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

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 x 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.