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((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.