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Determining the number of true different permutation polynomials of degrees up to five by Weng and Dong algorithm

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Abstract

Permutation polynomials (PPs) are used for interleavers in turbo codes, cryptography or sequence generation. The paper presents an algorithm for determining the number of true different PPs of degrees up to five. It is based on the algorithm from Weng and Dong (IEEE Trans Inf Theory 54(9):4388–4390, 2008) and on the null polynomials modulo the interleaver length.

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References

  1. Takeshita, O. Y. (2007). Permutation polynomial interleavers: An algebraic-geometric perspective. IEEE Transactions on Information Theory, 53(6), 2116–2132.

    Article  Google Scholar 

  2. Sun, J., & Takeshita, O. Y. (2005). Interleavers for turbo codes using permutation polynomials over integer rings. IEEE Transactions on Information Theory, 51(1), 101–119.

    Article  Google Scholar 

  3. Takeshita, O. Y. (2006). On maximum contention-free interleavers and permutation polynomials over integer rings. IEEE Transactions on Information Theory, 52(3), 1249–1253.

    Article  Google Scholar 

  4. Ryu, J., & Takeshita, O. Y. (2006). On quadratic inverses for quadratic permutation polynomials over integer rings. IEEE Transactions on Information Theory, 52(3), 1254–1260.

    Article  Google Scholar 

  5. Takeshita, O.Y. (2006). A new metric for permutation polynomial interleavers. In Proceedings of IEEE international symposium on information theory (ISIT), Seattle, USA, pp. 1983–1987.

  6. Rosnes, E., & Takeshita, O.Y. (2006). Optimum distance quadratic permutation polynomial-based interleavers for turbo codes. In Proceedings of IEEE international symposium on information theory (ISIT), Seattle, USA, pp. 1988–1992.

  7. Tarniceriu, D., Trifina, L., & Munteanu, V. (2009). About minimum distance for QPP interleavers. Annals of Telecommunications, 64(9–10), 745–751.

    Article  Google Scholar 

  8. Zhao, H., Fan, P., & Tarokh, V. (2010). On the equivalence of interleavers for turbo codes using quadratic permutation polynomials over integer rings. IEEE Communications Letters, 14(3), 236–238.

    Article  Google Scholar 

  9. Trifina, L., Tarniceriu, D., & Munteanu, V. (2011). Improved QPP interleavers for LTE standard. In Proceedings of IEEE international symposium on signals, circuits and systems (ISSCS), Iasi, Romania, pp. 403–406.

  10. Ryu, J. (2012). Efficient address generation for permutation polynomial based interleavers over integer rings. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E95–A(1), 421–424.

    Article  Google Scholar 

  11. Lahtonen, J., Ryu, J., & Suvitie, E. (2012). On the degree of the inverse of quadratic permutation polynomial interleavers. IEEE Transactions on Information Theory, 58(6), 3925–3932.

    Article  Google Scholar 

  12. Rosnes, E. (2012). On the minimum distance of turbo codes with quadratic permutation polynomial interleavers. IEEE Transactions on Information Theory, 58(7), 4781–4795.

    Article  Google Scholar 

  13. Trifina, L., Tarniceriu, D., & Munteanu, V. (2012). On dispersion and nonlinearity degree of QPP interleavers. Applied Mathemathics & Information Sciences, 6(3), 397–400.

    Google Scholar 

  14. Ryu, J. (2012). Permutation polynomials of higher degrees for turbo code interleavers. IEICE Transactions on Communications, E95–B(12), 3760–3762.

    Article  Google Scholar 

  15. Trifina, L., & Tarniceriu, D. (2014). Improved method for searching interleavers from a certain set using Garello’s method with applications for the LTE standard. Annals of Telecommunications, 69(5–6), 251–272.

    Article  Google Scholar 

  16. 3GPP TS 36.212 V8.3.0, 3rd Generation Partnership Project, Multiplexing and channel coding (Release 8), 2008. http://www.etsi.org/deliver/etsi_ts/136200136299/136212/08.03.0060/ts136212v080300p.pdf. Accessed June 5, 2015.

  17. Trifina, L., & Tarniceriu, D. (2013). Analysis of cubic permutation polynomials for turbo codes. Wireless Personal Communications, 69(1), 1–22.

    Article  Google Scholar 

  18. Chen, Y.-L., Ryu, J., & Takeshita, O. Y. (2006). A simple coefficient test for cubic permutation polynomials over integer rings. IEEE Communications Letters, 10(7), 549–551.

    Article  Google Scholar 

  19. Zhao, H., & Fan, P. (2007). A note on A simple coefficient test for cubic permutation polynomials over integer rings. IEEE Communications Letters, 11(12), 991.

    Article  Google Scholar 

  20. Weng, G., & Dong, C. (2008). A note on permutation polynomials over \(\mathbb{Z}_n\). IEEE Transactions on Information Theory, 54(9), 4388–4390.

    Article  Google Scholar 

  21. Ryu, J. (2007). Permutation polynomial based interleavers for turbo codes over integer rings. Ph.D. Thesis. https://etd.ohiolink.edu/rws_etd/document/get/osu1181139404/inline. Accessed February 1, 2011.

  22. Ryu, J., & Takeshita, O.Y. (2011). On inverses for quadratic permutation polynomials over integer rings. arxiv:1102.2223v1. Accessed March 18, 2011.

  23. Trifina, L., & Tarniceriu, D. (2017). A simple method to determine the number of true different quadratic and cubic permutation polynomial based interleavers for turbo codes. Telecommunication Systems, 64(1), 147–171.

    Article  Google Scholar 

  24. Nöbauer, W. (1965). Über permutationspolynome und permutationsfunktionen für primzahlpotenzen. Monatshefte für Mathematik, 69(3), 230–238.

  25. Mullen, G. L., & Stevens, H. (1984). Polynomial functions \((\text{ mod } m)\). Acta Mathematica Hungarica, 44(3–4), 237–241.

    Article  Google Scholar 

  26. Hardy, G. H., & Wright, E. M. (1975). An Introduction to the Theory of Numbers (4th ed.). Oxford: Oxford University Press.

    Google Scholar 

  27. Dickson, L. E. (1896–1897). The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group. Annals of Mathematics, 11(1/6), 65–120.

  28. Rivest, R. L. (2001). Permutation polynomials modulo \(2^w\). Finite Fields and Their Applications, 7(2), 287–292.

    Article  Google Scholar 

  29. Trifina, L., & Tarniceriu, D. (2017). A coefficient test for quintic permutation polynomials. Submitted to publication.

  30. Trifina, L., & Tarniceriu, D. (2016). The number of different true permutation polynomial based interleavers under Zhao and Fan sufficient conditions. Telecommunication Systems, 63(4), 593–623.

    Article  Google Scholar 

  31. Zhao, H., & Fan, P. (2007). Simple method for generating \(m\)th-order permutation polynomials over integer rings. Electronics Letters, 43(8), 449–451.

    Article  Google Scholar 

  32. http://telecom.etti.tuiasi.ro/tti/papers/Text_files/Number_of_true_diff_LPPs_QPPs_CPPs_4PPs_5PPs_N_000002_100000.txt.

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Authors and Affiliations

  1. Department of Telecommunications and Information Technologies, Faculty of Electronics, Telecommunications and Information Technology, “Gheorghe Asachi” Technical University, Bd. Carol I, no. 11, 700506, Iasi, Romania

    Lucian Trifina & Daniela Tarniceriu

Authors
  1. Lucian Trifina
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  2. Daniela Tarniceriu
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Correspondence to Lucian Trifina.

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Trifina, L., Tarniceriu, D. Determining the number of true different permutation polynomials of degrees up to five by Weng and Dong algorithm. Telecommun Syst 67, 211–215 (2018). https://doi.org/10.1007/s11235-017-0335-y

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  • DOI: https://doi.org/10.1007/s11235-017-0335-y

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