Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T22:11:05.815Z Has data issue: false hasContentIssue false
  • English
  • Français

EXPONENTIAL SUMS OVER POINTS OF ELLIPTIC CURVES WITH RECIPROCALS OF PRIMES

Part of: Finite fields and commutative rings (number-theoretic aspects) Arithmetic algebraic geometry Exponential sums and character sums

Published online by Cambridge University Press:  13 July 2011

Alina Ostafe  and
Igor E. Shparlinski
Alina Ostafe
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190 CH-8057, Zürich, Switzerland (email: alina.ostafe@math.uzh.ch)
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: igor.shparlinski@mq.edu.au)
Get access

Abstract

We consider exponential sums with x-coordinates of points qG and q−1G where G is a point of order T on an elliptic curve modulo a prime p and q runs through all primes up to N (with gcd (q,T)=1 in the case of the points q−1G). We obtain a new bound on exponential sums with q−1G and correct an imprecision in the work of W. D. Banks, J. B. Friedlander, M. Z. Garaev and I. E. Shparlinski on exponential sums with qG. We also note that similar sums with g1/q for an integer g with gcd (g,p)=1 have been estimated by J. Bourgain and I. E. Shparlinski.

MSC classification

Secondary: 11L07: Estimates on exponential sums 11T23: Exponential sums 11G20: Curves over finite and local fields
Type
Research Article
Information
Mathematika , Volume 58 , Issue 1 , January 2012 , pp. 21 - 33
Copyright
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Avanzi, R., Cohen, H., Doche, C., Frey, G., Lange, T., Nguyen, K. and Vercauteren, F., Elliptic and Hyperelliptic Curve Cryptography: Theory and Practice, CRC Press (Boca Raton, FL, 2005).Google Scholar
[2]Banks, W. D., Friedlander, J. B., Garaev, M. Z. and Shparlinski, I. E., Double character sums over elliptic curves and finite fields. Pure Appl. Math. Q. 2 (2006), 179–197.CrossRefGoogle Scholar
[3]Blake, I., Seroussi, G. and Smart, N., Elliptic Curves in Cryptography (London Mathematical Society Lecture Note Series 265), Cambridge University Press (Cambridge, MA, 1999).CrossRefGoogle Scholar
[4]Bombieri, E., On exponential sums in finite fields. Amer. J. Math. 88 (1966), 71–105.CrossRefGoogle Scholar
[5]Bourgain, J., More on the sum–product phenomenon in prime fields and its applications. Int. J. Number Theory 1 (2005), 1–32.CrossRefGoogle Scholar
[6]Bourgain, J. and Shparlinski, I. E., Distribution of consecutive modular roots of an integer. Acta Arith. 134 (2008), 83–91.Google Scholar
[7]Chen, Z., Elliptic curve analogue of Legendre sequences. Monatsh. Math. 154 (2008), 1–10.Google Scholar
[8]Davenport, H., Multiplicative Number Theory, 2nd edn., Springer (New York, 1980).Google Scholar
[9]El Mahassni, E. and Shparlinski, I. E., On the distribution of the elliptic curve power generator. In Proceedings of the 8th International Conference on Finite Fields and Applications (Contemporary Mathematics 461), American Mathematical Society (Providence, RI, 2008), 111–119.CrossRefGoogle Scholar
[10]Farashahi, R. R. and Shparlinski, I. E., Pseudorandom bits from points on elliptic curves. Preprint, 2009.CrossRefGoogle Scholar
[11]Fouvry, E. and Michel, P., Sur certaines sommes d’exponentielles sur les nombres premiers. Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), 93–130.Google Scholar
[12]Garaev, M. Z., An estimate of Kloosterman sums with prime numbers and an application. Mat. Zametki 88(3) (2010), 365–373 (in Russian).Google Scholar
[13]Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society (Providence, RI, 2004).Google Scholar
[14]Kohel, D. R. and Shparlinski, I. E., Exponential sums and group generators for elliptic curves over finite fields. In Proceedings of the 4th Algorithmic Number Theory Symposium (Lecture Notes in Computer Science 1838), Springer (Berlin, 2000), 395–404.Google Scholar
[15]Lange, T. and Shparlinski, I. E., Certain exponential sums and random walks on elliptic curves. Canad. J. Math. 57 (2005), 338–350.CrossRefGoogle Scholar
[16]Lange, T. and Shparlinski, I. E., Distribution of some sequences of points on elliptic curves. J. Math. Cryptol. 1 (2007), 1–11.CrossRefGoogle Scholar
[17]Ostafe, A. and Shparlinski, I. E., Twisted exponential sums over points of elliptic curves. Acta Arith. 148 (2011), 77–92.CrossRefGoogle Scholar
[18]Shparlinski, I. E., Bilinear character sums over elliptic curves. Finite Fields Appl. 14 (2008), 132–141.CrossRefGoogle Scholar
[19]Shparlinski, I. E., Pseudorandom number generators from elliptic curves. In Recent Trends in Cryptography (Contemporary Mathematics 477), American Mathematical Society (Providence, RI, 2009), 121–141.Google Scholar
[20]Shparlinski, I. E., Some special character sums over elliptic curves. Bol. Soc. Mat. Mexicana (3) 15 (2009), 37–40.Google Scholar
[21]Silverman, J. H., The Arithmetic of Elliptic Curves, Springer (Berlin, 1995).Google Scholar