Faster Factoring Algorithm?

[Citrix]

What do you think, does there exist a faster factoring algorithm, than current methods? When do you think humanity will find it (Year)?

Just looking for some thoughts from the experts.

Thanks!

[xilman]

Originally Posted by Citrix

What do you think, does there exist a faster factoring algorithm, than current methods? When do you think humanity will find it (Year)?

Just looking for some thoughts from the experts.

Thanks!

I'll take this question in the spirit I think it was asked in. That is, I'll indulge in vigorous hand-waving and give my gut feelings. Note that I do not consider myself an expert.

Personally, I think there is a reasonable chance that there is a deterministic polynomial time factoring algorithm which runs on Turing machines.

Some of the reasons for this optimism.

An expected polynomial time algorithm exists for quantum Turing machines.

Factoring is easily proved to be in NP --- hint, multiplication is in P

Although a P-time algorithm hasn't yet been found, neither has factoring been shown not to be in P, despite a lot of effort in each direction.

Forty years ago, only exp-time algorithms were known. Then came a bunch of algorithms (CFRAC, QS, ECM and others) which in a well-defined sense are half-way between polynomial and exponential time. Then came an algorithm (NFS) which, in the same sense, is one third of the way from polynomial to exponential time. Progress towards a P-time algorithm has been made --- indeed, we are already two thirds of the way to the destination.

Analysis of an exponential-time algorithm, Pollard's rho, shows that it works by computing highly composite integers. Unfortunately, the number of factors of those integers isn't large enough for Pollard-rho to factor in P-time.

If we could calculate x! mod N in polynomial time it could be used to produce a P-time factoring algorithm. Once more, no such algorithm has been found yet neither has it been proved that an algorithm can not exist. x!, of course, is a highly composite integer.


I make no prediction as to when a P-time algorithm may be discovered, assuming one exists. It may be years ago (though I doubt it) or it may be decades or centuries hence. It's quite possible, in my opinion, that it may not be discovered by a human mathematician.


Paul

[Uncwilly]

As a tag-along to the main question:

Is there a faster or better method that is known, but is awaiting some breakthrough in computers before it can become practical?
Quantum machines can right?

[jasonp]

Originally Posted by xilman

If we could calculate x! mod N in polynomial time it could be used to produce a P-time factoring algorithm. Once more, no such algorithm has been found yet neither has it been proved that an algorithm can not exist.

Hans Riesel also thought so, and mentioned this idea in one of his books. When I pointed this out in sci.crypt back in 1998, Bob said the idea was 'unconvincing'. Nothing has happened recently that gives anyone reason to change that view, unless you can take the AKS primality test as cause for hope that hard problems in number theory are susceptible to solution using simple tools.

jasonp

[xilman]

Originally Posted by jasonp

Hans Riesel also thought so, and mentioned this idea in one of his books. When I pointed this out in sci.crypt back in 1998, Bob said the idea was 'unconvincing'. Nothing has happened recently that gives anyone reason to change that view, unless you can take the AKS primality test as cause for hope that hard problems in number theory are susceptible to solution using simple tools.

jasonp

Knuth mentioned it long before Riesel (I've read it in each of their books). I got it from Knuth and the observation was probably old before he wrote TAOCP Vol 2.

Thanks for reminding me of AKS. That is indeed additional grounds for optimism.

Primality testing went from being as hard as factoring, to slightly superpolynomial to expected polynomial to deterministic polynomial over the course of a few decades. I'm optimistic that it can be brought back to being as hard as factoring again.

Bob and I discussed the very same question about difficulty of factoring and prospects of improvement when we met last September. I suspect that I'm a bit more optimistic than he is, but he'll have to make his own comments on that score.

We've certainly each thought about possible algorithmic improvements, in quite different ways, but neither of us has got anywhere. That last should be obvious --- you would have heard from one of us if we had!


Paul

is O(ln) fast?

[mgb]

Notions versus notations!* Personally I think it may need a new development in mathematics as radical as congruence theory or the advent of complex numbers. Then factoring might be routine.

*Gauss