Prime Numbers

7.5 Counting points on elliptic curves 355
if(P1 + P2 == O) return 0; // #E = p +1− t with t ≡ 0(modl).
P3 = P0;
for(1 ≤ k ≤ l/2) {
if(X-coordinates of (P1 + P2) and P3 match) {
if(Y -coordinates also match) return k; // Y -coordinate check.
return l − k;
}
P3 = P3 + P0;
}
In the addition tests above for matching of some coordinate between (P1 +P2)
and P3, one is asking generally whether
(N1/D1,YM1/C1)+(N2/D2,YM2/C2) =(N3/D3,YM3/C3),
and such a relation is to be checked, of course, using the usual elliptic addition
rules. The polynomial P1 + P2 on the left can be combined—using the elliptic
rules of Algorithm 7.2.2, with the coordinates in that algorithm being now, of
course, our polynomial ratios—into polynomial form (N ′ /D ′ ,YM ′ /C ′ ), and
this is compared with (N3/D3,YM3/C3).Forsuchcomparisoninturnone
checks whether the cross products (N3D ′ − N ′ D3) and(M3C ′ − M ′ C3) both
vanish mod (Ψl,p). As for the check on whether P1 + P2 = O, we are asking
whether M1/C1 = −M2/C2, and this is also an easy cross product relation.
The idea is that the entire implementation we are describing involves only
polynomial multiplication and the mod (Ψl,p) reductions throughout. And
as we have mentioned, both polynomial multiply and mod can be made quite
efficient.
In case an attempt is made by the reader to implement Algorithm 7.5.6,
we give here some small cases within the calculation, for purpose of, shall we
say, “algorithm debugging.” For p = 101 and the curve
Y 2 = X 3 +3X +4
over Fp, the algorithm gives, for l selections l =2, 3, 5, 7, the results t mod 2 =
0, t mod 3 = 1, t mod 5 = 0, t mod 7 = 3, from which we infer #E = 92.
(We might have skipped the prime l = 5, since the product of the other primes
exceeds 4 √ p.) Along the way we have, for example,
X p2
,Y p2
[2](X, Y )=
Ψ3 =98+16X +6X 2 + X 4 ,
= 32 + 17X +13X 2 +92X 3 , Y(74 + 96X +14X 2 +68X 3 ) ,
2
12 + 53X +89X
16 + 12X +4X3 , Y 74 + 10X +5X2 +64X3 27 + 91X +96X2 +37X3
,
(X p ,Y p )= 70 + 61X +83X 2 +44X 3 , Y(43 + 76X +21X 2 +25X 3 ) ,
where it will be observed that every polynomial appearing in the point
coordinates has been reduced mod (Ψ3,p). (Note that p in Step [Analyze