denotes the elliptic Group modulo
whose elements are
and
together with the pairs of
Integers
with
satisfying
![\begin{displaymath}
y^2\equiv x^3+ax+b\ \left({{\rm mod\ } {p}}\right)
\end{displaymath}](e_826.gif) |
(1) |
with
and
Integers such that
![\begin{displaymath}
4a^3+27b^2\not\equiv 0\ ({\rm mod\ }p).
\end{displaymath}](e_827.gif) |
(2) |
Given
, define
![\begin{displaymath}
(x_i,y_i)\equiv (x_1,y_1)^i\ \left({{\rm mod\ } {p}}\right).
\end{displaymath}](e_828.gif) |
(3) |
The Order
of
is given by
![\begin{displaymath}
h=1+\sum_{x=1}^p \left[{\left({x^3+ax+b\over p}\right)+1}\right],
\end{displaymath}](e_829.gif) |
(4) |
where
is the Legendre Symbol, although this Formula quickly becomes impractical. However, it
has been proven that
![\begin{displaymath}
p+1-2\sqrt{p} \leq h(E(a,b)/p)\leq p+1+2\sqrt{p}.
\end{displaymath}](e_831.gif) |
(5) |
Furthermore, for
a Prime
and Integer
in the above interval, there exists
and
such that
![\begin{displaymath}
h(E(a,b)/p)=n,
\end{displaymath}](e_833.gif) |
(6) |
and the orders of elliptic Groups mod
are nearly uniformly distributed in the interval.
© 1996-9 Eric W. Weisstein
1999-05-25