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?

  • $\begingroup$ You can't calculate a cardinality before choosing the finite field, do it is quite unclear how you got to the number 14. The difference between the number of elements in the field and in the curve is bound by Hasse's theorem (about the square root of the numbers). $\endgroup$ – tylo Sep 16 '19 at 15:05
  • $\begingroup$ I should've been more clear. If I choose a prime field to begin with, should I seek out curve parameters such that the cardinality is less that the size of the field? Will there be an issue with having the cardinality > finite field size? $\endgroup$ – Ellthegiant Sep 16 '19 at 15:15