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

though addition of integers is about the simplest group law that

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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