((Comments are appreciated. -Bruce))
Factoring large numbers is hard. Unfortunately for algorithm
designers, it is getting easier. Even worse, it is getting
easier faster than mathematicians expected. In 1976 Richard Guy
wrote: "I shall be surprised if anyone regularly factors numbers
of size 10^80 without special form during the present century."
In 1977 Ron Rivest said that factoring a 125-digit number would
take 40 quadrillion years. In 1994 a 129-digit number was
factored. If there is any lesson in all this, it is that making
predictions is foolish.
Table 1 shows factoring records over the past dozen years. The
fastest factoring algorithm during the time was the quadratic
sieve.
Table 1: Factoring Using the Quadratic Sieve
year # of decimal how many times harder to
digits factored factor a 512-bit number
1983 71 > 20 million
1985 80 > 2 million
1988 90 250,000
1989 100 30,000
1993 120 500
1994 129 100
These numbers are pretty frightening. Today it is not uncommon
to see 512-bit numbers used in operational systems. Factoring
them, and thereby completely compromising their security, is well
in the range of possibility: A weekend-long worm on the Internet
could do it.
Computing power is generally measured in mips-years: a one-
million-instruction-per-second computer running for one year, or
about 3*10^13 instructions. By convention, a 1 mips machine is
equivalent to the DEC VAX 11/780. Hence, a mips-year is a VAX
11/780 running for a year, or the equivalent.
The 1983 factorization of a 71-digit number required 0.1 mips-
years; the 1994 factorization of a 129-digit number required
5000. This dramatic increase in computing power resulted largely
from the introduction of distributed computing, using the idle
time on a network of workstations. The 1983 factorization used
9.5 CPU hours on a single Cray X-MP; the 1994 factorization used
the idle time on 1600 computers around the world for about 8
months. Modern factoring methods lend themselves to this kind of
distributed implementation.
The picture gets even worse. A new factoring algorithm has taken
over from the quadratic sieve: the general number field sieve.
In 1989 mathematicians would have told you that the general
number field sieve would never be practical. In 1992 they would
have told you that it was practical, but only faster than the
quadratic sieve for numbers greater than 130-150 digits or so.
Today it is known to be faster than the quadratic sieve for
numbers well below 116 digits. The general number field sieve
can factor a 512-bit number over 10 times faster than the
quadratic sieve. The algorithm would require less than a year to
run on an 1800-node Intel Paragon. Table 2 gives the number of
mips-years required to factor numbers of different sizes, given
current implementations of the general number field sieve.
Table 2: Factoring Using the General Number Field Sieve
# of bits mips-years required to factor
512 30,000
768 2*10^8
1024 3*10^11
1280 1*10^14
1536 3*10^16
2048 3*10^20
And the general number field sieve is still getting faster.
Mathematicians keep coming up with new tricks, new optimizations,
new techniques. There's no reason to think this trend won't
continue. A related algorithm, the special number field sieve,
can already factor numbers of a certain specialized form--numbers
not generally used for cryptography--must faster than the general
number field sieve can factor general numbers of the same size.
It is not unreasonable to assume that the general number field
sieve can be optimized to run this fast; it is possible that the
NSA already knows how to do this. Table 3 gives the number of
mips-years required for the special number field sieve to factor
numbers of different lengths.
Table 3: Factoring Using the Special Number Field Sieve
# of bits mips-years required to factor
512 < 200
768 100,000
1024 3*10^7
1280 3*10^9
1536 2*10^11
2048 4*10^14
At a European Institute for System Security workshop in 1992, the
participants agreed that a 1024-bit modulus should be sufficient
for long-term secrets through 2002. However, they warned:
"Although the participants of this workshop feel best qualified
in their respective areas, this statement [with respect to
lasting security] should be taken with caution." This is good
advice.
The wise cryptographer is ultra-conservative when choosing
public-key key lengths. To determine how long a key you need
requires you to look at both the intended security and lifetime
of the key, and the current state-of-the-art of factoring. Today
you need a 1024-bit number to get the level of security you got
from a 512-bit number in the early 1980s. If you want your keys
to remain secure for 20 years, 1024 bits is likely too short.
Even if your particular secrets aren't worth the effort required
to factor your modulus, you may be at risk. Imagine an automatic
banking system that uses RSA for security. Mallory can stand up
in court and say: "Did you read in the newspaper in 1994 that
RSA-129 was broken, and that 512-bit numbers can be factored by
any organization willing to spend a few million dollars and wait
a few months? My bank uses 512-bit numbers for security, and by
the way I didn't make these seven withdrawals." Even if Mallory
is lying, the judge will probably put the onus on the bank to
prove it.
Earlier I called making predictions foolish. Now I am about to
make some. Table 4 gives my recommendations for public-key
lengths, depending on how long you require the key to be secure.
There are three key lengths for each year, one secure against an
individual, one secure against a major corporation, and the third
secure against a major government.
Here are some assumptions from the mathematicians who factored
RSA-129:
We believe that we could acquire 100 thousand machines
without superhuman or unethical efforts. That is, we would
not set free an Internet worm or virus to find resources for
us. Many organizations have several thousand machines each
on the net. Making use of their facilities would require
skillful diplomacy, but should not be impossible. Assuming
the 5 mips average power, and one year elapsed time, it is
not too unreasonable to embark on a project which would
require half a million mips years.
The project to factor the 129-digit number harnesses an estimated
0.03% of the total computing power of the Internet, and they
didn't even try very hard. It isn't unreasonable to assume that
a well-publicized project can harness 0.1% of the world's
computing power for a year.
Assume a dedicated cryptanalyst can get his hands on 10,000 mips-
years, a large corporation can get 10^7 mips-years, and that a
large government can get 10^9 mips-years. Also assume that
computing power will increase by a factor of ten every five
years. And finally, assume that advances in factoring
mathematics allows us to factor general numbers at the speeds of
the special number field sieve. Table 4 recommends different key
lengths for security during different years.
Table 4: Recommended public-key key lengths (in bits)
Year vs. I vs. C vs. G
1995 768 1280 1536
2000 1024 1280 1536
2005 1280 1536 2048
2010 1280 1536 2048
2015 1536 2048 2048
Remember to take the value of the key into account. Public keys
are often used to secure things of great value for a long time:
the bank's master key for a digital cash system, the key the
government uses to certify its passports, a notary public's
digital signature key. It probably isn't worth the effort to
spend months of computing time to break an individual's private
key, but if you can print your own money with a broken key the
idea becomes more attractive. A 1024-bit key is long enough to
sign something that will be verified within the week, or month,
or even a few years. But you don't want to stand up in court
twenty years from now with a digitally signed document, and have
the opposition demonstrate how to forge documents with the same
signature.
Making predictions beyond the near future is even more foolish.
Who knows what kind of advances in computing, networking, and
mathematics are going to happen by 2020? However, if you look at
the broad picture, in every decade we can factor numbers twice as
long as in the previous decade. This leads to Table 5.
Table 5: Long-range factoring predictions
Year Key length (in bits)
1995 1024
2005 2048
2015 4096
2025 8192
2035 16,384
2045 32,768
Not everyone will agree with my recommendations. The NSA has
mandated 512-bit to 1024-bit keys for their Digital Signature
Standard--far less than I recommend for long-term security. PGP
has a maximum RSA key length of 1280 bits. Lenstra, the world's
most successful factorer, refuses to make predictions past ten
years. And Table 6 gives Ron Rivest's key-length
recommendations, originally made in 1990, which I consider much
too optimistic. While his analysis looks fine on paper, recent
history illustrates that surprises regularly happen. It makes
sense to choose your keys to be resilient against future
surprises.
Table 6: Rivest's Optimistic Key-Length Recommendations (In
Bits)
Year Low Avg High
1990 398 515 1289
1995 405 542 1399
2000 422 572 1512
2005 439 602 1628
2010 455 631 1754
2015 472 661 1884
2020 489 677 2017
Low estimates assume a budget of $25,000, the quadratic
sieve algorithm, and a technology advance of 20% per year.
Average estimates assume a budget of $25 million, the
general number field sieve algorithm, and a technology
advance of 33% per year. High estimates assume a budget of
$25 billion, a general quadratic sieve algorithm running at
the speed of the special number field sieve, and a
technology advance of 45% per year.
There is always the possibility that an advance in factoring will
surprise me as well, but I think that unlikely. But why trust
me? I just proved my own foolishness by making predictions.
