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New Time and Space Based Key Size Equivalents for RSA and Diffie- Hellman

2000-12-13 22:34:03
I am sorry for the multiple postings, but I thought this particular subject,
although probably quite controversial, might be of interest to the many
peoples following these mailing lists, especially because of the upcoming
adoption of the AES algorithm by many IETF protocols.

As symmetric keys grow, they can be attacked by more processors without a
change in processor technology since the memory requirements for breaking
symmetric keys remain trivial.  However, for the Number Field Sieve (NFS)
algorithm, which is currently the most efficient method to break RSA keys,
this is not true.  Based on this premise, the "time and space" based RSA key
size equivalents previously published in the RSA Labs Bulletin #13 of April
2000 by Robert Silverman ( have recently
been extended to cover all the AES symmetric key sizes in the latest draft
of ANSI X9.44, which will eventually become the ANSI standard for RSA key

                        Time and Space
Symmetric               Equivalent
Key Size                RSA Key Size
(in bits)               (in bits)

64                      450
128                     1620
192                     2500
256                     4200

These "time and space" based key sizes equivalents assume that both time and
memory are binding constraints in order to break RSA keys.  This same draft
also indicates that beyond RSA key sizes of 768 bits one can no longer
effectively utilize 32-bit processors with the NFS algorithm because the
required memory exceeds what can be addressed in 32 bits; one is forced to
use 64-bit machines.  Beyond RSA key sizes of about 2500 bits, the memory
requirements for the NFS algorithm exceed what can be addressed even on 64
bit machines.

For your information, here are also the estimated "time" only based RSA key
size equivalents for solving the NFS problem from the same ANSI draft:

                        Time Only
Symmetric               Equivalent
Key Size                RSA Key Size
(in bits)               (in bits)

64                      512
128                     2550
192                     6700
256                     13500

Note that either of these sets of RSA key size equivalents could be used
with Diffie-Hellman for solving the value of "p" since the NFS algorithm is
also the most efficient method to break Diffie-Hellman algorithm today.
Note also that these time only equivalents numbers are slightly smaller than
those from ANSI X9.42 for Diffie-Hellman (i.e. 2550 vs 3072 for 128 bits,
6700 vs 7680 for 192 bits and 13500 vs 15360 for 256 bits) and the numbers
in Hilarie Orman's Internet Draft (i.e.

Shouldn't IETF standards mention these new "time and space" based key size
equivalents in addition to existing "time" only based key size equivalents,
and possibly even suggest that "time and space" based key size equivalents
be used for RSA and Diffie-Hellman?  Why mandate larger equivalent key sizes
when smaller equivalent key sizes can probably suffice?

Food for thought!


Francois Rousseau
Director of Standards and Conformance
1688 Woodward Drive
Ottawa, Ontario, CANADA, K2C 3R7
frousseau(_at_)chrysalis-its(_dot_)com      Tel. (613) 723-5076 ext. 419     Fax. (613) 723-5078
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