336-2342, 2014. DOI: 10.1587/transfun.E97.A.2336.
  10. R. Lidl and H. Niederreiter, Finite Fields. Boston, MA: Addison-Wesley, 1983.