Prime Numbers

7.7 Exercises 377
(some Bj equals Ai). Give the computational complexity of this polynomial
method for finding A ∩ B. How does one handle duplicate matches in this
polynomial setting? Note the related material in Sections 5.5, 9.6.3.
7.14. By analyzing the trend of “record” ECM factorizations, estimate in
what calendar year we shall be able to discover 70-digit factors via ECM.
([Zimmermann 2000] has projected the year 2010, for example.)
7.15. Verify claims made in reference to Algorithm 7.5.10, as follows. First,
show how the tabulated parameters r, s were obtained. For this, one uses the
fact of the class polynomial being at most quadratic, and notes also that a
defining cubic y 2 = x 3 + Rx/S + T/S can be cleared of denominator S by
multiplying through by S 6 . Second, use quadratic reciprocity to prove that
every explicit square root in the tabulated parameters does, in fact, exist. For
this, one presumes that a representation 4p = u 2 +|D|v 2 has been found for p.
Third, show that 4a 3 +27b 2 cannot vanish (mod p). This could be done case
by case, but it is easier to go back to Algorithm 7.5.9 and see how the final a, b
parameters actually arise. Finally, factor the s values of the tabulated data
to verify that they tend to be highly smooth. How can this smoothness be
explained?
7.16. Recall that for elliptic curve Ea,b(Fp) atwistcurveE ′ of E is governed
by a cubic
y 2 = x 3 + g 2 ax + g 3 b,
where g
p = −1. Show that the curve orders are related thus:
#E +#E ′ =2p +2.
7.17. Suppose the largest order of an element in a finite abelian group G is
m. Show there is an absolute constant c>0 (that is, c does not depend on
m or G) such that the proportion of elements of G with order m is at least
c/ ln ln(3m). (The presence of the factor 3 is only to ensure that the double
log is positive.) This result is relevant to the comments following Theorem
7.5.2 and also to some results in Chapter 3.
7.18. Consider, for p = 229, the curves E,E ′ over Fp governed respectively
by
y 2 = x 3 − 1,
y 2 = x 3 − 8,
the latter being a twist curve of the former. Show that #E = 252, #E ′ = 208
with respective group structures
E ∼ = Z42 × Z6,
E ′ ∼ = Z52 × Z4.
Argue thus that every point P ∈ E has [252]P = [210]P = O, and similarly
every point P ∈ E ′ has [208]P = [260]P = O, and therefore that for any point