Abstract
We extend a result of Ding and Helleseth on the autocorrelation of a cyclotomic generator in several ways. We define and analyze cyclotomic generators of arbitrary orders and over arbitrary finite fields, and we consider two, in general, different definitions of autocorrelation. Cyclotomic generators are closely related to the discrete logarithm. Hence, the results of this paper do not only describe interesting cryptographic properties of cyclotomic generators and their generalizations but also desirable features of the discrete logarithm.
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References
Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi sums. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1998)
Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic curves in cryptography. Reprint of the 1999 original. London Mathematical Society Lecture Note Series, vol. 265. Cambridge University Press, Cambridge (2000)
Burgess, D.A.: On Dirichlet characters of polynomials. Proc. London Math. Soc. 13(3), 537–548 (1963)
Cusick, T.W., Ding, C., Renvall, A.: Stream ciphers and number theory. North- Holland Mathematical Library, vol. 55. North-Holland Publishing Co., Amsterdam (1998)
Dai, Z., Yang, J., Gong, G., Wang, P.: On the linear complexity of generalized Legendre sequence. Sequences and their applications (Bergen), pp. 145–153 (2001); Discrete Math. Theor. Comput. Sci. (Lond.). Springer, London (2002)
Davenport, H., Lewis, D.J.: Character sums and primitive roots in finite fields. Rend. Circ. Mat. Palermo 12(2), 129–136 (1963)
Ding, C., Helleseth, T., Shan, W.: On the linear complexity of Legendre sequences. IEEE Transactions on Information Theory 44, 1276–1278 (1998)
Ding, C., Helleseth, T.: On cyclotomic generator of order r. Inform. Process. Lett. 66(1), 21–25 (1998)
Helleseth, T.: On the crosscorrelation of m-sequences and related sequences with ideal autocorrelation. Sequences and their applications (Bergen), pp. 34–45 (2001); Discrete Math. Theor. Comput. Sci. (Lond.). Springer, London (2002)
Helleseth, T., Yang, K.: On binary sequences of period n = pm −1 with optimal autocorrelation. Sequences and their applications (Bergen), pp. 209–217 (2001); Discrete Math. Theor. Comput. Sci. (Lond.). Springer, London (2002)
Jungnickel, D.: Finite fields. Structure and arithmetics. Bibliographisches Institut, Mannheim (1993)
Konyagin, S., Lange, T., Shparlinski, I.: Linear complexity of the discrete logarithm. Designs, Codes, and Cryptography 28(2), 135–146 (2003)
Kyureghyan, G.M., Pott, A.: On the linear complexity of the Sidelnikov-Lempel- Cohn-Eastman sequences. Designs, Codes, and Cryptography 29, 149–164 (2003)
Lempel, A., Cohn, M., Eastman, W.L.: A class of balanced binary sequences with optimal autocorrelation properties. IEEE Trans. Information Theory IT-23(1), 38–42 (1977)
Lenstra, A.K., Verheul, E.R.: The XTR public key system. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 1–19. Springer, Heidelberg (2000)
Lenstra, A.K., Verheul, E.R.: An overview of the XTR public key system. In: Public-key cryptography and computational number theory, Warsaw, 2000, pp. 151–180. de Gruyter, Berlin (2001)
Lüke, H.D., Schotten, H.D., Hadinejad-Mahram, H.: Generalized Sidelnikov sequences with optimal autocorrelation properties. Electronic Letters 36(6), 525–527 (2000)
Meidl, W., Winterhof, A.: Lower bounds on the linear complexity of the discrete logarithm in finite fields. IEEE Transactions on Information Theory 47, 2807–2811 (2001)
Menezes, A.: Elliptic curve public key cryptosystems. In: Communications and Information Theory. The Kluwer International Series in Engineering and Computer Science, vol. 234. Kluwer Academic Publishers, Boston (1993)
Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of applied cryptography. In: With a foreword by Rivest, R.L. (ed.). CRC Press Series on Discrete Mathematics and its Applications. CRC Press, Boca Raton (1997)
Naranjani, A.M.: On Dirichlet characters of polynomials. Acta Arith. 43, 245–251 (1984)
Perelḿuter, G.I., Shparlinskiĭ, I.E.: Distribution of primitive roots in finite fields (Russian) Uspekhi Mat. Nauk. 45(1)(271), 185–186 (1990); translation in Russian Math. Surveys 45(1), 223–224 (1990)
Perron, O.: Bemerkungen über die Verteilung der quadratischen. Reste. Math. Z 56, 122–130 (1952)
Shparlinski, I.E.: Number Theoretic Methods in Cryptography. Birkhäuser, Basel (1999)
Shparlinski, I.E.: Cryptographic Applications of Analytic Number Theory. Birkhäuser, Basel (2003)
Sidelńikov, V.M.: Some k-valued pseudo-random sequences and nearly equidistant codes. Problems of Information Transmission 5(1), 12–16 (1969); translated from Problemy Peredači Informacii 5(1), 16–22 (1969) (Russian)
Tietäväinen, A.: Vinogradov’s method and some applications. In: Number theory and its applications, Ankara, 1996. Lecture Notes in Pure and Appl. Math., vol. 204, pp. 261–282. Dekker, New York (1999)
Winterhof, A.: On the distribution of powers in finite fields. Finite Fields Appl. 4(1), 43–54 (1998)
Winterhof, A.: Some estimates for character sums and applications. Des. Codes Cryptogr. 22(2), 123–131 (2001)
Winterhof, A.: Incomplete additive character sums and applications. Finite fields and applications, Augsburg 1999, pp. 462–476. Springer, Berlin (2001)
Winterhof, A.: Character sums, primitive elements, and powers in finite fields. J. Number Theory 91(1), 153–163 (2001)
Winterhof, A.: A note on the linear complexity profile of the discrete logarithm in finite fields (preprint)
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Meidl, W., Winterhof, A. (2004). On the Autocorrelation of Cyclotomic Generators. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds) Finite Fields and Applications. Fq 2003. Lecture Notes in Computer Science, vol 2948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24633-6_1
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DOI: https://doi.org/10.1007/978-3-540-24633-6_1
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