Tuesday, March 20, 2007
What surrogate factoring theory now says
I want to emphasize that there can be a new factoring method that just takes a while to be fully engineered, as there is the theory and there is the engineering into a practical solution.
Like Isaac Newton knew difference of squares, but he didn't have the Number Field Sieve.
Surrogate factoring theory says that you can turn factoring a hard target T, into a problem of factoring an indefinite number of surrogates S_1, S_2, S_3, … making the problem potentially tractable.
My own target for my research is factoring an RSA sized number—of any bit length feasible—within ten minutes on a home computer.
That has been my research target for years now. The theory says that once the engineering is figured out that is achievable.
You can personally check the very simple underlying mathematics yourself.
(Web search on surrogate factoring, stay away from the old failed stuff though.)
Mainly I just added one more congruence to the difference of squares.
So it's not like the algebra is hard, or it's difficult to follow.
But just like with just the difference of squares, figuring out a practical solution could take a while, where I don't think it'll take centuries like with difference of squares to the NFS.
I am in the process of trying to turn what could take years of research from lots of people around the world into months or days of research where I am the primary engine, but I could fail, and others could succeed.
So, say, Russia could succeed. Or China could succeed. Or Iran could succeed. Or, maybe even North Korea could succeed.
Would my own country the United States?
Sure, but history says that people here might not bother because we're on the top of the heap.
People at the top tend to ignore "crackpot" ideas.
But I could be wrong, right? Lots of math people say I'm a crackpot and I've been babbling about surrogate factoring for YEARS, including in the past having said that I'd solved the factoring problem, when I hadn't.
Yup. I've failed a lot. I admit it. But I've succeeded a lot, and "mathematicians" won't admit it.
Roll the dice and the fate of the world could change.
That's how it's happened before…people like you under-rate the power of ideas despite thinking you're idea people, and you ignore something you DECIDE is dinky and worthless, and civilization itself changes.
If that didn't happen, no dominant country would ever lose that position. We might be under the Persian Empire, or the Roman or the Egyptian or some other if people just learned not to underestimate the power of ideas.
Then again, I could be wrong. I don't think I am, but I have been wrong before.
But hey, it's mathematics!!! I say, don't trust me. I don't trust you, or I wouldn't be making this post. I think most of you are complacent idiots who would let the world go up in flames because you're too small-minded and maybe corrupt to really care, and I don't trust you.
Go with the math.
If I'm right, it says I'm right. If I'm wrong, it says I'm wrong.
If it says I'm right, and you think you can just play the odds that no one in the world will figure this out, not Russia, not China, not anybody, and you're wrong…well, welcome then to a Brave New World, and yet another example that history repeats…
[A reply to someone who wrote that “there can be a new factoring method that never becomes practical no matter how much engineering you put into it”.]
The equations are fundamental being congruence of squares plus one more congruence:
x^2 = y^2 mod T
and
α*k^2 = 2xk mod T
with T the target composite, where the second congruence is necessary to introduce surrogate factoring, and now it can be seen that prior factoring techniques can be said to be using k=0.
So the two main factoring congruences encompass all prior known factoring techniques that are based on congruence of squares so the Number Field Sieve among others is just part of the smaller mathematical knowledge base.
And now in the 21st century we know more.
One way to look at it is that humanity was not quite as advanced with its mathematics in this area as it could have been, where it is easy to prove that the second congruence while including EVERYTHING known with just the previous also gives you a few extras.
It took hundreds of years to develop congruence of squares, while I'm working at developing the full theory, as I now know that there are no more useful factoring congruences, in just a few months.
There are only two main factoring congruences. For centuries only one was known, and now the second is known as well.
Hundreds of years were needed with just the one, but in the 21st century, it will be a lot faster with the two, I hope.
As then the Math Wars will be over, and the world will finally understand how close it came to losing Progress in its most important discipline, and maybe the future of the human race as well.
Like Isaac Newton knew difference of squares, but he didn't have the Number Field Sieve.
Surrogate factoring theory says that you can turn factoring a hard target T, into a problem of factoring an indefinite number of surrogates S_1, S_2, S_3, … making the problem potentially tractable.
My own target for my research is factoring an RSA sized number—of any bit length feasible—within ten minutes on a home computer.
That has been my research target for years now. The theory says that once the engineering is figured out that is achievable.
You can personally check the very simple underlying mathematics yourself.
(Web search on surrogate factoring, stay away from the old failed stuff though.)
Mainly I just added one more congruence to the difference of squares.
So it's not like the algebra is hard, or it's difficult to follow.
But just like with just the difference of squares, figuring out a practical solution could take a while, where I don't think it'll take centuries like with difference of squares to the NFS.
I am in the process of trying to turn what could take years of research from lots of people around the world into months or days of research where I am the primary engine, but I could fail, and others could succeed.
So, say, Russia could succeed. Or China could succeed. Or Iran could succeed. Or, maybe even North Korea could succeed.
Would my own country the United States?
Sure, but history says that people here might not bother because we're on the top of the heap.
People at the top tend to ignore "crackpot" ideas.
But I could be wrong, right? Lots of math people say I'm a crackpot and I've been babbling about surrogate factoring for YEARS, including in the past having said that I'd solved the factoring problem, when I hadn't.
Yup. I've failed a lot. I admit it. But I've succeeded a lot, and "mathematicians" won't admit it.
Roll the dice and the fate of the world could change.
That's how it's happened before…people like you under-rate the power of ideas despite thinking you're idea people, and you ignore something you DECIDE is dinky and worthless, and civilization itself changes.
If that didn't happen, no dominant country would ever lose that position. We might be under the Persian Empire, or the Roman or the Egyptian or some other if people just learned not to underestimate the power of ideas.
Then again, I could be wrong. I don't think I am, but I have been wrong before.
But hey, it's mathematics!!! I say, don't trust me. I don't trust you, or I wouldn't be making this post. I think most of you are complacent idiots who would let the world go up in flames because you're too small-minded and maybe corrupt to really care, and I don't trust you.
Go with the math.
If I'm right, it says I'm right. If I'm wrong, it says I'm wrong.
If it says I'm right, and you think you can just play the odds that no one in the world will figure this out, not Russia, not China, not anybody, and you're wrong…well, welcome then to a Brave New World, and yet another example that history repeats…
[A reply to someone who wrote that “there can be a new factoring method that never becomes practical no matter how much engineering you put into it”.]
The equations are fundamental being congruence of squares plus one more congruence:
x^2 = y^2 mod T
and
α*k^2 = 2xk mod T
with T the target composite, where the second congruence is necessary to introduce surrogate factoring, and now it can be seen that prior factoring techniques can be said to be using k=0.
So the two main factoring congruences encompass all prior known factoring techniques that are based on congruence of squares so the Number Field Sieve among others is just part of the smaller mathematical knowledge base.
And now in the 21st century we know more.
One way to look at it is that humanity was not quite as advanced with its mathematics in this area as it could have been, where it is easy to prove that the second congruence while including EVERYTHING known with just the previous also gives you a few extras.
It took hundreds of years to develop congruence of squares, while I'm working at developing the full theory, as I now know that there are no more useful factoring congruences, in just a few months.
There are only two main factoring congruences. For centuries only one was known, and now the second is known as well.
Hundreds of years were needed with just the one, but in the 21st century, it will be a lot faster with the two, I hope.
As then the Math Wars will be over, and the world will finally understand how close it came to losing Progress in its most important discipline, and maybe the future of the human race as well.
# posted by James Harris @ 21:36 
