1280-Bit RSA

[Brandon Enright]

On Fri, 9 Jul 2010 21:16:30 -0400 (EDT)
Jonathan Thornburg <jthorn at astro.indiana.edu> wrote:

> The following usenet posting from 1993 provides an interesting bit
> (no pun itended) of history on RSA key sizes.  The key passage is the
> last paragraph, asserting that 1024-bit keys should be ok (safe from
> key-factoring attacks) for "a few decades".  We're currently just
> under 1.75 decades on from that message.  I think the take-home lesson
> is that forecasting progress in factoring is hard, so it's useful to
> add a safety margin...

This is quite interesting.  The post doesn't say but I suspect at the
factoring effort was based on using Quadratic Sieve rather than GNFS.
The difference in speed for QS versus GNFS starts to really diverge with
larger composites.  Here's another table:

RSA       GNFS      QS
===========================
256      43.68     43.73
384      52.58     55.62
512      59.84     65.86
664      67.17     76.64
768      71.62     83.40
1024     81.22     98.48
1280     89.46    111.96
1536     96.76    124.28
2048     109.41   146.44
3072     129.86   184.29
4096     146.49   216.76
8192     195.14   319.63
16384    258.83   469.80
32768    342.05   688.62

Clearly starting at key sizes of 1024 and greater GNFS starts to really
improve over QS.  If the 1993 estimate for RSA 1024 was assuming QS
then that was roughly equivalent to RSA 1536 today.  Even improving the
GNFS constant from 1.8 to 1.6 cuts off the equivalent of about 256 bits
from the modulus.

The only certainty in factoring techniques is that they won't get worse
than what we have today.

Brandon

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