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Correlation of m-Sequences and Related Topics

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Sequences and their Applications

Part of the book series: Discrete Mathematics and Theoretical Computer Science ((DISCMATH))

Abstract

We present a survey of many connections between the auto and crosscorrelation function of m-sequences and related topics. We show connections to perfect (PN) and almost perfect (APN) nonlinear power mappings which are of interest in differential cryptanalysis. Further, we give connections to difference sets, sequences with two-level ideal auto-correlation and families of sequences with optimal correlation properties, as well as to hyperovals and permutation polynomials.

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References

  1. T. Beth and C. Ding, On almost perfect nonlinear permutations, Advances in Cryptology - EUROCRYPT’93. Lecture Notes in Computer Science, Vol. 765, pp. 65–76, Springer-Verlag, 1994.

    MathSciNet  Google Scholar 

  2. A. Canteaut, P. Charpin and H. Dobbertin, Binary m-sequences with three-valued crosscorrelation: A proof of Welch’s conjecture, preprint, 1998.

    Google Scholar 

  3. F. Chabaud and S. Vaudenay, Links between differential and linear cryptanalysis, Advances in Cryptology - EUROCRYPT’94. Lecture Notes in Computer Science, Vol. 950, pp. 356–365, Springer-Verlag, 1995.

    MathSciNet  Google Scholar 

  4. A. Chang, S.W. Golomb, G. Gong and P.V. Kumar, Trace expansion and linear span of ideal autocorrelation sequences associated to the Segre hyperoval, preprint, May 1998.

    Google Scholar 

  5. R. S. Coulter and R. Matthews, Planar functions and planes of the Lenz-Barlotti class II, Designs, Codes and Cryptography, Vol. 10, pp. 165–185, 1997.

    Article  MathSciNet  Google Scholar 

  6. T. W. Cusick, Constructing differentially uniform permutations via crosscorrelation functions, unpublished manuscript.

    Google Scholar 

  7. T. W. Cusick and H. Dobbertin, Some new three-valued crosscorrelation functions for binary m-sequences, IEEE Trans. Inform. Theory, Vol. 42, pp. 1238–1240, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  8. . J.F. Dillon, Multiplicative difference sets via additive characters, to appear in Designs, Codes and Cryptography.

    Google Scholar 

  9. J.F. Dillon and H. Dobbertin, New cyclic difference sets with Singer parameters, preprint, February 1999.

    Google Scholar 

  10. H. Dobbertin, Almost perfect nonlinear power functions on GF(2n), preprint, July 1997.

    Google Scholar 

  11. R. Evans, H. Hollmann, C. Krattenthaler and Q. Xiang, Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets, submitted for publication.

    Google Scholar 

  12. R. Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions, IEEE Trans. Inform. Theory, Vol. 14, pp. 154–156, 1968.

    Article  MATH  Google Scholar 

  13. T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discrete Math., Vol. 16, pp. 209–232, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  14. T. Helleseth, A note on the cross-correlation function between two binary maximal length linear sequences, Discrete Math., Vol. 23, pp. 301–307, 1978.

    MathSciNet  Google Scholar 

  15. T. Helleseth and P. Vijay Kumar, Sequences with low correlation, Chapter in: Handbook of Coding Theory (Eds. V.S. Pless and W. C. Huffman), North-Holland 1998.

    Google Scholar 

  16. T. Helleseth, P.Vijay Kumar and H. Martinsen, A new family of ternary sequences with ideal two-level autocorrelation function, submitted for publication.

    Google Scholar 

  17. T. Helleseth, C. Rong and D. Sandberg, New families of almost perfect nonlinear power mappings, IEEE Transactions on Inform. Theory, Vol. 45, pp. 475–485, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  18. T. Helleseth and D. Sandberg, Some power mappings with low differential uniformity, Applicable Algebra in Engineering, Communication and Computing, Vol. 8, pp. 363–370, 1997.

    MathSciNet  MATH  Google Scholar 

  19. T. Kasami, Weight distribution formula for some class of cyclic codes, Technical Report R-285 (AD 632574), Coordinated Science Laboratory, University of Illinois, Urbana, April 1966.

    Google Scholar 

  20. P. Kumar and O. Moreno, Prime-phase sequences with periodic correlation properties better than binary sequences, IEEE Trans. Inform. Theory, Vol. 37, pp. 603–616, 1991.

    Article  MATH  Google Scholar 

  21. G. Lachaud and J. Wolfmann, The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans. Inform. Theory Vol. 36, pp. 686–692, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Lidl and H. Niederreiter, Finite Fields. Vol. 20 Encyclopedia of Mathematics and its Applications, Reading, MA: Addison-Wesley, 1983.

    Google Scholar 

  23. A. Maschietti, Difference sets and hyperovals, Designs, Codes and Cryptography, Vol. 14, pp. 89–98, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  24. G. McGuire and A.R. Calderbank, Proof of a conjecture of Sarwate and Pursley regarding pairs of binary m-sequences, IEEE Trans. Inform. Theory, Vol. 41, pp. 1153–1155, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  25. J.S. No, H. Chung and M.S. Yun, Binary pseudorandom sequences of period 2’ —1 with ideal autocorrelation generated by the polynomial z d + ( z + 1) d, IEEE Trans. Inform. Theory, Vol. 44, pp. 1278–1282, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  26. J.S. No, S.W. Golomb, G. Gong, H.K. Lee and P. Gaal, Binary pseudorandom sequences of period 2’ —1 with ideal autocorrelation, IEEE Trans. Inform. Theory,Vol. 44, pp. 814–817, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  27. Y. Niho, Multi-valued cross-correlation functions between two maximal linear recursive sequences,. Ph.D. Thesis, University of Southern California, 1972.

    Google Scholar 

  28. K. Nyberg and L. R. Knudsen, Provable security against a differential attack, Journal of Cryptology, Vol. 8, pp. 27–37, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  29. K. Nyberg, Differentially uniform mappings for cryptography, Advances in Cryptology - EUROCRYPT’g3. Lecture Notes in Computer Science, Vol. 765, pp. 55–64, Springer-Verlag, 1994.

    MathSciNet  Google Scholar 

  30. D. Sandberg, On the differential uniformity of power mappings, Master Thesis, Department of Informatics, University of Bergen, 1997.

    Google Scholar 

  31. D.V. Sarwate and M.B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE 68, pp. 593–619, 1980.

    Article  Google Scholar 

  32. H. M. Trachtenberg, On the cross-correlation functions of maximal linear sequences, Ph. D. Thesis, University of Southern California, Los Angeles, 1970.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen Høyteknologisenteret, Bergen, N-5020, Norway

    Tor Helleseth

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  1. Tor Helleseth
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Editor information

Editors and Affiliations

  1. Department of Computer Science, School of Computing, National University of Singapore, Lower Kent Ridge Road, Singapore, 119260

    C. Ding

  2. Department of Informatics, University of Bergen, N-5020, Bergen, Norway

    T. Helleseth

  3. Institute of Information Processing, Austrian Academy of Sciences, Sonnenfelsgasse 19, A01010, Vienna, Austria

    H. Niederreiter

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© 1999 Springer-Verlag London

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Helleseth, T. (1999). Correlation of m-Sequences and Related Topics. In: Ding, C., Helleseth, T., Niederreiter, H. (eds) Sequences and their Applications. Discrete Mathematics and Theoretical Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-0551-0_4

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  • DOI: https://doi.org/10.1007/978-1-4471-0551-0_4

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-196-2

  • Online ISBN: 978-1-4471-0551-0

  • eBook Packages: Springer Book Archive

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