An elliptic group modulo a prime 
is the abelian group
of rational points on an elliptic
curve over the finite field 
. For integers 
and 
, let 
denote the group consisting
of the pairs of integers 
with 
satisfying
 |
(1)
|
together with one point at infinity 
, which is the identity element.
Assume
 |
(2)
|
For a positive integer 
and 
, the notation ![[i]P](/images/equations/EllipticGroupModulop/Inline11.svg)
denotes the sum of 
copies of 
under the group operation. When
![[i]P!=O](/images/equations/EllipticGroupModulop/Inline14.svg)
,
write
![[i]P=(x_i,y_i).](/images/equations/EllipticGroupModulop/NumberedEquation3.svg) |
(3)
|
These are coordinates in 
(Sutherland 2023a, p. 1). 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 (Sutherland 2023b, p. 2).
The group order satisfies
 |
(5)
|
For every integer 
in this interval, there exist 
and 
such that
 |
(6)
|
(Sutherland 2023b, pp. 1-2).
