Skip to main content Accessibility help
×
Home
Hostname: page-component-568f69f84b-8fhp6 Total loading time: 0.168 Render date: 2021-09-19T03:14:09.883Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Article contents

EXPONENTIAL SUMS OVER POINTS OF ELLIPTIC CURVES WITH RECIPROCALS OF PRIMES

Published online by Cambridge University Press:  13 July 2011

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)

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.

Type
Research Article
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), 179197.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), 71105.CrossRefGoogle Scholar
[5]Bourgain, J., More on the sum–product phenomenon in prime fields and its applications. Int. J. Number Theory 1 (2005), 132.CrossRefGoogle Scholar
[6]Bourgain, J. and Shparlinski, I. E., Distribution of consecutive modular roots of an integer. Acta Arith. 134 (2008), 8391.CrossRefGoogle Scholar
[7]Chen, Z., Elliptic curve analogue of Legendre sequences. Monatsh. Math. 154 (2008), 110.CrossRefGoogle Scholar
[8]Davenport, H., Multiplicative Number Theory, 2nd edn., Springer (New York, 1980).CrossRefGoogle 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), 111119.CrossRefGoogle Scholar
[10]Farashahi, R. R. and Shparlinski, I. E., Pseudorandom bits from points on elliptic curves. Preprint, 2009.Google Scholar
[11]Fouvry, E. and Michel, P., Sur certaines sommes d’exponentielles sur les nombres premiers. Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), 93130.CrossRefGoogle Scholar
[12]Garaev, M. Z., An estimate of Kloosterman sums with prime numbers and an application. Mat. Zametki 88(3) (2010), 365373 (in Russian).Google Scholar
[13]Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society (Providence, RI, 2004).CrossRefGoogle 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), 395404.CrossRefGoogle Scholar
[15]Lange, T. and Shparlinski, I. E., Certain exponential sums and random walks on elliptic curves. Canad. J. Math. 57 (2005), 338350.CrossRefGoogle Scholar
[16]Lange, T. and Shparlinski, I. E., Distribution of some sequences of points on elliptic curves. J. Math. Cryptol. 1 (2007), 111.CrossRefGoogle Scholar
[17]Ostafe, A. and Shparlinski, I. E., Twisted exponential sums over points of elliptic curves. Acta Arith. 148 (2011), 7792.CrossRefGoogle Scholar
[18]Shparlinski, I. E., Bilinear character sums over elliptic curves. Finite Fields Appl. 14 (2008), 132141.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), 121141.CrossRefGoogle Scholar
[20]Shparlinski, I. E., Some special character sums over elliptic curves. Bol. Soc. Mat. Mexicana (3) 15 (2009), 3740.Google Scholar
[21]Silverman, J. H., The Arithmetic of Elliptic Curves, Springer (Berlin, 1995).Google Scholar
2
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

EXPONENTIAL SUMS OVER POINTS OF ELLIPTIC CURVES WITH RECIPROCALS OF PRIMES
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

EXPONENTIAL SUMS OVER POINTS OF ELLIPTIC CURVES WITH RECIPROCALS OF PRIMES
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

EXPONENTIAL SUMS OVER POINTS OF ELLIPTIC CURVES WITH RECIPROCALS OF PRIMES
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *