Prime Numbers

316 Chapter 6 SUBEXPONENTIAL FACTORING ALGORITHMS
can simply be incremented/decremented to x ± 1, yielding a whole new flock
of factors. Is there some way to exploit this phenomenon for more gain?
Incidentally, there are other identities that require, for a desired product of
terms, fewer operations than one might expect. For example, we have another
general identity which reads:
(n + 8)!
n!
= 204 + 270n + 111n 2 +18n 3 + n 4 2 − 16(9 + 2n) 2 ,
allowing for a product of 8 consecutive integers to be effected in 5 multiplies
(not counting multiplications by constants). Thus, even if the pure-squaring
ladder at the beginning of this exercise fails to allow generalization, there are
perhaps other ways to proceed.
Theoretical work on such issues does exist; for example, [Dilcher 1999]
discourses on the difficulty of creating longer squaring ladders of the indicated
kind. Recently, D. Symes has discovered a (k = 4) identity, with coefficients
(a1,a2,a3,a4) as implied in the construct
(((x 2 −67405) 2 −3525798096) 2 −533470702551552000) 2 −469208209191321600 2
which, as the reader may wish to verify via symbolic processing, is indeed
the product of 16 monomials! P. Carmody recently reports that many such
4-squarings cases are easy to generate via, say, a GP/Pari script.
6.18. Are there yet-unknown ways to extract square roots in number fields,
as required for successful NFS? We have discussed in Section 6.2.5 some stateof-the-art
approaches, and seen in Exercise 6.15 that some elementary means
exist. Here we enumerate some further ideas and directions.
(1) The method of Hensel lifting mentioned in Section 6.2.5 is a kind of padic
Newton method. But are there other Newton variants? Note as in
Exercise 9.14 that one can extract, in principle, square roots without
inversion, at least in the real-number field. Moreover, there is such a thing
as Newton solution of simultaneous nonlinear equations. But a collection
of such equations is what one gets if one simply writes down the relations
for a polynomial squared to be another polynomial (there is a mod f
complication but that can possibly be built into the Newton–Jacobian
matrix for the solver).
(2) In number fields depending on polynomials of the simple form f(x) =
x d +1, one can actually extract square roots via “negacyclic deconvolution”
(see Section 9.5.3 for the relevant techniques in what follows). Let the
entity for which we know there exists a square root be written
γ 2 d−1
= zjα j
j=0
where α is a d-th root of (−1) (i.e., a root of f). Now, in signal
processing terminology, we are saying that for some length-d signal γ to