Must the prime finite field, Fp, an elliptic curve is defined over always have a greater number of elements than the cardinality of an elliptic curve.
For example, If I have
Elliptic Curve defined by y^2 = x^3 + x + 1 and I can calculate that the
E.cardinality() = 14. Must the prime field I chose be >14?
Would there be any loss in security if I chose 11 as the prime field?
Would this be different if I chose the prime field beforehand?