On elements of large order of elliptic curves and multiplicative\n dependent images of rational functions over finite fields

2020/08/02 by Bryce Kerr, Jorge Mello, Kerr, Bryce +3
Computer Science · Mathematics · Social Sciences · #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Historical and Political Studies #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2008.00433

openalex publication_date 2020/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E₁ and E₂ be elliptic curves in Legendre form with integer\nparameters. We show there exists a constant C such that for almost all\nprimes, for all but at most C pairs of points on the reduction of E₁ \×\nE₂ modulo p having equal x coordinate, at least one among P₁ and P₂\nhas a large group order. We also show similar abundance over finite fields of\nelements whose images under the reduction modulo p of a finite set of\nrational functions have large multiplicative orders\n

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