Construction of Pseudo-random Binary Sequences from Elliptic Curves by Using Discrete Logarithm

  • Zhixiong Chen
  • Shengqiang Li
  • Guozhen Xiao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4086)


An upper bound is established for certain exponential sums with respect to multiplicative characters defined on the rational points of an elliptic curve over a prime field. The bound is applied to investigate the pseudo-randomness of a large family of binary sequences generated from elliptic curves by using discrete logarithm. That is, we use this estimate to show that the resulting sequences have the advantages of ‘small’ well-distribution measure and ‘small’ multiple correlation measure.


Elliptic Curve Rational Point Elliptic Curf Binary Sequence Discrete Logarithm 
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  1. 1.
    Beelen, P.H.T., Doumen, J.M.: Pseudorandom Sequences from Elliptic Curves. In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 37–52. Springer, Heidelberg (2002)Google Scholar
  2. 2.
    Bombieri, E.: On Exponential Sums in Finite Fields. Amer. J. Math. 88, 71–105 (1966)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cassaigne, J., Mauduit, C., Sárközy, A.: On Finite Pseudorandom Binary Sequences, VII: The Measures of Pseudorandomness. Acta Arithmetica 103, 97–118 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Enge, A.: Elliptic Curves and Their Applications to Cryptography: an Introduction. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  5. 5.
    Gong, G., Berson, T., Stinson, D.: Elliptic Curve Pseudorandom Sequence Generator. Technical Reports, No. CORR1998-53 (1998), Available at:
  6. 6.
    Gong, G., Lam, C.Y.: Linear Recursive Sequences over Elliptic Curves. In: Proceedings of Sequences and Their Applications-SETA 2001. DMTCS series, pp. 182–196. Springer, Heidelberg (2001)Google Scholar
  7. 7.
    Goubin, L., Mauduit, C., Sárközy, A.: Construction of Large Families of Pseudorandom Binary Sequences. J. Number Theory 106(1), 56–69 (2004)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gyarmati, K.: On a Family of Pseudorandom Binary Sequences. Periodica Mathematica Hungarica 49(2), 45–63 (2004)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hallgren, S.: Linear Congruential Generators over Elliptic Curves. Technical Report, No. CS-94-143, Cornegie Mellon University (1994)Google Scholar
  10. 10.
    Hess, F., Shparlinski, I.E.: On the Linear Complexity and Multidimensional Distribution of Congruential Generators over Elliptic Curves. Designs, Codes and Cryptography 35(1), 111–117 (2005)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kohel, D., Shparlinski, I.E.: On Exponential Sums and Group Generators for Elliptic Curves over Finite Fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 395–404. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  12. 12.
    Lachaud, G.: Artin-Schreier Curves, Exponential Sums and the Carlitz-Uchiyama Bound for Geometric Codes. J. Number Theory 39(1), 18–40 (1991)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lam, C.Y., Gong, G.: Randomness of Elliptic Curve Sequences. Technical Reports, No. CORR 2002-18 (2002), Available at:
  14. 14.
    Lange, T., Shparlinski, I.E.: Certain Exponential Sums and Random Walks on Elliptic Curves. Canad. J. Math. 57(2), 338–350 (2005)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Lee, L., Wong, K.: An Elliptic Curve Random Number Generator. In: Communications and Multimedia Security Issues of the New Century, Fifth Joint Working Conference on Communications and Multimedia Security-CMS 2001, pp. 127–133 (2001)Google Scholar
  16. 16.
    Mauduit, C., Rivat, J., Sárközy, A.: Construction of Pseudorandom Binary Sequences Using Additive Characters. Mh. Math. 141(3), 197–208 (2004)MATHCrossRefGoogle Scholar
  17. 17.
    Mauduit, C., Sárközy, A.: On Finite Pseudorandom Binary Sequences I: Measures of Pseudorandomness, the Legendre Symbol. Acta Arithmetica 82, 365–377 (1997)MATHMathSciNetGoogle Scholar
  18. 18.
    Mauduit, C., Sárközy, A.: On Finite Pseudorandom Binary Sequences II: The Champernowne, Rudin-Shapiro, and Thue-Morse Sequences, A Further Construction. J. Number Theory 73(2), 256–276 (1998)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    El Mahassni, E., Shparlinski, I.E.: On the Uniformity of Distribution of Congruential Generators over Elliptic Curves. In: Proc. Intern. Conf. on Sequences and Their Applications-SETA 2001, pp. 257–264. Springer, Heidelberg (2002)Google Scholar
  20. 20.
    Perret, M.: Multiplicative Character Sums and Nonlinear Geometric Codes. In: Charpin, P., Cohen, G. (eds.) EUROCODE 1990. LNCS, vol. 514, pp. 158–165. Springer, Heidelberg (1991)Google Scholar
  21. 21.
    Perret, M.: Multiplicative Character Sums and Kummer Coverings. Acta Arithmetica 59, 279–290 (1991)MATHMathSciNetGoogle Scholar
  22. 22.
    Shparlinski, I.E.: On the Naor-Reingold Pseudo-random Number Function from Elliptic Curves. Appl. Algebra Engng. Comm. Comput. 11(1), 27–34 (2000)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Shparlinski, I.E.: Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness. In: Progress in Computer Science and Applied Logic, vol. 22, Birkhauser, Basel (2003)Google Scholar
  24. 24.
    Shparlinski, I.E., Silverman, J.H.: On the Linear Complexity of the Naor-Reingold Pseudo-random Function from Elliptic Curves. Designs Codes and Cryptography 24(3), 279–289 (2001)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Vlăduţ, S.G.: Cyclicity Statistics for Elliptic Curves over Finite Fields. Finite Fields and Their Applications 5(1), 13–25 (1999)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Vlăduţ, S.G.: On the Cyclicity of Elliptic Curves over Finite Field Extensions. Finite Fields and Their Applications 5(3), 354–363 (1999)MATHMathSciNetGoogle Scholar
  27. 27.
    Voloch, J.F., Walker, J.L.: Euclidean Weights of Codes from Elliptic Curves over Rings. Trans. Amer. Math. Soc. 352(11), 5063–5076 (2000)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Zhixiong Chen
    • 1
    • 2
  • Shengqiang Li
    • 1
    • 3
  • Guozhen Xiao
    • 1
  1. 1.National Key Lab. of I.S.NXidian Univ.Xi’anChina
  2. 2.Depart. of Math.Putian Univ.Putian, FujianChina
  3. 3.University of Electronic Science and Technology of ChinaChengduChina

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