### Friday, March 14, 2008

## JSH: Upside down situation

For those who missed it, a poster who has routinely critiqued my factoring research made a post noting that my latest research gave an algorithm better than Fermat's and I guess that was cool of that person to admit.

I am currently working on my own practical implementation of what I call surrogate factoring that follows from the latest theory, and it is pleasingly fast, while I still have to work at getting it to factor really big numbers to my satisfaction. Oh, my stated goal is the ability to factor an RSA encryption sized number in under 10 minutes on a standard desktop.

Now there is no doubt that I've discovered a new factoring method then. And that factoring method at its heart relies on the use of two congruences where mathematicians have traditionally used one:

x^2 = y^2 mod p

and

z^2 = y^2 mod T

where T is your target composite to factor and p is what I call a helper prime, as it is just there to help you factor, and otherwise does nothing. So it's just kind of like your buddy. There to help you, then kind of wander off, other things to do…

There are rules for p though, as, with T factored into two positive factors, p must be LESS than the smaller factor or p minus the smaller factors must be less than it, which will make more sense later with the congruence relations for those factors modulo p.

But incredibly, as in monster math that just kind of boggles the mind, surrogate factoring tells you that there is a deeper structure to z, where

z = (1 + 2α^2)k/(2α)

where

k^2 = (1 + α^2)^{-1}(nT) mod p

and 'n' is a helper variable allowing you to shift things around when needed, and α is found such that k exists.

And finally, with f_1*f_2 = nT:

f_1 = αk mod p

and

f_2 = α^{-1}(1 + α^2)k mod p.

So there is this whole zoo of new relations for the classical difference of squares that follow from just adding one more congruence!!! And it just gets better.

It turns out that for solutions in all integers, k will be near what I call k_0, where k_0 is the value of k such that

abs(nT - (1 + α^2)k_0)

is a minimum. Which is just one of the most beautiful pieces of mathematics that you will ever see, and it helps a lot.

So you can factor T, modulo p, where to find p you look for big primes that are not too big, and solve for k modulo p, and then find a k with that residue near k_0 and then you can look for the k that will factor modulo p.

Great!!! Easy!!! Factoring made easy!!! Pushing the easy button on factoring!!!

So why am I still babbling on newsgroups about this and math society isn't hollering to the public about it?

Because we are in an upside down situation.

I liken it to if Olympic runners faced races with people who were unethical actors who fantasized at being Olympic runners, who were in the stands, and were the judges and were everywhere, so that when the Olympic runner ran the race and won, they'd just say he didn't and award the gold medal to one of their own.

You see, I have other major discoveries. Math people just lie about them. And call me names. Nasty names, like crackpot and crank. And even NASTIER names.

They can get away with this as they have what I call, critical mass.

MOST mathematicians around the world are like those wannabe runners. So they can just as a group, lie, and who can challenge them then?

Only factoring can which is where this situation gets really, really, really strange, as hey, if the surrogate factoring theory is correct, then quite a few people around the world can now, um, probably, um, factor really huge numbers, but if math people ARE wannabes like I say, they have NO INCENTIVE to tell the truth, except being decent human beings, but I digress.

So game theory says they will lie and claim that nothing has changed, nothing is wrong, nothing to see here, and will explain any security breaches if they happen, away, and will do so indefinitely.

So game theory says they will do their best to collapse human civilization as we know it, and we are traveling down that scenario now.

End of this tale, may be that history is being ended now. The front story to troubles with stock markets will carry things so far, but eventually, oh, eventually civilization as we know it will just collapse, people will turn to endless wars, there will be mass starvation, disease, lots of nasty flies and things, Armageddon and all that stuff, but…

Yes, there is a but. I didn't like that end to the story.

So I changed it.

Oh, but I had to lose two countries.

I am currently working on my own practical implementation of what I call surrogate factoring that follows from the latest theory, and it is pleasingly fast, while I still have to work at getting it to factor really big numbers to my satisfaction. Oh, my stated goal is the ability to factor an RSA encryption sized number in under 10 minutes on a standard desktop.

Now there is no doubt that I've discovered a new factoring method then. And that factoring method at its heart relies on the use of two congruences where mathematicians have traditionally used one:

x^2 = y^2 mod p

and

z^2 = y^2 mod T

where T is your target composite to factor and p is what I call a helper prime, as it is just there to help you factor, and otherwise does nothing. So it's just kind of like your buddy. There to help you, then kind of wander off, other things to do…

There are rules for p though, as, with T factored into two positive factors, p must be LESS than the smaller factor or p minus the smaller factors must be less than it, which will make more sense later with the congruence relations for those factors modulo p.

But incredibly, as in monster math that just kind of boggles the mind, surrogate factoring tells you that there is a deeper structure to z, where

z = (1 + 2α^2)k/(2α)

where

k^2 = (1 + α^2)^{-1}(nT) mod p

and 'n' is a helper variable allowing you to shift things around when needed, and α is found such that k exists.

And finally, with f_1*f_2 = nT:

f_1 = αk mod p

and

f_2 = α^{-1}(1 + α^2)k mod p.

So there is this whole zoo of new relations for the classical difference of squares that follow from just adding one more congruence!!! And it just gets better.

It turns out that for solutions in all integers, k will be near what I call k_0, where k_0 is the value of k such that

abs(nT - (1 + α^2)k_0)

is a minimum. Which is just one of the most beautiful pieces of mathematics that you will ever see, and it helps a lot.

So you can factor T, modulo p, where to find p you look for big primes that are not too big, and solve for k modulo p, and then find a k with that residue near k_0 and then you can look for the k that will factor modulo p.

Great!!! Easy!!! Factoring made easy!!! Pushing the easy button on factoring!!!

So why am I still babbling on newsgroups about this and math society isn't hollering to the public about it?

Because we are in an upside down situation.

I liken it to if Olympic runners faced races with people who were unethical actors who fantasized at being Olympic runners, who were in the stands, and were the judges and were everywhere, so that when the Olympic runner ran the race and won, they'd just say he didn't and award the gold medal to one of their own.

You see, I have other major discoveries. Math people just lie about them. And call me names. Nasty names, like crackpot and crank. And even NASTIER names.

They can get away with this as they have what I call, critical mass.

MOST mathematicians around the world are like those wannabe runners. So they can just as a group, lie, and who can challenge them then?

Only factoring can which is where this situation gets really, really, really strange, as hey, if the surrogate factoring theory is correct, then quite a few people around the world can now, um, probably, um, factor really huge numbers, but if math people ARE wannabes like I say, they have NO INCENTIVE to tell the truth, except being decent human beings, but I digress.

So game theory says they will lie and claim that nothing has changed, nothing is wrong, nothing to see here, and will explain any security breaches if they happen, away, and will do so indefinitely.

So game theory says they will do their best to collapse human civilization as we know it, and we are traveling down that scenario now.

End of this tale, may be that history is being ended now. The front story to troubles with stock markets will carry things so far, but eventually, oh, eventually civilization as we know it will just collapse, people will turn to endless wars, there will be mass starvation, disease, lots of nasty flies and things, Armageddon and all that stuff, but…

Yes, there is a but. I didn't like that end to the story.

So I changed it.

Oh, but I had to lose two countries.