[Cryptography] New attacks on discrete logs?

[Bear]

On Sat, 2014-05-24 at 08:07 -0400, Jerry Leichter wrote:
> On May 23, 2014, at 5:03 PM, Bear <bear at sonic.net> wrote:
 
> > Would anyone like to clarify what exactly they mean by 
> > "small" and "large" characteristic here?  Please?
> Ah, time for some fun math I haven't thought about in years.
> 
> A (commutative) group G is a bunch of elements with a binary operation . with the following properties:
> 

Thanks Jerry, for that lesson on group theory.  It was genuinely
helpful.  It permits me, at least, to formulate the question in 
a more specific way. 

What I really wanted to know was how to adjust security 
estimates for modular groups as a function of the bit length
of the prime factor of the modulus.

Using modular addition as the operation, modular groups in the 
integers are, for reasons you explained, necessarily integers 
modulo some number which has exactly one prime factor.  

And if that factor is "small" this algorithmic advance makes 
operations in those groups somewhat easier to reverse, and if 
that factor is "large" this algorithmic advance does not make 
operations in those groups significantly easier to reverse.

Because reversing group operations is provably at least as 
hard as factoring, (though in practice, at least as far as we 
know how to do it now, much harder) I could conservatively 
interpret "large" as meaning >4096 bits, or out of range of 
current factoring technology, and "small" as <2048 bits.  

But that's a very dire interpretation, and would effectively
destroy Elliptic Curve cryptosystems outright, which no one 
so far is claiming that this advance does.  

So what bit lengths (of the prime factor of the modulus) are 
we talking about when we say "small" and "large" in this 
context?

			Bear