Surrogate factoring, update

From: James Harris (jstevh_at_msn.com)
Date: 07/10/04 

Date: 10 Jul 2004 09:15:55 -0700

It's been a while since I mentioned surrogate factoring, and I'll
admit that I'm still just trying to figure out basics. I haven't seen
evidence of interest in the idea, so I think I'll talk a bit more
about why I think there should be a LOT of interest.

A little while back I discovered the factorization:

(jk - Tk + T)(jk + Tk + T) = T^4

and I've emphasized that it's of interest because solving for k, gives

k = (-jT +/- T^2 sqrt(j^2 - T^2 + 1))/(j^2 - T^2)

so that rational k's depend on sqrt(j^2 - T^2 + 1), so if you're
trying to factor T, the target, then you can use rational factors of
T^2 - 1, the surrogate.

Now then, what about j? Well the easiest thing is just to solve for
it to get

j = (-T +/- T sqrt(k^2 + T^2))/k

where now you see that getting a rational k depends on rational
factors of T itself.

Another way to think of it is that if you find a rational k with the
solution for k using rational factors of T^2 - 1, then necessarily,
you are getting solutions dependent on the rational factors of T, as
seen by the solution for j.

You see I deliberately went looking for factorizations like

(jk - Tk + T)(jk + Tk + T) = T^4

where the solution for k is warped slightly by the asymmetry, so that
it couldn't be defined in a direct manner by the rational factors of
T, while j is.

Now then, how does the math pick?

If T were prime, then of course the only positive *integer* k that you
could have with positive integer T is T-1. That's easy.

But if T is composite, say, T = p_1 p_2, then k can equal p_1 - p_2,
as well as T - 1.

So how does the math pick between those possibilities?

I don't know.

My original hope was that it would split down the middle.

That is, that half the time you'd get p_1 - p_2, and the other half
you'd get T-1, but if I'd found that to be true, I'd have factored the
RSA Challenge by now, and wouldn't be making this post!!!

So what gives? What decides which difference of rational factors of T
get picked?

One problem is that practically you end up with k's that are
fractions, though if someone figured out a way to get integer k's then
they'd have a potent solution.

Now then, at this point in time, the *reasonable* position, unless
someone can show otherwise, is that mathematically there may indeed be
a way to factor a composite using surrogate factoring a fairly high
percentage of the time.

If that is true, then it will affect public key encryption schemes.

I'm just one person. While I'm the only one talking publicly about
this, others may be busily working in private, or more than likely, it
just sits while I fiddle with it.

But ignoring this idea may be the most dangerous thing that mainstream
mathematicians do.

Personally, given the talk about "pure math" I find it extraordinary
that I have to even push this idea. You'd think there'd be some
curious mathematicians out there, just because.

Maybe I'm missing something. If someone can quickly explain why this
idea is no threat, then that'd be appreciated, so that I don't worry
anyone else!!!

If no one can, then I think mathematicians may one day have to answer
some hard questions about what they really believe and value, versus
what they claim is important to them.

Like, what if "pure math" is just an excuse for mathematicians to do
nothing important but get paid for it, when they don't even believe in
it themselves?

What if it's just a scam?

What if they left the world unprotected, trusting and open for selfish
reasons in a story as old as man, and one that often ends tragically?

What if you have no security at all?

I'm painting a dire picture because I want answers. If someone can
rule out value to this idea, then I hope they'll do it now. No need
for unnecessary panic.

James Harris

Relevant Pages

  • JSH: Lets recrap
    ... Surrogate factoring preferentially yanks out small prime factors. ... but that was without modern computing technology, ... There is no way that I can see that mathematicians ignoring this ...
    (sci.math)
  • Surrogate factoring, update
    ... It's been a while since I mentioned surrogate factoring, ... given the talk about "pure math" I find it extraordinary ... curious mathematicians out there, just because. ...
    (sci.crypt)
  • Proper evaluation of surrogate factoring
    ... Surrogate factoring is just kind of a wild idea that means you have to ... I am being very serious here, modern mathematicians lie a lot. ...
    (sci.crypt)
  • Re: Surrogate factoring, a fascinating idea
    ... >> If mathematicians just ignore the result, however, and someone out ... I'm a theoretician who works on "pure math" problems, ... This factoring paper is just a minor result in comparison with my ... James Harris ...
    (sci.crypt)
  • JSH: Good news!
    ... Rather than just talk about my latest surrogate factoring idea and ... by instead factoring some other number that I call the ... So now there is no doubt--supposedly top mathematicians around the ... It is a magnet for cons. ...
    (sci.crypt)