The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 11

In most of this topic, we will develop the theory using the Weierstrass
equation, occasionally pointing out what modifications need to be made in
characteristics 2 and 3. In Section 2.8, we discuss the case of characteristic 2 in
more detail, since the formulas for the (nongeneralized) Weierstrass equation
do not apply. In contrast, these formulas are correct in characteristic 3 for
curves of the form y 2 = x 3 + Ax + B , but there are curves that are not of
this form. The general case for characteristic 3 can be obtained by using the
present methods to treat curves of the form y 2 = x 3 + Cx 2 + Ax + B .
Finally, suppose we start with an equation

cy 2 = dx 3 + ax + b

with c, d = 0. Multiply both sides of the equation by c 3 d 2 to obtain

( c 2 dy ) 2 =( cdx ) 3 +( ac 2 d )( cdx )+( bc 3 d 2 ) .

The change of variables

y 1 = c 2 dy, x 1 = cdx

yields an equation in Weierstrass form.
Later in this chapter, we will meet other types of equations that can be
transformed into Weierstrass equations for elliptic curves. These will be useful
in certain contexts.
For technical reasons, it is useful to add a point at infinity to an elliptic
curve. In Section 2.3, this concept will be made rigorous. However, it is
easiest to regard it as a point ( ∞,∞ ), usually denoted simply by ∞ , sitting
at the top of the y -axis. For computational purposes, it will be a formal
symbol satisfying certain computational rules. For example, a line is said to
pass through ∞ exactly when this line is vertical (i.e., x =constant). The
point ∞ might seem a little unnatural, but we will see that including it has
very useful consequences.
We now make one more convention regarding . It not only is at the top of

the y -axis, it is also at the bottom of the y -axis. Namely, we think of the ends
of the y -axis as wrapping around and meeting (perhaps somewhere in the back
behind the page) in the point

∞

. This might seem a little strange. However,
if we are working with a field other than the real numbers, for example, a
finite field, then there might not be any meaningful ordering of the elements
and therefore distinguishing a top and a bottom of the y -axis might not make
sense. In fact, in this situation, the ends of the y -axis do not have meaning
until we introduce projective coordinates in Section 2.3. This is why it is best
to regard ∞ as a formal symbol satisfying certain properties. Also, we have
arranged that two vertical lines meet at ∞ . By symmetry, if they meet at the
top of the y -axis, they should also meet at the bottom. But two lines should
intersect in only one point, so the “top ∞ ” and the “bottom ∞ ” need to be
the same. In any case, this will be a useful property of ∞ .

∞