Re: Factorization polynomially reducible to discrete log - known fact or not?

2006-07-12 Thread Max A.

On 7/9/06, Ondrej Mikle [EMAIL PROTECTED] wrote:


I believe I have the proof that factorization of N=p*q (p, q prime) is
polynomially reducible to discrete logarithm problem. Is it a known fact
or not? I searched for such proof, but only found that the two problems
are believed to be equivalent (i.e. no proof).


Take a look at this paper: http://portal.acm.org/citation.cfm?id=894497

Eric Bach  Discrete Logarithms and Factoring

ABSTRACT: This note discusses the relationship between the two
problems of the title. We present probabilistic polynomial-time
reduction that show: 1) To factor n, it suffices to be able to compute
discrete logarithms modulo n. 2) To compute a discrete logarithm
modulo a prime power p^E, it suffices to know It mod p. 3) To compute
a discrete logarithm modulo any n, it suffices to be able to factor
and compute discrete logarithms modulo primes. To summarize: solving
the discrete logarithm problem for a composite modulus is exactly as
hard as factoring and solving it modulo primes.

Max

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Re: Factorization polynomially reducible to discrete log - known fact or not?

2006-07-12 Thread Peter Kosinar

The algorithm is very simple:
1. Choose a big random value x from some very broad range
  (say, {1,2,..,N^2}).
2. Pick a random element g (mod N).
3. Compute y = g^x (mod N).
4. Ask for the discrete log of y to the base g, and get back some
  answer x' such that y = g^x' (mod N).
5. Compute x-x'.  Note that x-x' is a multiple of phi(N), and
  it is highly likely that x-x' is non-zero.  It is well-known
  that given a non-zero multiple of phi(N), you can factor N in
  polynomial time.


Not exactly. Consider N = 3*7 = 21, phi(N) = 12, g = 4, x = 2, x' = 5. 
You'll only get a multiple of phi(N) if g was a generator of the 
multiplicative group Z_N^*.


Peter

--
[Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI)  [ICQ] 134813278

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Re: Factorization polynomially reducible to discrete log - known fact or not?

2006-07-11 Thread Ondrej Mikle

Charlie Kaufman wrote:

I believe this has been known for a long time, though I have never seen the 
proof. I could imagine constructing one based on quadratic sieve.

I believe that a proof that the discrete log problem is polynomially reducible 
to the factorization problem is much harder and more recent (as in sometime in 
the last 20 years). I've never seen that proof either.

--Charlie



OK, I had the proof checked. I put it here: 
http://www.ms.mff.cuni.cz/~miklo1am/Factorization_to_DLog.pdf


Warning: it may be not what you'd expect.

First of all, it reduces the factorization to a discrete log in a group 
of unknown order (or put in another words: you'd need to factorize to 
learn the group order). It has been proven by V. Shoup that when group 
operation and the inverse are the only operations that can be done with 
group elements, then the best algorithm can be O(sqrt(n)), where n is 
the number of elements. I guess then the group of Z_N* (where N=pq) of 
unknown order qualifies for this if we don't want to use factorization 
(actually you can't compute inverse group operation here). In the light 
of this fact, is this proof of any use?


Even if the proof is not useful, is the generator picking lemma (lemma 
2) anything new? It states basically this:
In any cyclic group of order n there is at least 1/log2(n) probability 
of picking a generator randomly and thus generator can be found in 
polynomial time with overwhelming probability of success.


The only facts close to this lemma I found were:
1) Product phi(p_i)/p_i for consecutive primes p_i approaches zero as 
more and factors are added to the product (phi is Euler phi function). 
The lemma states a lower bound for the product.
2) If the generalized Riemann hypothesis is true, then for every prime 
number p, there exists a primitive root modulo p that is less than 70 
(ln(p))^2. (http://en.wikipedia.org/wiki/Primitive_root_modulo_n)


Charlie:
Thanks for answering my second question which I have not asked yet :-) 
(the reduction in opposite direction). I'm also working on the opposite 
reduction, but I'm at best halfway through (and not sure if I am able to 
finish it).


Last question:
Joseph Ashwood mentioned someone who claimed to have algorithm for 
factorization and had only the reduction to DLP. Anyone knows where I 
could find the algorithm? Or maybe name of the person, so I could search 
the web.


Thanks
  O. Mikle

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RE: Factorization polynomially reducible to discrete log - known fact or not?

2006-07-10 Thread Charlie Kaufman
I believe this has been known for a long time, though I have never seen the 
proof. I could imagine constructing one based on quadratic sieve.

I believe that a proof that the discrete log problem is polynomially reducible 
to the factorization problem is much harder and more recent (as in sometime in 
the last 20 years). I've never seen that proof either.

--Charlie

-Original Message-
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of Ondrej Mikle
Sent: Sunday, July 09, 2006 12:22 PM
To: cryptography@metzdowd.com
Subject: Factorization polynomially reducible to discrete log - known fact or 
not?

Hello.

I believe I have the proof that factorization of N=p*q (p, q prime) is
polynomially reducible to discrete logarithm problem. Is it a known fact
or not? I searched for such proof, but only found that the two problems
are believed to be equivalent (i.e. no proof).

I still might have some error in the proof, so it needs to be checked by
someone yet. I'd like to know if it is already known (in that case there
would be no reason to bother with it).

Thanks
   O. Mikle

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