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On the number of irreducible linear transformation shift registers

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Abstract

We deal with the problem of counting the number of irreducible linear transformation shift registers (TSRs) over a finite field. In a recent paper, Ram reduced this problem to calculate the cardinality of some set of irreducible polynomials and got explicit formulae for the number of irreducible TSRs of order two. We find a bijection between Ram’s set to another set of irreducible polynomials which is easier to count, and then give a conjecture about the number of irreducible TSRs of any order. We also get explicit formulae for the number of irreducible TSRs of order three.

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References

  1. Chen E., Tseng D.: The splitting subspace conjecture. Finite Fields Appl. 24, 15–28 (2013).

  2. Cohen S.D., Hasan S.U., Panario D., Wang Q.: An asymptotic formula for the number of irreducible transformation shift registers. Linear Algebra Appl. 484, 46–62 (2015).

  3. Dewar M., Panario D.: Linear transformation shift registers. IEEE Trans. Inf. Theory 49(8), 2047–2052 (2003).

  4. Dewar M., Panario D.: Mutual irreducibility of certain polynomials. In: Mullen G.L., Poli A., Stichtenoth H. (eds.) Finite Fields and Applications. Lecture Notes in Computer Science, vol. 2948, pp. 59–68. Springer, Heidelberg (2004).

  5. Ghorpade S.R., Hasan S.U., Kumari M.: Primitive polynomials, singer cycles and word-oriented linear feedback shift registers. Des. Codes Cryptogr. 58(2), 123–134 (2011).

  6. Ghorpade S.R., Ram S.: Block companion singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields. Finite Fields Appl. 17(5), 461–472 (2011).

  7. Hasan S.U., Panario D., Wang Q.: Word-oriented transformation shift registers and their linear complexity. In: Helleseth T., Jedwab J. (eds.) Sequences and Their Applications—SETA 2012. Lecture Notes in Computer Science, vol. 7280, pp. 190–201. Springer, Heidelberg (2012).

  8. Krishnaswamy S., Pillai, H.K.: On multisequences and their extensions. arXiv:1208.4501v1 (2012).

  9. Krishnaswamy S., Pillai H.K.: On the number of linear feedback shift registers with a special structure. IEEE Trans. Inf. Theory 58(3), 1783–1790 (2012).

  10. Kuz’min E.N.: Irreducible polynomials over a finite field and an analogue of gauss sums over a field of characteristic \(2\). Sib. Math. J. 32(6), 982–989 (1991).

  11. Lidl R., Niederreiter H.: Finite Fields Number 20 in Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1997).

  12. Niederreiter, H.: The multiple-recursive matrix method for pseudorandom number generation. Finite Fields Appl. 1(1), 3–30 (1995).

  13. Preneel B.: Introduction. In: Preneel B. (ed.) Fast Software Encryption. Lecture Notes in Computer Science, vol. 1008, pp. 1–5. Springer, Heidelberg (1995).

  14. Ram S.: Enumeration of linear transformation shift registers. Des. Codes Cryptogr. 75(2), 301–314 (2015).

  15. Stepanov S.A.: The number of irreducible polynomials of a given form over a finite field. Math. Notes Acad. Sci. USSR 41(3), 165–169 (1987).

  16. Tsaban B., Vishne U.: Efficient linear feedback shift registers with maximal period. Finite Fields Appl. 8(2), 256–267 (2002).

  17. Turnwald G.: A new criterion for permutation polynomials. Finite Fields Appl. 1(1), 64–82 (1995).

  18. Zeng G., Han W., He K.: High efficiency feedback shift register: \(\sigma \)-LFSR. http://eprint.iacr.org/2007/114 (2007).

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Acknowledgments

The authors thank the referees for pointing out the recent Reference [2] in which Cohen et al. proved Conjecture 2 for the odd characteristic case. In an earlier version, we gave a self-contained proof for the explicit number of irreducible polynomials of degree 3 in Section 3. We thank a referee for introducing us the Reference [17]. Applying the results about value sets, Section 3 is considerably shortened. This work was supported by the National Natural Science Foundation of China under Grants 61379139, 61502483 and 11526215, and the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant XDA06010701.

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Authors and Affiliations

  1. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, People’s Republic of China

    Yupeng Jiang

  2. Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Jiangshuai Yang

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  1. Yupeng Jiang
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  2. Jiangshuai Yang
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Correspondence to Yupeng Jiang.

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Communicated by D. Panario.

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Jiang, Y., Yang, J. On the number of irreducible linear transformation shift registers. Des. Codes Cryptogr. 83, 445–454 (2017). https://doi.org/10.1007/s10623-016-0240-5

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