Factoring problem and the SFT

jstevh_at_msn.com
Date: 04/27/05 

Date: 27 Apr 2005 04:18:57 -0700

The SFT provides a solution to

sqrt(x^2 - 4A^2(A^2 - B^2)y^2)

with all non-zero integer, where x/y is determined by the rational
factorization of B^2(A^2 - B^2).

That gives you what the SFT does in a nutshell.

And that reality is not in doubt. The SFT does in fact give a solution
to the square root shown.

I used integers this time to help in explaining and also to point out
that discussion on this subject has not been rational, why that may be
the case, and why that may be the worst case scenario.

For instance, if you can find x/y such that y=1, then you'll have a
finite set of solutions that must factor A.

I haven't seen any discussion mentioning that possibility.

Posters aren't rational on this subject. The raw emotion is clear in
the heat of their replies, but emotion does not change mathematics.

For the factoring problem A would be some number you wished to factor,
and B would be a number picked just to use the SFT, where you can use
any non-zero integer.

And I can say that if you can find all solutions such that y=1, then
you are guaranteed to factor A, having used the factorization of
another number, which I call the surrogate.

There is no mathematical reason for the equation to be picky.

Look again at

sqrt(x^2 - 4A^2(A^2 - B^2)y^2)

and the position that it will mostly give trivial factors of A depends
on the assertion that factorizations that don't factor A non-trivially
are selected.

But why?

In answer posters have replied with a deluge of pseudo-math, which may
be sufficient for policymakers, cryptologists with an investment in RSA
and people who just want to sleep at night, but is it ok just to be
comforted against the facts?

The reality is there is no mathematical reason to believe that the SFT
is picky.

BUT with posters comforting people who are supposed to worry about such
things with the idea that if it were possible it might work so many
people wouldn't argue against it, that just leaves the people willing
to check, capable of exploiting the mathematics in secret.

Some of you may live in the fantasy world where that's not possible.

But in the real world, RSA is worth trillions of dollars in economic
potential, and quite a few companies depend on the idea that the
Internet is secure.

Acknowledging pesky mathematics that isn't even complicated looking and
doesn't require an advanced degree to understand could seem like some
weird nightmare, best ignored.

That leaves the people who do not care about how much value is wrapped
up in the idea of RSA, who would happily exploit anything they can
find.

I'm stuck. I contacted cryptologists and even the NSA, months ago with
my early research, and recently tried to contact the NSA again with the
paper that covers the surrogate factoring theorem.

I got not reply.

I'm stuck here explaining the SFT in public on Usenet dealing with a
lot of ridicule.

That's the system.

People believe that if there were some simple way with something they
know is so important in human terms then it'd be some "great"
mathematician at some major university and not some Usenet "crank" who
figures it out.

They are certain that anything so important to people must also be
extremely difficult or mathematically impossible just because people
are so important, you know?

But just because you don't believe in mathematics doesn't make it not
so.

Generalized Surrogate Factoring Theorem:

Given non-zero integers A and B, let

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

then

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

and

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

and

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

where

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

James Harris

Relevant Pages

  • SF: World politics and mathematicians
    ... So, I can explain that the SFT maps hyperbolas, so it's not ... which shows how you connect hyperbolas. ... In the social world, it's a factoring theorem. ... If you believe in mathematics, ...
    (sci.crypt)
  • Re: SF: World politics and mathematicians
    ... > factorization from the start the SFT gives you. ... which shows how you connect hyperbolas. ... it's a factoring theorem. ... then you don't believe in mathematics so you'll wait. ...
    (sci.crypt)
  • James Harris still at it
    ... The simplest way to think of the surrogate factoring theorem is ... First off, if you actually try the SFT, it's quite natural to start by ... mathematics community at this time is a massive puzzle. ...
    (sci.crypt)
  • SF: Symmetry, practical matters, social realities
    ... The surrogate factoring theorem is mathematics at its most beautiful, ... Abstracted Surrogate Factoring Theorem: ... What the SFT is, is a connection betwee factorizations that covers the ... As we speak, if the idea has been exploited, and if Internet security ...
    (sci.crypt)
  • Re: Gist of surrogate factoring theorem
    ... while the surrogate factoring ... > a difference of squares with your target. ... mathematics requirement for physics majors is algebra, calculus, ... of people with this view of mathematics. ...
    (sci.crypt)
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