Advertisement

A Result on the Distribution of Quadratic Residues with Applications to Elliptic Curve Cryptography

  • Muralidhara V.N.
  • Sandeep Sen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4859)

Abstract

In this paper, we prove that for any polynomial function f of fixed degree without multiple roots, the probability that all the (f(x + 1), f(x + 2), ..., f(x + κ)) are quadratic non-residue is \(\approx \frac{1}{2^\kappa}\). In particular for f(x) = x 3 + ax + b corresponding to the elliptic curve y 2 = x 3 + ax + b, it implies that the quadratic residues (f(x + 1), f(x + 2), ... in a finite field are sufficiently randomly distributed. Using this result we describe an efficient implementation of El-Gamal Cryptosystem. that requires efficient computation of a mapping between plain-texts and the points on the elliptic curve.

Keywords

Elliptic Curve Multiple Root Elliptic Curve Cryptography Random Integer Quadratic Residue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences 1: Measure of pseudorandomness, the Legendre symbol. Acta Arith. 82, 365–377 (1997)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Peralta, R.: On the distribution of quadratic residues and nonresidues modulo a prime number. Mathematics of Computation 58(197), 433–440 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Babai, L., G’al, A., Koll’ar, J., R’onyai, L., Szab’o, T., Wigderson, A.: Extremal Bipartite Graphs and Superpolynomial Lowerbounds for Monotone Span Programs. In: Proc. ACM STOC 1996, pp. 603–611 (1996)Google Scholar
  4. 4.
    Gallant, R., Lambert, R., Vanstone, S.: Improving the parallelized Pollard lambda search on binary anomalous curves. Mathematics of Computation 69, 1699–1705 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Koblitz, N.: A Course in Number theory and Cryptography. Springer, New York (1994)zbMATHGoogle Scholar
  6. 6.
    Lidl, R., Niederreiter, H., Cohn, P.M.: Encyclopedia of Mathematics and its Applications20-Finite Fields. Cambridge University Press, Cambridge (1997)Google Scholar
  7. 7.
    Menezes, A.: Elliptic Curve Public Key Cryptosystems. Kluwer Academic Publishers, Dordrecht (1996)Google Scholar
  8. 8.
    Pollard, J.: Monte Carlo methods for index computation mod p. Mathematics of computation 32, 918–924 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Radhakrishnan, J.: Private CommunicationGoogle Scholar
  10. 10.
    Van Oorschot, P., Wiener, M.: Parallel collision search with cryptanalytic applications. Journal of Cryptology 12, 1–28 (1999)zbMATHCrossRefGoogle Scholar
  11. 11.
    Wiener, M., Zuccherato, R.: Faster attacks on elliptic curve cryptosystems. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 190–200. Springer, Heidelberg (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Muralidhara V.N.
    • 1
  • Sandeep Sen
    • 1
  1. 1.Department of Computer Science and Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110 016India

Personalised recommendations