Harristotelian Logic

IN arguing about research I call surrogate factoring, I bump into this weird thing where posters seem to be lost on what is actually going on, so I thought I'd start a thread informing, yet again, what surrogate factoring is.

Years ago, while thinking about RSA encryption, I wondered to myself if instead of directly attacking a large number that you wanted to factor, you might instead factor some other number and in that way factor the target.

I termed the concept: surrogate factoring.

So, to repeat, years ago, as in about four years ago I think it was, I was just kind of wondering about factoring because I was thinking about RSA encryption, and I wondered if you might go after a large composite that was otherwise hard to factor, by instead factoring some other number.

To me a good name for the concept was "surrogate factoring", so it was called surrogate factoring.

Now years later I have finally settled in my own mind that mathematically the concept reduces to considering

x^2 = y^2 mod T

and

k = 2x mod T

and equations that result from those two basic congruences, where T is the target, which took me about three years to figure out.

With those two relations I found that my surrogate to factor is given by deriving

(x+k)^2 = y^2 + 2k^2 + nT

as then the surrogate S, is

S = 2k^2 + nT

and the big question is, how do you pick k and n?

For those of you who wonder how it works from there, it's trivial algebra that if you let

4f_1*f_2 = 2k^2 + nT

then

x+y = 2f_1 - k and x-y = 2f_2 - k

so once you factor the surrogate S, you just loop through solutions for x+y and x-y, by going through the various possible values for f_1 and f_2, and check the gcd with T.

So, to recap, about four years ago I was wondering whether or not you could go after an RSA sized number to try and factor it by instead factoring another number. For years I tried various approaches and last year I boiled down the idea to two congruence relations, which lead through some simple algebra to a way to factor the target T, by factoring the surrogate S.

So I had an idea, and after four years I have the math that implements the idea.

That is the "pure math" aspect of it all where a person just pursues a mathematical problem for the hell of it, you might say, while, of course, I had practical reasons for picking the factoring problem.

Now then, from the realm of mathematical curiosity to a world changing idea requires that surrogate factoring be a way to actually factor a large composite faster than the other known methods, which is where the arguing comes in with people who want to be certain that no one believes it can be, or who are getting on my case for declaring it is, and then not delivering by factoring some large number.

But that is secondary, as it is a practical matter that can move stock markets and scare people because if surrogate factoring makes factoring easy, then a lot of industries around the world would be impacted.

But what real mathematician cares about practical crap anyway?

So there is the "pure math" of being curious about this way to factor.

And to the extent that math people act more like business people who care about the practical side than math people who would care about the curiosity side, I point out a contradiction!

Maybe they are not math people after all, eh?

As there is the practical and political reality of possibly changing the world with a simple concept.

Understand surrogate factoring now?

Oh yeah, so recently I came up with a detailed analysis of when and why surrogate factoring works, which has some very complicated looking equations in it, so it is a massive puzzle.

A MASSIVE puzzle.

Some have done experiments where they claim that surrogate factoring works worse than random!

And the world hangs in the balance on the answer, or maybe not, if it's just a crap idea, but for some reason, supposedly brilliant mathematicians have not settled the question so that the stock markets can rest easy.

And your fate may depend on the answer, so the math world cannot keep looking, so Google and Yahoo! search results move accordingly, as if this concept is viable, then it ends the modern math world as it currently operates.

But, on the other hand, it is also just a 'pure math' idea in a lot of ways.

Two ways of looking at it, and entire economies can be destroyed if people do not do the right thing here, and guess wrong.