Re: Experimentation starts
- From: "Tim Peters" <tim.one@xxxxxxxxxxx>
- Date: Wed, 4 May 2005 23:45:34 -0400
[cut the distribution list to sci.math alone]
[JSH]
> Well I finally couldn't resist the impulse to check out the SFT, and
> hey, the equations actually do work!
Well, f_1 f_2 = A^2 (A^2 - B^2) regardless of what z is, so that part isn't
mysterious. It's mildly interesting that the z's defined by the other
equations are in fact rational, although Rick Decker derived much simpler
expressions for them, which makes that part obvious.
> It's always nice to see that at least you got the equations right.
Yup!
> In any event, I did verify though that as you use *integer* factors to
> get your surrogate, the factoring percentage drops as the size of the
> number increases.
I reported statistically significant test results for that last Friday, with
message ID:
ysGdncDNfpMK6O_fRVn-qA@xxxxxxxxxxx
Note that success rate _vastly_ depends on choice of B (as noted there, and
for all the same reasons choice of j made a vast difference in the February
and March algorithms).
> So one of my assumptions was wrong, as I felt that it wouldn't care and
> factor about 50% of the time without regard to size, which was not the
> case--with integer factors of the surrogate.
>
> However, in checking that result, I also looked at what I call z in the
> generalized SFT and found that it dropped in size relative to the
> number I call x.
>
> So there seems to be a definite movement in one direction when only
> integers are used for the surrogate, and that direction is to smaller
> z's and a smaller factoring percentage.
>
> However, the theorem works over all rationals, and focusing on integers
> is a human choice, where it looks like I can narrow down mathematical
> reasons for it mattering, which is what I'll probably move towards as I
> experiment.
>
> There are some simple reasons why the SFT cannot in general care about
> what factor it gives you of your target--in rationals--but also, it
> does seem to know integers, and behave differently when integers are
> involved.
It behaves exactly the way everyone (except possibly you) expected it to
behave. It's really a shame you can't hear that!
> Further experimentation and theorizing should reveal why.
Or just reading old messages about this. The reasons for why "the infinity
of non-trivial rational factors" won't increase the success rate above
chance level have already been explained clearly.
[... GSFT repetition snipped ...]
> and x is given by
>
> +/- (g_1 - g_2) + 2B^2
Will you at least drop the +/- here? It serves no purpose. Given that
g_1 g_2 = g_2 g_1
and you consider *all* g_1 and g_2 such that g_1 g_2 = B^2(A^2 - B^2), the
minus choice in the above is just a needlessly complicated way of swapping
g_1 and g_2 (-(g_1-g_2)=g_2-g1). You consider them in swapped order anyway
(and, BTW, dropping "the minus choice" here is an easy way to speed your
testing by a factor 2).
[...]
[from another message]
[...]
> So, say, if M is the largest public key known, there are an *infinite*
> number of rationals in the range from 1/2 to 1 that will factor M non-
> trivially.
Much more is true: for every non-zero integer M, and for every positive
integer divisor i of M (including i=1 and i=|M|), and for every rational r1
and r2 such that r1 < r2, there are an infinite number of rationals r
strictly between r1 and r2 for which gcd(numerator(r), M) = i.
Alas, the GSFT doesn't appear to be of any use in finding an interesting r.
[...]
> It's about time and mental effort, and luck, at this point.
No argument there, except 2/3rds of it can be dropped <wink>.
.
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