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June 2021 On elements of large order on elliptic curves and multiplicative dependent images of rational functions over finite fields
Bryce Kerr, Jorge Mello, Igor E. Shparlinski
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Illinois J. Math. 65(2): 499-514 (June 2021). DOI: 10.1215/00192082-9043478

Abstract

Let E1 and E2 be elliptic curves in Legendre form with integer parameters. We show there exists a constant C such that for almost all primes, for all but at most C pairs of points on the reduction of E1×E2 modulo p having equal x coordinate, at least one among P1 and P2 has a large group order. We also show similar abundance over finite fields of elements whose images under the reduction modulo p of a finite set of rational functions have large multiplicative orders.

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Bryce Kerr. Jorge Mello. Igor E. Shparlinski. "On elements of large order on elliptic curves and multiplicative dependent images of rational functions over finite fields." Illinois J. Math. 65 (2) 499 - 514, June 2021. https://doi.org/10.1215/00192082-9043478

Information

Received: 10 August 2020; Revised: 21 January 2021; Published: June 2021
First available in Project Euclid: 7 April 2021

Digital Object Identifier: 10.1215/00192082-9043478

Subjects:
Primary: 11C08
Secondary: 11G05 , 11G07

Rights: Copyright © 2021 by the University of Illinois at Urbana–Champaign

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Vol.65 • No. 2 • June 2021
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