Thursday, May 05, 2005


JSH: Letting it drop

It's been fun, this hobby of mine of fiddling with simple math equations, and arguing about my work, but I'm increasingly concerned that maybe it's gotten a bit out of hand.

Like, I actually had a family member who didn't realize I'd deleted the mathforprofit blog and someone took it over, who actually thought that maybe it was me, ranting and raving on the blog, like that I'd completely lost it.

And yes, I am an intense person, and I've often taken this Quixotic quest of pushing against "math society" and finding my own major discoveries very seriously, but I've also had the luxury of looking at it as an odd hobby, possible because of our high tech tools.

But increasingly I'm getting a feeling that others have gone way overboard, with the webpages, and especially with the blog thing, and it's just not nearly as much fun as it used to be, and not nearly as easy for me to just dismiss Usenet antics as just, Usenet antics.

The other fear, of course, is that others won't just let go, but I can't control what other people do, but I can control what I do, and to me, it looks like it's time to hang up the "JSH", move on to other things, like focusing more on open source, and on other personal projects, to occupy my free time.

Math was fun for a while, occupied my attention for years, but now, it's "been there, done that" time.

It has been a wild ride. It has been often truly crazy, as in, actually crazy, with a bizarre cast of characters, and some massive adventures as far as I'm concerned, from meetings with mathematicians (yes I've met with more than one) to contacts with journals around the world, and wacky editor replies (and you didn't get them all) to
hearing from people from other countries who have been, yes, at times, very oddly hostile--especially Brits and Aussies, as well as people from the Netherlands for some odd reason--to some very nice people who have been very supportive.

Actually there have been quite a few very supportive people over the years, from quite a few countries, which just goes to show you how connected the world actually is, not just as an abstraction, but as easy as going out on Usenet, and posting.

It is an addictive experience to be sure, and one that doesn't often seem to be substantive, like I often wonder what I was really doing.

But it is something to do. You know?

And thinking about stepping away doesn't bother me, at all, while I wonder if I'll ever look back, and see it as something more.

Was there really much to it at all? Did it all really matter?

Who knows?

Now it's on to more adventures. Maybe I'll find some other Quixotic quest to satisfy my thirst for intellectual adventure, and maybe I'll find some other people to argue with, as I do love to argue.

Adventure is where you find it.

The future is out there, and my path moves on, beyond...

Wednesday, May 04, 2005


JSH: SFT is not easy

Well actually verifying the SFT equations myself was very useful in multiple ways.

For one thing, it revealed the role of experimentation, and convinced me that the theorem itself is just a step along the path to a practical factoring method.

That is good and it probably means that it's not trivial to figure out all the in's and out's to getting it to work, but there are simple reasons why it must be possible to find a way to get it to work.

In trying to explain before I ran into a flurry of hostile and disparaging postings from people who were making it their business to try and distract from the actual issues.

Now that I realize that it takes some time to figure out how to get from the SFT to a practical factoring algorithm, I realize just how dangerous those people are.

It is quite reasonable that there has been a delay up until now, and it's possible that there will be an indefinite delay while the mechanics of using the SFT are figured out.

So, why do I know it must work?

Well, given

ab = M

where 'a' and 'b' are rationals, and M is an integer with, say, two prime factors, the number of factors that will give a non-trivial factorization is infinite, as is the number of factors that will give a trivial factorization.

For some odd reason, posters have gotten away with arguing a relative size difference between these twin infinities, to push the argument that as M gets larger and its prime factors get larger, you have a lower probability of getting 'a' and 'b' such that M is non-trivially factored.

That makes no sense though, as in even a small range, like from 1/2 to 1 in rationals, you have an infinity of solutions to 'a' that would non-trivially factor M, without regard to the size of M.

That's important. 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.

There are an infinite number that will trivially factor it as well.

The reality is that the SFT allows you to do what has never been possible before, which is, if you wished, check in the range from 1/2 to 1, to get one of your factors of the surrogate, and see what happens.

You can experiment.

That means that the people who check may figure out what the people who do not, can't.

And someone might just get lucky and stumble across something.

It's already clear that using integer factors of the surrogate doesn't work well--so easiest is out--and I'll probably research that area as I'm curious.

But who knows when I'll move on to fractions.

I kind of like you people. You're so...calm...and as you're calm, things can progress slowly, which I think may just save us all.

Eventually, yeah, I think that RSA is toast, but it might take a few months, as the research progresses.

It's about time and mental effort, and luck, at this point.

And everybody has a chance.


SFT: Experimentation starts

Well I finally couldn't resist the impulse to check out the SFT, and hey, the equations actually do work!

It's always nice to see that at least you got the equations right.

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.

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.

Further experimentation and theorizing should reveal why.

I'll include the generalized SFT for reference.

Generalized Surrogate Factoring Theorem:

Given non-zero integers A and B, let

f_1 f_2 = A^2 (A^2 - B^2)


f_1 = (-(z - 2A^2)+ sqrt((z - 2A^2)^2 - 4A^2(A^2 - B^2)))/2


f_2 = (-(z - 2A^2) - sqrt((z - 2A^2)^2 - 4A^2(A^2 - B^2)))/2


z = x(x +/- sqrt((x - 2B^2)^2 + 4B^2 (A^2 - B^2)))/(2x - 2A^2)

and x is given by

x = +/- (g_1 - g_2) + 2B^2


g_1 g_2 = B^2(A^2 - B^2).

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