  Let $E_1$ and $E_2$ 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 $E_1 \times
E_2$ modulo $p$ having equal $x$ coordinate, at least one among $P_1$ and $P_2$
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
