Monday, March 26, 2007
Surrogate factoring, top to bottom
The best way to understand surrogate factoring may be to to start with
x^2 = y^2 mod T
where T is the target composite to be factored, so you just have the familiar congruence of squares, also called the difference of squares, where you MAY factor T non-trivially by checking for prime factors in common between it and x-y or x+y.
Now for some solution set {x,y} let
k = 2x mod T
so
2x = k mod T, and you can multiply both sides by k, to get
2xk = k^2 mod T
and now you can add back to the original congruence of squares to get
x^2 + 2xk = y^2 + k^2 mod T
and, of course, you can complete the square and simplify to get
(x+k)^2 = y^2 + 2k^2 mod T
and that gives you most of surrogate factoring as you now know that for ANY solutions with a difference of squares you have a solution to that congruence.
So, trivially, every factorization of any given target composite T is connected to a factorization of some other composite, as of course, looking explicitly—introducing the integer omega—I have
(x+k)^2 = y^2 + 2k^2 + omega*T
and y is determined by the factorization of 2k^2 + omega*T, with again, trivial algebra.
Note then that y is determined by two factorizations as also with the original difference of squares, again going explicit, this time using alpha, I have
x^2 = y^2 + alpha*T
and here y is defined by a factorization of T, and SIMULTANEOUSLY y is defined by the factorization of omega*T, which means that at a very fundamental level, factorizations of numbers are bound by some very basic congruence relationships.
So it becomes a simple step to ask, what if I start with k, can I factor T from the surrogate factorization of 2k^2 + omega*T?
And the answer is, yes, you can.
In terms of actually trying to factor, you have to pick k, which forces the residue modulo T of x, and you also have to pick omega, and that is it.
You can only pick k and omega.
And you know that for any x you can pick there is a k and omega that have to work, but maybe the two don't match well somehow? Maybe your pick of k and omega on your end is unlikely to find a welcoming x and y on the other?
Well, good questions, and shouldn't there be answers? Shouldn't there be some mathematics that would tell you if that were true?
I've checked. It's trivial algebra, and the mathematics does not say that at all.
It says that for a given k and omega, outside of a narrow size range where k is too small and the surrogate is too small to give factors, you have a 50% chance of factoring T, with SOME combination of factors of the surrogate 2k^2 + omega*T.
(If I am wrong, it'd be nice for someone to post a nice analysis of the very trivial algebra that DOES show that it's hard to pick k and omega.)
So then if this idea might be worth something, why wouldn't it be a big deal by now?
Well you have to factor SOMETHING. If you can't factor that well, then you can't use surrogate factoring as you still need to factor 2k^2 + omega*T, and that's why this method could be a stealth one that takes the world by surprise because to check it against really big numbers, you need to be able to factor some really big numbers.
Easy math then that says that you can factor one number through another, and on the "pure math" side, easy proof of a connection between composites where otherwise it might not be clear that a connection existed.
For the "pure math" fun of it, what if you play around with the equations? Can you find more congruence relations?
Well, I've done that and you can do some other stuff, but it doesn't seem to change much, like you can have
ak = 2x mod T
and get to
(x+k)^2 = y^2 + (a+1)k^2 mod T
which is a variable I like enough to add to my own main research page on the subject, but that doesn't change much…hmmm…what about something like
k^2 = 3xk mod T?
Well, that can be shown to be equivalent to what I already have (do the math for yourself if you wish).
Ok then, what about something like
k^2 = 2xk + 3 mod T
or something else of that nature?
Well, that isn't always true, as it's not necessarily the case that you have a solution for k, so I haven't played with anything like that, and if you just go all out to add another variable, something weird happens, like with
k^2 = 2xk + z mod T
you completely de-couple T from the equations anyway and it behaves like random.
It's like some kind of quantum mechanics thing, where if you add another variable, you blow up surrogate factoring as the mathematics kind of throws up its hands and asks, how do I know what T is now?
You know, it's like you collapse the wave function or something, and it doesn't work any more.
That is freaky if you see it, so I suggest you play with these equations.
The other way to de-couple T, is, of course, to have k = 0, or have omega = 0, as then you just get random.
So yes, the mathematics here may relate back to how our very reality works, but the mathematics here is VERY troubling for mathematicians out there who have made it their business to ignore my research.
You think they give a damn if somehow composites and factorizations could be related to physics?
Nope, not if it helps me they don't. These are what I call the Math Wars for a reason.
And like with any war you do not help the enemy, no matter what.
The human species can rot and die before any of those people will lift a finger to support any of my research as they rightly see it as an issue of survival. Their survival against the Progress of the species, and given that choice they keep picking themselves.
After all, I say they are fakes and cons. If I win, they are probably out of jobs.
If you were a Ph.D in mathematics with decades of living rather well, would you really accept having to go find a new job, maybe working at McDonald's or whatever you could get just to tell the truth about some basic mathematics?
Just for the fate of the human species? Like anybody even gives a damn any more.
It's a cold, cruel world, and we're all doomed anyway, right?
Make no mistake, each day while I chat about this to the world through this medium while the "mathematicians" ignore it, while you may notice, still promoting themselves and their research, is another day they choose themselves against Progress, and the future.
They hate everything, but protecting narrow interests, and deep down, they hate mathematics.
x^2 = y^2 mod T
where T is the target composite to be factored, so you just have the familiar congruence of squares, also called the difference of squares, where you MAY factor T non-trivially by checking for prime factors in common between it and x-y or x+y.
Now for some solution set {x,y} let
k = 2x mod T
so
2x = k mod T, and you can multiply both sides by k, to get
2xk = k^2 mod T
and now you can add back to the original congruence of squares to get
x^2 + 2xk = y^2 + k^2 mod T
and, of course, you can complete the square and simplify to get
(x+k)^2 = y^2 + 2k^2 mod T
and that gives you most of surrogate factoring as you now know that for ANY solutions with a difference of squares you have a solution to that congruence.
So, trivially, every factorization of any given target composite T is connected to a factorization of some other composite, as of course, looking explicitly—introducing the integer omega—I have
(x+k)^2 = y^2 + 2k^2 + omega*T
and y is determined by the factorization of 2k^2 + omega*T, with again, trivial algebra.
Note then that y is determined by two factorizations as also with the original difference of squares, again going explicit, this time using alpha, I have
x^2 = y^2 + alpha*T
and here y is defined by a factorization of T, and SIMULTANEOUSLY y is defined by the factorization of omega*T, which means that at a very fundamental level, factorizations of numbers are bound by some very basic congruence relationships.
So it becomes a simple step to ask, what if I start with k, can I factor T from the surrogate factorization of 2k^2 + omega*T?
And the answer is, yes, you can.
In terms of actually trying to factor, you have to pick k, which forces the residue modulo T of x, and you also have to pick omega, and that is it.
You can only pick k and omega.
And you know that for any x you can pick there is a k and omega that have to work, but maybe the two don't match well somehow? Maybe your pick of k and omega on your end is unlikely to find a welcoming x and y on the other?
Well, good questions, and shouldn't there be answers? Shouldn't there be some mathematics that would tell you if that were true?
I've checked. It's trivial algebra, and the mathematics does not say that at all.
It says that for a given k and omega, outside of a narrow size range where k is too small and the surrogate is too small to give factors, you have a 50% chance of factoring T, with SOME combination of factors of the surrogate 2k^2 + omega*T.
(If I am wrong, it'd be nice for someone to post a nice analysis of the very trivial algebra that DOES show that it's hard to pick k and omega.)
So then if this idea might be worth something, why wouldn't it be a big deal by now?
Well you have to factor SOMETHING. If you can't factor that well, then you can't use surrogate factoring as you still need to factor 2k^2 + omega*T, and that's why this method could be a stealth one that takes the world by surprise because to check it against really big numbers, you need to be able to factor some really big numbers.
Easy math then that says that you can factor one number through another, and on the "pure math" side, easy proof of a connection between composites where otherwise it might not be clear that a connection existed.
For the "pure math" fun of it, what if you play around with the equations? Can you find more congruence relations?
Well, I've done that and you can do some other stuff, but it doesn't seem to change much, like you can have
ak = 2x mod T
and get to
(x+k)^2 = y^2 + (a+1)k^2 mod T
which is a variable I like enough to add to my own main research page on the subject, but that doesn't change much…hmmm…what about something like
k^2 = 3xk mod T?
Well, that can be shown to be equivalent to what I already have (do the math for yourself if you wish).
Ok then, what about something like
k^2 = 2xk + 3 mod T
or something else of that nature?
Well, that isn't always true, as it's not necessarily the case that you have a solution for k, so I haven't played with anything like that, and if you just go all out to add another variable, something weird happens, like with
k^2 = 2xk + z mod T
you completely de-couple T from the equations anyway and it behaves like random.
It's like some kind of quantum mechanics thing, where if you add another variable, you blow up surrogate factoring as the mathematics kind of throws up its hands and asks, how do I know what T is now?
You know, it's like you collapse the wave function or something, and it doesn't work any more.
That is freaky if you see it, so I suggest you play with these equations.
The other way to de-couple T, is, of course, to have k = 0, or have omega = 0, as then you just get random.
So yes, the mathematics here may relate back to how our very reality works, but the mathematics here is VERY troubling for mathematicians out there who have made it their business to ignore my research.
You think they give a damn if somehow composites and factorizations could be related to physics?
Nope, not if it helps me they don't. These are what I call the Math Wars for a reason.
And like with any war you do not help the enemy, no matter what.
The human species can rot and die before any of those people will lift a finger to support any of my research as they rightly see it as an issue of survival. Their survival against the Progress of the species, and given that choice they keep picking themselves.
After all, I say they are fakes and cons. If I win, they are probably out of jobs.
If you were a Ph.D in mathematics with decades of living rather well, would you really accept having to go find a new job, maybe working at McDonald's or whatever you could get just to tell the truth about some basic mathematics?
Just for the fate of the human species? Like anybody even gives a damn any more.
It's a cold, cruel world, and we're all doomed anyway, right?
Make no mistake, each day while I chat about this to the world through this medium while the "mathematicians" ignore it, while you may notice, still promoting themselves and their research, is another day they choose themselves against Progress, and the future.
They hate everything, but protecting narrow interests, and deep down, they hate mathematics.
# posted by James Harris @ 20:39 
