Harristotelian Logic

I ended up with a longer post than I would have liked as I went over the functional equation behind surrogate factoring—like, if someone wanted to implement an algorithm without worrying about theory they could just go by that post.

But some may still wonder why this idea might have potential and, how would they know?

Well, I invented a factoring method that relies on VERY fundamental equations:

x^2 = y^2 mod T

and

k^2 = 2xk mod T

and you can solve to get

(x+k)^2 = y^2 + 2k^2(mod T)

which is then the fundamental factoring congruence.

And you have as a surrogate to factor 2k^2 - wT, where I subtract just because I like to do so, as w is a non-zero integer.

So factoring a target T is turned into factoring a series of surrogates, S_1, S_2, S_3, ... and checking with each one to see if you've factored which may seem like a lot, but it doesn't have to be, and it is a direct assault against methods like RSA encryption because it allows you to go after smaller numbers that might be easily
factored to get your target, and you get as many tries as you want where if you fully factor your surrogates you have a 50% chance of getting your target as well, versus just banging your head against one big number.

Example:

T = 732367903, k=24412263

Surrogate: 1191915704826532 = ( 4 )( 7 )( 73 )( 583129014103 )

f_1 = 7/2 and f_2 = 85136836059038

and 4f_1*f_2 = 2k^2 - 2T

y=-170273672118069/2 and x=170273623293557/2

so, x+y=-24412256, which has 223 as a factor.

T = 732367903 = (223)(3284161)

Iterations: 1

Notice that with the surrogate factored there was not even a lot of iterating through combinations of those factors, as it took one try.

That example is the potential, as it could be a much larger target, where the primary issue is factoring the surrogate.

With the surrogate factored, looping through all the iterations of factors the theory says you have a 50% chance of factoring, so, if I got the mathematical theory right, anybody with some serious factoring power already could convert maybe even an RSA sized number into several surrogates, go at it, and just walk away with a factorization, just like that.

It is a fascinating invention to me, because it is amazing that such simple mathematics could have been missed, while it is so simple it's hard to see where the math can sneak in something that ruins it, like actually the only thing is if w's that work get scarcer as T increases in size, where there is no mathematical indication that they do, as you are multiplying T, and there are an infinity of w's that must work. So how do you concentrate that infinity?

How do you force mathematically a smaller infinite subset of working w's against a bigger infinite majority that don't?

Do the math. There is no way. So you get the same factoring potential as is already known for the difference of squares—50%—as that is just a variation with k=0 anyway.

And if I'm right, people with the know-how all over the world can immediately start factoring much larger numbers than they did before. Immediately, all over the world, with some very simple mathematics that is inexplicably still just out there, with barely a murmur from the mathematical community.

Maybe they just want to wait and see what happens…