Cryptography Reference
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with A, B
R an d assu m e there exists r
R su ch that
(4 A 3 +27 B 2 ) r
1
I.
Then the m ap
red I : E ( R )
−→
E ( R/I )
( x : y : z )
( x : y : z )mod I
is a group hom om orphism .
PROOF The proof is the same as for Corollary 2.33, with R in place of
Z and mod I in place of mod n . The condition that (4 A 3 +27 B 2 ) r − 1 ∈ I
for some r is the requirement that 4 A 3 +27 B 2 is a unit in R/I ,whichwas
required in the definition of an elliptic curve over the ring R/I .
Exercises
2.1
(a) Show that the constant term of a monic cubic polynomial is the
negative of the product of the roots.
(b) Use (a) to derive the formula for the sum of two distinct points
P 1 ,P 2 in the case that the x -coordinates x 1 and x 2 are nonzero, as
in Section 2.2. Note that when one of these coordinates is 0, you
need to divide by zero to obtain the usual formula.
2.2 The point (3 , 5) lies on the elliptic curve E : y 2 = x 3
2, defined over
Q . Find a point (not
) with rational, nonintegral coordinates in ( Q ).
2.3 The points P =(2 , 9), Q =(3 , 10), and R =(
4 ,
3) lie on the elliptic
curve E : y 2 = x 3 + 73.
(a) Compute P + Q and ( P + Q )+ R .
(b) Compute Q + R and P +( Q + R ). Your answer for P +( Q + R )
should agree with the result of part (a). However, note that one
computation used the doubling formula while the other did not use
it.
2.4 Let E be the elliptic curve y 2
= x 3
34 x + 37 defined over Q .Let
P =(1 , 2) and Q =(6 , 7).
(a) Compute P + Q .
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