Elliptic Group Modulo p

denotes the elliptic Group modulo whose elements are and together with the pairs of Integers with satisfying

 (1)

with and Integers such that
 (2)

Given , define
 (3)

The Order of is given by
 (4)

where is the Legendre Symbol, although this Formula quickly becomes impractical. However, it has been proven that
 (5)

Furthermore, for a Prime and Integer in the above interval, there exists and such that
 (6)

and the orders of elliptic Groups mod are nearly uniformly distributed in the interval.