Prime Numbers

384 Chapter 7 ELLIPTIC CURVE ARITHMETIC
neither floating-point class-polynomial calculations nor massive polynomial
storage nor sophisticated root-finding routines would be required.
7.32. There is a way to simplify somewhat the elliptic curve computations
for ECPP. Argue that Montgomery parameterization (as in Algorithm 7.2.7)
can certainly be used for primality proofs of some candidate n in the
ECPP Algorithms 7.6.2 or 7.5.9, provided that along with the conditions of
nonvanishing for multiples (X ′ ,Z ′ )=[m/q](X, Z), we always check gcd(Z ′ ,n)
for possible factors of n.
Describe, then, some enhancements to the ECPP algorithms that we enjoy
when Montgomery parameterization is in force. For example, finding a point
on a curve is simpler, because we only need a valid x-coordinate, and so on.
7.33. Here is a peculiar form of “rapid ECPP” that can—if one has sufficient
luck—work to effect virtually instantaneous primality proofs. Recall, as in
Corollary 4.1.4, that if a probable prime n has n − 1=FR where the factored
part F exceeds √ n (or in various refinements exceeds an even lesser bound),
then a primality proof can be effected quickly. Consider instead a scenario in
which the same “FR” decomposition is obtained, but we are lucky to be able
to write
R = αF + β,
with a representation 4α = β 2 + |D|γ 2 existing for fundamental discriminant
−|D|. Show that, under these conditions, if n is prime, there then exists a
CM curve E for discriminant −|D|, with curve order given by the attractive
relation
#E = αF 2 .
Thus, we might be able to have F nearly as small as n 1/4 , and still effect an
ECPP result on n.
Next, show that a McIntosh–Wagstaff probable prime of the form n =
(2 q +1)/3 always has a representation with discriminant D = −8, and give the
corresponding curve order. Using these ideas, prove that (2 313 +1)/3 isprime,
taking account of the fact that the curve order in question is #E =(2/3)h 2 ,
where h is
3 2 ·5·7·13 2 ·53·79·157·313·1259·1613·2731·3121·8191·21841·121369·22366891.
Then prove another interesting corollary: If
n =2 2r+2m +2 r+m+1 +2 2r +1
is prime, then the curve E in question has
#E =2 2r (2 2m +1).
In this manner, and by analyzing the known algebraic factors of 2 2m +1 when
m is odd, prove that
n =2 576 +2 289 +2 2 +1