denotes the elliptic group modulo 
whose elements are 1 and 
together with the pairs of integers 
with 
satisfying
 |
(1)
|
with 
and 
integers such that
 |
(2)
|
Given 
,
define
 |
(3)
|
The group order 
of 
is given by
![h=1+sum_(x=1)^p[((x^3+ax+b)/p)+1],](/images/equations/EllipticGroupModulop/NumberedEquation4.svg) |
(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.
