The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 70

This yields a bijection between triples in Z n 1 n 2 and pairs of triples, one in
Z n 1 and one in Z n 2 . It is not hard to see that primitive triples for Z n 1 n 2
correspond to pairs of primitive triples in Z n 1 and Z n 2 .Moreover,

y 2 z ≡ x 3 + Axz 2 + Bz 3

(mod n 1 n 2 )

y 2 z

x 3 + Axz 2 + Bz 3

≡ (mod n 1 )

⇐⇒

y 2 z x 3 + Axz 2 + Bz 3

≡ (mod n 2 )

Therefore, there is a bijection

ψ : E ( Z n 1 n 2 ) −→ E ( Z n 1 ) ⊕ E ( Z n 2 ) .

It remains to show that ψ is a homomorphism. Let P 1 ,P 2 ∈ E ( Z n 1 n 2 )andlet

P 3 = P 1 + P 2 . This means that there is a linear combination of the outputs
of formulas I, II, III that is primitive and yields P 3 . Reducing all of these
calculations mod n i (for i =1 , 2) yields exactly the same result, namely the
primitive point P 3 (mod n i )isthesumof P 1 (mod n i )and P 2 (mod n i ).
This means that ψ ( P 3 )= ψ ( P 1 )+ ψ ( P 2 ), so ψ is a homomorphism.

COROLLARY 2.33
Let E be an elliptic curve over Q given by

y 2 = x 3 + Ax + B

with A, B ∈ Z .Let n be a positive odd integer such that gcd( n, 4 A 3 +27 B 2 )=
1 .Representthe elem entsof E ( Q ) as primitive triples ( x : y : z ) ∈ P 2 ( Z ) .
Themap

red n : E ( Q )

−→ E ( Z n )

( x : y : z ) → ( x : y : z )(mod n )

is a group hom om orphism .

PROOF E ( Q )and P 1 + P 2 = P 3 ,then P 3 is a primitive point

that can be expressed as a linear combination of the outputs of formulas I, II,
III. Reducing all of the calculations mod n yields the result.

If P 1 ,P 2 ∈

Corollary 2.33 can be generalized as follows.

COROLLARY 2.34
Let R be a ring and let I be an ideal of R . A ssu m e that both R and R/I
satisfy conditions (1) and (2) on page 66. Let E be given by

y 2 z = x 3 + Axz 2 + Bz 3