Saturday, January 19, 2008

 

JSH: Result is for real, so is corruption

So the scenario that would occur if I factored an RSA number is most likely I'd be killed, and then the information would just kind of disappear as people worked to clean up the situation and most of you would just go on about your lives.

And soon enough it'd all be forgotten.

That is why Usenet is important in this scenario to spread the information widely enough that by the time certain unethical people who don't give a damn about laws figure out what it is, just killing me and cleaning up by suppression is not an option.

Given George W. Bush's current history of lack of love of the law, I don't think I can really be challenged on my fear that this government in the US is quite capable of breaking any law which could mean me getting murdered for my great achievement of discovery.

So there will be no absolute demonstration by factoring of an RSA public key though I dare any of you to do it, as then at least I wouldn't die alone:

Given a nonzero target composite T and integer factors f_1 and f_2, such that f_1*f_2 = nT, and any prime p, the following relations must be true:

f_1 = ak mod p

and

f_2 = a^{-1}(1 + a^2)k mod p

where

k^2 = (a^2+1)^{-1}(nT) mod p.

Those represent the fundamental factoring relations that underpin ALL composite factorizations.

Example: n=1, T=119, p=11, and a=2 gives

k^2 = (5)^{-1}(119) mod 11 = 9(9) mod 11 = 4 mod 11.

And k = 2 mod 11 is a solution, so f_1 = 4 mod 11, and

f_2 = 2^{-1}(1+4)(2) mod 11 = 5 mod 11.

In this case, subtracting 11 gives one prime factor as f_1 = -7 mod 11, and f_2 = -6 mod 11. Subtracting 11 again from f_2 gives -17.





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