Abstract
Cross-correlation functions are determined for a large class of geometric sequences based on m-sequences in odd characteristic. These sequences are shown to have low cross-correlation values in certain cases. They also have significantly higher linear spans than previously studied geometric sequences. These results show that geometric sequences are candidates for use in spread-spectrum communications systems in which cryptographic security is a factor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Antweiller and L. Bohmer, Complex sequences over GF(p M) with a two-level autocorrelation function and a large linear span, IEEE Trans. Inform. Theory, Vol. 38, (1992) pp. 120–130.
A. H. Chan, and R. Games, On the linear span of binary sequences from finite geometries, q odd, IEEE Trans. Inform. Theory, Vol. 36, (1990) pp. 548–552.
A.H. Chan, M. Goresky, and A. Klapper, Correlation functions of geometric sequences, Advances in Cryptology: Proc. Eurocrypt '90, (I. B. Damgard, ed.), Lecture Notes in Computer Science, Springer-Verlag, Berlin, 473 (1991) pp. 214–221.
S. Golomb, Shift Register Sequences, Aegean Park Press: Laguna Hills, CA (1982).
B. Gordon, W. H. Mills, and L. R. Welch, Some new difference sets, Canad. J. Math., Vol. 14 (1962) pp. 614–625.
C. Kim and J. Komo, A new family of binary sequences with large linear span, Proceedings of Finite Fields, Coding Theory, and Advances in Communications and Computing, (G. Mullen and P. Shiue, ed.), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 141 (1993) pp. 123–130.
A. Klapper. Cross-correlations of geometric sequences in characteristic two, Designs, Codes, and Cryptography, Vol. 3, (1993) pp. 347–377.
A. Klapper, A. H. Chan, and M. Goresky, Cross-correlations of linearly and quadratically related geometric sequences and GMW sequences, Discrete Applied Mathematics, Vol. 46, (1993) pp. 1–20.
A. Klapper, A. H. Chan, and M. Goresky, Cascaded GMW sequences, IEEE Trans. Inform. Theory, Vol. 39, (1993) pp. 177–183.
R. Lidl and H. Niederreiter, Finite fields, in Encyclopedia of Mathematics Cambridge University Press, Cambridge, Vol. 20 (1983).
R. McEliece Finite Fields for Computer Scientists and Engineers, Kluwer Academic Publishers, Boston (1987).
J. No and P. V. Kumar, A new family of binary pseudorandom sequences having optimal periodic correlation properties and large linear span, IEEE Trans. Inform. Theory, Vol. 35, (1989) pp. 371–379.
O. Rothaus, On bent functions, J. Combin. Theory, Ser. A, Vol. 20, (1976) pp. 300–305.
M. Simon, J. Omura, R. Scholtz, and B. Levitt, Spread-Spectrum Communications, Computer Science Press, Vol. 1 (1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Klapper, A. Cross-Correlations of Quadratic Form Sequences in Odd Characteristic. Designs, Codes and Cryptography 11, 289–305 (1997). https://doi.org/10.1023/A:1008250313089
Issue Date:
DOI: https://doi.org/10.1023/A:1008250313089