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)


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.


Elliptic Curve Multiple Root Elliptic Curve Cryptography Random Integer Quadratic Residue 
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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

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