Wednesday, February 27, 2008
JSH: Stepping back
I've been pushing myself as hard as I can go to get to a practical implementation and I'm facing that I can't get it done tonight, but I'm hoping to be finished within the next couple of weeks with a working solution to the factoring problem fully programmed.
But I want to keep raising the stakes, but I think I shouldn't so I'm stepping back.
Problem solving is about finding what's necessary to get the solution and I think I have it now where getting to the answer was more than just figuring out the math, it was also about facing a steady stream of negativity, character assassination and questions about my sanity from people fighting their own battle to prevent the knowledge from being found.
And what knowledge.
The factoring problem can be attacked through what I call surrogate factoring by leveraging one factorization against another.
With
z^2 = y^2 + nT
where T is the target to be factored, z itself can be approximated, surprisingly enough by looking at a maximum value for a variable I call k, for which
abs(nT - (α^2+1)k^2)
is a minimum where '=E1' is yet another variable, which is chosen such that
k^2 = (α^2+1)^{-1}(nT) mod p
exists, where p is an odd prime of your choice.
It is preferred that z have 3 as a factor as in general it can be shown to have
(2=α^2+1)
as a factor, so for most values of 'α'—2 out of 3—it will at least have 3 as a factor, and who knew that factoring had all these relationships available?
But that's what can make mathematics exciting!!!
Interested readers can figure out the derivations on their own.
Key is letting 2αx = k + pr_2, and z = x + αk, and considering what happens if you move k about with k = k_0 + jp, and substituting into
z^2 = y^2 + nT
and then you can re-derive everything I have shown here. Easily.
x^2 = y^2 + nT - (1 + α^2)(k_0^2 + 2pjk_0 + p^2*j^2) - (k_0 + pj)pr_2
Seemingly complex it is the result of just doing the substitution and simplifying a bit with the given relations.
It shows that if you move k around with j, you will have a minimum absolute value for
nT - (1 + α^2)(k_0^2 + 2pjk_0 + p^2*j^2)
because x, y and nT are constant, as is α and k_0, so because the j^2 term will dominate r_2 will tend to be negative to compensate, and that allows you to get an idea of where k_0 is.
And if j=0, then z = (1+2α^2)k/(2α), so you have it explicitly.
The mathematics is what is commonly called elementary methods.
And I like simple.
I think about researchers around the world claiming to be working on the factoring problem who will say nothing about this research.
Weaklings.
Fakes.
That's me stepping back a bit. But I don't want these people staying in positions that they clearly are not filling later. So the warning to them is, yes, I want this research acknowledged before I fully force the situation, but when I do force it, if that is necessary, then the next problem I will solve will be making sure that none of them remain in the field as mathematicians or cryptographers, so they need to start thinking about what they will be doing for work, in the aftermath.
Or, show some goddamn sense, and just acknowledge the research now.
[A reply to someone who asked James whether or not he was thinking about doing some violent action.]
I'll admit that I'm still wary of the impact of a sudden change precipitated by me just factoring some really large number so I'd prefer buy-in from the cryptographic industry ahead of that event.
But make no mistake, if this impasse ends with me, say, factoring a large enough number to show this research must be viable, WITH the growing history now of an improper response from the cryptographic community then the likely impact will be a sharp loss of confidence in that industry.
But that industry would next be tasked with resolving the issue along with the high tech community so that secure transmissions could continue.
So the first major step would be just figuring out which of you are in on this cover-up or not, and even if you're not part of this particular cover-up, do you actually have real mathematical skills or are you a fake?
That could mean a snarl on looking for solutions, so yes, my putting out the theory now and talking about the research in-depth is methodical AND important.
Posters challenging me to just factor an RSA number are pushing the snarl, when if the research is viable the theory would show lots of indications that I could do so with plenty of time for proper industry action.
Given what I know about the current corruption in the mathematical community I'm not surprised by the behavior, but I'm still hopeful that there are some members of the cryptographic community who are legit.
Otherwise the snarl is what we will see down the line with the world—and I'm sure all major world leaders—facing the big issue of trying to figure out what to do when RSA encryption goes away, probably literally overnight, as it's potentially broken now by others, but we don't know that so it still works if only out of faith in it.
But I want to keep raising the stakes, but I think I shouldn't so I'm stepping back.
Problem solving is about finding what's necessary to get the solution and I think I have it now where getting to the answer was more than just figuring out the math, it was also about facing a steady stream of negativity, character assassination and questions about my sanity from people fighting their own battle to prevent the knowledge from being found.
And what knowledge.
The factoring problem can be attacked through what I call surrogate factoring by leveraging one factorization against another.
With
z^2 = y^2 + nT
where T is the target to be factored, z itself can be approximated, surprisingly enough by looking at a maximum value for a variable I call k, for which
abs(nT - (α^2+1)k^2)
is a minimum where '=E1' is yet another variable, which is chosen such that
k^2 = (α^2+1)^{-1}(nT) mod p
exists, where p is an odd prime of your choice.
It is preferred that z have 3 as a factor as in general it can be shown to have
(2=α^2+1)
as a factor, so for most values of 'α'—2 out of 3—it will at least have 3 as a factor, and who knew that factoring had all these relationships available?
But that's what can make mathematics exciting!!!
Interested readers can figure out the derivations on their own.
Key is letting 2αx = k + pr_2, and z = x + αk, and considering what happens if you move k about with k = k_0 + jp, and substituting into
z^2 = y^2 + nT
and then you can re-derive everything I have shown here. Easily.
x^2 = y^2 + nT - (1 + α^2)(k_0^2 + 2pjk_0 + p^2*j^2) - (k_0 + pj)pr_2
Seemingly complex it is the result of just doing the substitution and simplifying a bit with the given relations.
It shows that if you move k around with j, you will have a minimum absolute value for
nT - (1 + α^2)(k_0^2 + 2pjk_0 + p^2*j^2)
because x, y and nT are constant, as is α and k_0, so because the j^2 term will dominate r_2 will tend to be negative to compensate, and that allows you to get an idea of where k_0 is.
And if j=0, then z = (1+2α^2)k/(2α), so you have it explicitly.
The mathematics is what is commonly called elementary methods.
And I like simple.
I think about researchers around the world claiming to be working on the factoring problem who will say nothing about this research.
Weaklings.
Fakes.
That's me stepping back a bit. But I don't want these people staying in positions that they clearly are not filling later. So the warning to them is, yes, I want this research acknowledged before I fully force the situation, but when I do force it, if that is necessary, then the next problem I will solve will be making sure that none of them remain in the field as mathematicians or cryptographers, so they need to start thinking about what they will be doing for work, in the aftermath.
Or, show some goddamn sense, and just acknowledge the research now.
[A reply to someone who asked James whether or not he was thinking about doing some violent action.]
I'll admit that I'm still wary of the impact of a sudden change precipitated by me just factoring some really large number so I'd prefer buy-in from the cryptographic industry ahead of that event.
But make no mistake, if this impasse ends with me, say, factoring a large enough number to show this research must be viable, WITH the growing history now of an improper response from the cryptographic community then the likely impact will be a sharp loss of confidence in that industry.
But that industry would next be tasked with resolving the issue along with the high tech community so that secure transmissions could continue.
So the first major step would be just figuring out which of you are in on this cover-up or not, and even if you're not part of this particular cover-up, do you actually have real mathematical skills or are you a fake?
That could mean a snarl on looking for solutions, so yes, my putting out the theory now and talking about the research in-depth is methodical AND important.
Posters challenging me to just factor an RSA number are pushing the snarl, when if the research is viable the theory would show lots of indications that I could do so with plenty of time for proper industry action.
Given what I know about the current corruption in the mathematical community I'm not surprised by the behavior, but I'm still hopeful that there are some members of the cryptographic community who are legit.
Otherwise the snarl is what we will see down the line with the world—and I'm sure all major world leaders—facing the big issue of trying to figure out what to do when RSA encryption goes away, probably literally overnight, as it's potentially broken now by others, but we don't know that so it still works if only out of faith in it.
# posted by James Harris @ 02:13 
