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Tangent-chebyshev maps over finite fields: New properties and functional graphs

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Abstract

In a recent work, a family of rational maps identified as tangent-Chebyshev maps over finite fields was introduced. Such maps have potential applicability in cryptography and channel coding. In this paper, we present new properties for the referred maps: their symmetry and fixed points are determined, their relationships with other maps are described and their functional graphs are characterized. Illustrative examples related to the developed properties are given; in particular, we show that the use of the referred maps to interleave in a turbo code may provide coding gains, when compared with usual approaches.

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Notes

  1. We use “=” as a notation abuse in \(x_{d/2}=\infty\), as a reference to the possibility of having an undefined value for the finite field tangent function computed with respect to \(\zeta\), when its argument equals \(\frac{\text { ord }(\zeta )}{4}\).

  2. In similar contexts, the functional graph is called state-mapping network (SMN). In [17], for example, SMNs are used to analyze the dynamic of digital chaotic maps.

  3. The number 1 in parentheses indicates that the code is systematic and the numbers 15 and 13 are octal representations of the polynomials used to generate the convolutional encoder employed to design the corresponding turbo code [28].

References

  1. Bae, S., Cha, B., Jung, H.: Möbius function in short intervals for function fields. Finite Fields and Their Applications 34, 235–249 (2015)

    Article  MathSciNet  Google Scholar 

  2. Bae, S., Hu, S., Jung, H.: The generalized Rédei-matrix for function fields. Finite Fields and Their Applications 18(4), 760–780 (2012)

    Article  MathSciNet  Google Scholar 

  3. Barbier, M., Cheballah, H., Bars, J.M.: On the computation of the Möbius transform. Theoretical Computer Science 809, 171–188 (2020)

    Article  MathSciNet  Google Scholar 

  4. Bellini, E., Murru, N.: An efficient and secure RSA-like cryptosystem exploiting Rédei rational functions over conics. Finite Fields and Their Applications 39, 179–194 (2016)

    Article  MathSciNet  Google Scholar 

  5. Burton, D.M.: Elementary Number Theory, 7th edn. McGraw-Hill Science/Engineering/Math (2010)

  6. Capaverde, J., Masuda, A.M., Rodrigues, V.M.: Rédei permutations with cycles of the same length. Designs, Codes and Cryptography 88(12), 2561–2579 (2020)

    Article  MathSciNet  Google Scholar 

  7. Çeşmelioğlu, A., Meidl, W., Topuzoğlu, A.: Permutations of finite fields with prescribed properties. Journal of Computational and Applied Mathematics 259, 536–545 (2014)

    Article  MathSciNet  Google Scholar 

  8. Charpin, P., Mesnager, S., Sarkar, S.: Involutions over the Galois field \(\mathbb{F}_{2^n}\). IEEE Trans. Inf. Theory 62(4), 2266–2275 (2016)

    Article  Google Scholar 

  9. Dassios, I., Tzounas, G., Milano, F.: The Möbius transform effect in singular systems of differential equations. Applied Mathematics and Computation 361, 338–353 (2019)

    Article  MathSciNet  Google Scholar 

  10. Figueiredo, R.B.D., Lima, J.B.: Fractional angular transform: a number-theoretic approach. Electronics Letters 55(6), 322–325 (2019)

    Article  Google Scholar 

  11. Fu, S., Feng, X., Lin, D., Wang, Q.: A recursive construction of permutation polynomials over \(\mathbb{F}_{q^2}\) Rédei functions. Designs, Codes and Cryptography 87(7), 1481–1498 (2019)

    Article  MathSciNet  Google Scholar 

  12. Gassert, T.A.: Chebyshev action on finite fields. Discrete Mathematics 315–316, 83–94 (2014)

    Article  MathSciNet  Google Scholar 

  13. Gravel, C., Panario, D., Thomson, D.: Unicyclic strong permutations. Cryptography and Communications 11(6), 1211–1231 (2019)

    Article  MathSciNet  Google Scholar 

  14. Hosseinalipour, S., Sakzad, A., Sadeghi, M.R.: Performance of Möbius interleavers for turbo codes. Wireless Personal Communications 98(1), 271–291 (2018)

    Article  Google Scholar 

  15. Jelínek, V., Kantor, I., Kyncl, J., Tancer, M.: On the growth of the Möbius function of permutations. J. Comb. Theor, Ser. A 169, 105–121 (2020)

    Article  Google Scholar 

  16. Konyagina, S.V., Luca, F., Mans, B., Mathieson, L., Sha, M., Shparlinski, I.E.: Functional graphs of polynomials over finite fields. J. Comb. Theory Ser. B 116, 87–122 (2016)

    Article  MathSciNet  Google Scholar 

  17. Li, C., Feng, B., Li, S., Kurths, J., Chen, G.: Dynamic analysis of digital chaotic maps via state-mapping networks. IEEE Trans. Circuits Syst. I, Reg. Papers 66(6), 2322–2335 (2019)

    Article  MathSciNet  Google Scholar 

  18. Liao, X., Chen, F., Wong, K.W.: On the security of public-key algorithms based on Chebyshev polynomials over the finite field \(Z_{N}\). IEEE Trans. Comput. 59(10), 1392–1401 (2010)

    Article  MathSciNet  Google Scholar 

  19. Lima, J.B., Panario, D., Campello de Souza, R.M.: Public-key encryption based on Chebyshev polynomials over GF(q). Information Processing Letters 111(2), 51–56 (2010)

    Article  MathSciNet  Google Scholar 

  20. Lima, J.B., Panario, D., de Souza, R.M.C.: A trigonometric approach for Chebyshev polynomials over finite fields. In: Larcher, G., Pillichshammer, F., Winterhof, A., Xing, C. (eds.) Applied Algebra and Number Theory, pp. 255–279. Cambridge University Press, Cambridge (2014)

    Chapter  Google Scholar 

  21. Lima, J.B., Campello de Souza, R.M.: Fractional cosine and sine transforms over finite fields. Linear Algebra and its Applications 438(8), 3217–3230 (2013)

    Article  MathSciNet  Google Scholar 

  22. Lima, J.B., de Souza, R.M.C.: Tangent function and Chebyshev-like rational maps over finite fields. IEEE Trans. Circuits Syst. II, Exp. Briefs 67(4), 775–779 (2020)

    Article  Google Scholar 

  23. LTE: Evolved universal terrestrial radio access (e-utra); multiplexing and channel coding (3gpp ts 36.212 version 8.8.0 release 8) (2010–01). Technical Specification, ETSI (2010)

  24. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall/CRC, Boca Raton (2003)

    MATH  Google Scholar 

  25. Mikhail, M., Abouelseoud, Y., ElKobrosy, G.: Two-phase image encryption scheme based on FFCT and fractals. Security and Communication Networks pp. 1–13 (2017)

  26. More, W.: Fast evaluation of Rédei functions. Applicable Algebra in Engineering, Communication and Computing 6(3), 171–173 (1995)

    Article  MathSciNet  Google Scholar 

  27. Mullen, G.L., Panario, D.: Handbook of Finite Fields. Chapman & Hall/CRC (2013)

  28. Nour, C.A.: Enhanced turbo codes for NR: Performance evaluation for eMBB and URLLC. Technical report, 3GPP TSG-RAN WG1 Meeting #87 (2016)

  29. Panario, D., Reis, L.: The functional graph of linear maps over finite fields and applications. Designs, Codes and Cryptography 87(2), 437–453 (2019)

    Article  MathSciNet  Google Scholar 

  30. Qureshi, C., Panario, D.: Rédei actions on finite fields and multiplication map in cyclic group. SIAM Journal on Discrete Mathematics 29(3), 1486–1503 (2015)

    Article  MathSciNet  Google Scholar 

  31. Qureshi, C., Panario, D.: The graph structure of chebyshev polynomials over finite fields and applications. Designs, Codes and Cryptography 87(2), 393–416 (2019)

    Article  MathSciNet  Google Scholar 

  32. Sakzad, A., Panario, D., Sadeghi, M.R., Eshghi, N.: Self-inverse interleavers based on permutation functions for turbo codes. In: 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton) pp. 22–28 (2010)

  33. da Silva Neto, E.F., Lima, J.B.: Audio encryption based on the cosine number transform. Multimedia Tools and Applications 75(14), 8403–8418 (2016)

    Article  Google Scholar 

  34. Campello de Souza, R.M., de Oliveira, H.M., Kauffman, A., Paschoal, A.J.A.: Trigonometry in finite fields and a new Hartley transform. In: Proc. IEEE Int. Symp. Inf. Theory, p. 293. IEEE (1998)

  35. Yoshioka, D.: Periodic properties of Chebyshev polynomial sequences over the residue ring \(\mathbb{Z}/2^{k}\mathbb{Z}\). IEEE Trans. Circuits Syst. II, Exp. Briefs 63(8), 778–782 (2016)

  36. Yoshioka, D.: Properties of Chebyshev polynomials modulo \(p^{k}\). IEEE Trans. Circuits Syst. II, Exp. Briefs 65(3), 386–390 (2018)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Electronics and Systems, Federal University of Pernambuco, Recife, Brazil

    Ravi B. D. Figueiredo & Juliano B. Lima

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  1. Ravi B. D. Figueiredo
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  2. Juliano B. Lima
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Correspondence to Juliano B. Lima.

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This work was supported by CNPq under Grants 309598/2017-6 and 409543/2018-7, and FACEPE under grant IBPG-0071-3.04/17.

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Figueiredo, R.B.D., Lima, J.B. Tangent-chebyshev maps over finite fields: New properties and functional graphs. Cryptogr. Commun. 14, 897–908 (2022). https://doi.org/10.1007/s12095-022-00565-8

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