Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 212

Such an f exists since there is a unique quadratic polynomial whose graph
passes through any three points that have distinct x -coordinates. In fact

1
( e 1 − e 2 )( e 1 − e 3 ) ( T

f ( T )= v 1 − e 2 )( T − e 3 )

1

+ v 2

e 3 ) ( T − e 1 )( T − e 3 )

( e 2 − e 1 )( e 2 −

1
( e 3 − e 1 )( e 3 − e 2 ) ( T

+ v 3 − e 1 )( T − e 2 ) .

f ( T ) 2 .Then g ( e i ) = 0 for all i ,so

Let g ( T )= x − T −

T 3 + AT + B =( T

− e 1 )( T − e 2 )( T − e 3 ) divides g ( T ) .

Therefore, g ( T ) ≡ 0(mod T 3 + AT + B ), so

x − T ≡ ( u 0 + u 1 T + u 2 T 2 ) 2 (mod T 3 + AT + B ) .

(We say that two polynomials P 1 ,P 2 are congruent mod P 3 if P 1 − P 2 is a
multiple of P 3 .) This congruence for x − T can be thought of as a way of
simultaneously capturing the information that x − e i is a square for all i .
Mod T 3 + AT + B ,wehave

T 3 T 4 ≡ T · T 3 ≡−AT 2

≡−AT − B, − BT.

Therefore,

x − T ≡ ( u 0 + u 1 T + u 2 T 2 ) 2
≡ u 0 +2 u 0 u 1 T +( u 1 +2 u 0 u 2 ) T 2 +2 u 1 u 2 T 3 + u 2 T 4
≡

( u 0 − Bu 2 ) T 2 Bu 1 u 2 )+(2 u 0 u 1 − 2 Au 1 u 2 −

+( u 1 +2 u 0 u 2 − Au 2 ) T 2 .

If two polynomials P 1 and P 2 of degree at most two are congruent mod a
polynomial of degree three, then their difference P 1 − P 2 is a polynomial of

degree at most two that is divisible by a polynomial of degree three. This can
only happen if P 1 = P 2 . In our case, this means that

x = u 0 −

2 Bu 1 u 2 (8.1)

Bu 2 − 1=2 u 0 u 1 − 2 Au 1 u 2 − (8.2)

0= u 1 +2 u 0 u 2 − Au 2 . (8.3)

If u 2 = 0 then (8.3) implies that also u 1 =0. Then f ( T ) is constant, so
v 1 = v 2 = v 3 . This means that e 1 = e 2 = e 3 , contradiction. Therefore,

= 0. Multiply (8.3) by u 1 /u 2 and multiply (8.2) by 1 /u 2 , then subtract to
obtain

u 2

1
u 2

2

= u 1
u 2

3

+ A u 1
u 2

+ B.