full-disclosure - Rapid integer factorization = end of RSA?

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Date: Thu, 26 Apr 2007 10:53:56 +0400
From: Eugene Chukhlomin <chukh29ru@...oline.su>
To: full-disclosure@...ts.grok.org.uk
Subject: Rapid integer factorization = end of RSA?

Hi list!
I discovered a new method of integer factorization for any precision 
numbers, probable it should be an end of RSA era.
Details:
Let N - the ring and N = p*q
Then, (-p) in terms of ring(N) is equal (N-p)
Lemma:
p*(-q)=p*q*(-p)
and respective:
(-p)*q=p*q*(-q)
Proof:
p*(-q)=p*(N-q) - by the data, then 
p*(-q)=p*(p*q-q)=p*pq-p*q=p*q*p-p*q=(p-1)*(p*q)
(-p)*q=q*(N-p) - by the data, then 
(-p)*q=(p*q-p)*q=p*q*q-p*q=p*q*q-p*q=(q-1)*(p*q)
Q. E. D.
Gypothesis:
Let N = p*q = A1*B1 + A2*B2... + An*Bn
Then exists some subset(A1...An) and respective subset(B1...Bn), which 
satisfies for equality:
A1*(-B1)+A2*(-B2)...+An*(-Bn) = p*(-q)=p*q*(p-1)
or
A1*(-B1)+A2*(-B2)...+An*(-Bn) = (-p)*q=p*q*(q-1)

If found such (A1...An) and (B1...Bn), we can find p or q by dividing 
p*(q-1) on p*q:
p*(q-1)=p*q*(p-1) => (p*(q-1))/(p*q)=(p-1) => (p-1)+1 = p
or
(p-1)*q=p*q*(q-1)=>((-p)*q)/(p*q)=(q-1) => (q-1)+1 = q

Sample: 21 = 3*7
Let's view a binary representation of this number: 10101 => 2^4 + 2^2 + 
1 => 4*4+2*2+1*1
Then, we can try to find 7*(-3) in terms of ring(21):
4*(-4) + 2(-2) + 1*(-1) => 4*(21-4)+2*(21-2)+1*(21-1)=>4*17+2*19+1*20 = 
68+38+20=>
68+38+20 = 126 = 6*21
6+1=7
This implementation of my gypothesis has very hard complexity (about a 
log2(N)! comparations), but exists a short way with fixed complexity for 
implementation of hypothesis ("plan B") - but, by ethical reason, I'll 
not post it here.
Regards,
Eugene Chukhlomin

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