2.6. Elliptic curves over finite fields 35

2.6. Elliptic curves over finite fields

Let p ≥ 2 be a prime and let Fp be the finite field with p elements,

i.e.,

Fp = Z/pZ = {a mod p : a = 0,1,2,...,p − 1}.

Fp is a field and we may consider elliptic curves defined over Fp. As

for elliptic curves over Q, there are two conditions that need to be

satisfied: the curve needs to be given by a cubic equation, and the

curve needs to be smooth.

Example 2.6.1. For instance, E :

y2

≡

x3

+ 1 mod 5 is an ellip-

tic curve defined over F5. It is clearly given by a cubic equation

(zy2

≡

x3 +z3

mod 5 in the projective plane

P2(F5))

and it is smooth,

because for F ≡

zy2

−

x3

−

z3

mod 5, the partial derivatives are:

∂F

∂x

≡

−3x2,

∂F

∂y

≡ 2yz,

∂F

∂z

≡

y2

−

3z2

mod 5.

Thus, if the partial derivatives are congruent to 0 modulo 5, then

x ≡ 0 mod 5 and yz ≡ 0 mod 5. The latter congruence implies that

y or z ≡ 0 mod 5, and ∂F/∂z ≡ 0 implies that y ≡ z ≡ 0 mod 5.

Since [0,0,0] is not a point in the projective plane, we conclude that

there are no singular points on E/F5.

However, C/F3 :

y2

≡

x3

+ 1 mod 3 is not an elliptic curve be-

cause it is not smooth. Indeed, the point P = (2 mod 3,0 mod 3) ∈

C(F3) is a singular point:

∂F

∂x

(P) ≡ −3 ·

22

≡ 0,

∂F

∂y

(P) ≡ 2 · 0 · 1 ≡ 0, and

∂F

∂z

(P) ≡

02

− 3 ·

12

≡ 0 mod 3.

Let E/Q be an elliptic curve given by a Weierstrass equation

y2 = x3 + Ax + B with integer coeﬃcients A, B ∈ Z, and let p ≥ 2

be a prime number. If we reduce A and B modulo p, then we obtain

the equation of a curve E given by a cubic curve and defined over

the field Fp. Even though E is smooth as a curve over Q, the curve

E may be singular over Fp. In the previous example, we saw that

E/Q :

y2

=

x3

+ 1 is smooth over Q and F5 but it has a singularity

over F3. If the reduction curve E is smooth, then it is an elliptic

curve over Fp.