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ELLIPTIC CURVES 107
rarely write it like that. So the correct answer is that
E(F
5
) ={(0, 1), (0, 4), (2, 3), (2, 2), (4, 0), O},
and
E(F
7
) ={(0, 1), (0, 6), (1, 3), (1, 4), (2, 3), (2, 4), (3, 0),
(4, 3), (4, 4), (5, 0), (6, 0), O}.
What is the reason to add this extra solution? We need it in order
to make the set of solutions to the Z-equation into a group. In fact,
O is the neutral element for the group law.
The Group Law on an Elliptic Curve
What is the deﬁnition of the group operation? It is pretty compli-
cated, because it has a lot of parts. Let P = (x
1
, y
1
) and Q = (x
2
, y
2
)
be two points in E(R), for some ﬁeld R. We are going to tell you
how to combine them. Even though what we are going to do is
much more complicated algorithmically than anything we have
done so far, mathematicians still just use the symbol + to stand
for the group law on an elliptic curve. We use this symbol even
there is and combining points on an elliptic curve may be the most
complicated group law that you will ever encounter. The symbol
+ reminds us that the commutative law holds when we combine
points on an elliptic curve.
This is a group law that is easily worked on a computer. What is
the rule for combining two points P and Q on an elliptic curve E to
get another point on that same elliptic curve?
1. If P = O, then P + Q = Q.
2. If Q = O, then P + Q = P.
3. If x
1
= x
2
and y
1
+ y
2
= 0, then P + Q = O. Here, y
1
+ y
2
= 0
means equality in the ﬁeld R, and similarly in the
remainder of our description.
4. If x
1
= x
2
, then compute the numbers λ =
y
2
y
1
x
2
x
1
and
ν = y
1
λx
1
. (As usual, the arithmetic operations take

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