Re: Rapid integer factorization = end of RSA?

[Stanislaw Klekot <dozzie () dynamit im pwr wroc pl>]

On Thu, Apr 26, 2007 at 10:53:56AM +0400, Eugene Chukhlomin wrote:
> 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.

Funny way to pull the -1 out from the parenthesis.
p * (-q) = p * (-1) * q = p * q * (-1)       (mod pq)
That is, p * (-q) = 0      (mod pq).

> 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)

For example, whole A_k and B_k, k = {1..n} sets? Second and third
expressions in both lines are congruent to 0 mod pq.

> 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
  ^^^^^^^^^^^^^^^^^
This is untrue.
  p * (q - 1) = p * q - p = -p != 0   (mod pq)
  p * q * (p - 1) = 0 * (p - 1) = 0   (mod pq)

> p*(q-1)=p*q*(p-1) => (p*(q-1))/(p*q)=(p-1) => (p-1)+1 = p
                       ^^^^^^^^^^^^^^^
Dividing by zero in _any ring_ is illegal.

By the way, if you find x = p * (q - 1) you can use Euclidean algorithm
to find GCD(x, pq). Since GCD(q - 1, q) = 1, you get GCD(x, p), and that
would be p as p divides x.

> 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

OK, but where did you get 7 and -3 (from underscored expression) from?
3*7 is public, but both 3 and 7, as elements of multiplication, are
private. And if you get (7, -3) pair, why didn't you simply multiplicate
the second element of this pair by -1?

> 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

-- 
Stanislaw Klekot

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