Recently I have been hearing some odd “Twofish has been broken” rumors. I thought I’d quell them once and for all.

Rumors of the death of Twofish has been greatly exaggerated.

The analysis in question is by Shiho Moriai and Yiqun Lisa Yin, who published their results in Japan in 2000. Recently, someone either got a copy of the paper or heard about the results, and rumors started spreading.

Here’s the actual paper. It presents no cryptanalytic attacks, only some hypothesized differential characteristics. Moriai and Yin discovered byte-sized truncated differentials for 12- and 16-round Twofish (the full cipher has 16 rounds), but were unable to use them in any sort of attack. They also discovered a larger, 5-round truncated differential. No one has been able to convert these differentials into an attack, and Twofish is nowhere near broken. On the other hand, they are excellent and interesting results — and it’s a really good paper.

In more detail, here are the paper’s three results:

- The authors show a 12-round truncated differential characteristic that predicts that the 2nd byte of the ciphertext difference will be 0 when the plaintext difference is all-zeros except for its last byte. They say the characteristic holds with probability 2
^{-40.9}. Note that for an ideal cipher, we expect the 2nd byte of ciphertext to be 0 with probability 2^{-8}, just by chance. Of course, 2^{-8} is much, much larger than 2^{-40.9}. Therefore, this is not particularly useful in a distinguishing attack.
One possible interpretation of their result would be to conjecture that the 2nd byte of ciphertext difference will be 0 with probability 2^{-8} + 2^{-40.9} for Twofish, but only 2^{-8} for an ideal cipher. Their characteristic is just one path. If one is lucky, perhaps all other paths behave randomly and contribute an additional 2^{-8} factor to the total probability of getting a 0 in the 2nd byte of ciphertext difference. Perhaps. One might conjecture that, anyway.

It is not at all clear whether this conjecture is true, and the authors are careful not to claim it. If it were true, it might lead to a theoretical distinguishing attack using 2^{75} chosen plaintexts or so (very rough estimate). But I’m not at all sure that the conjecture is true.

- They show a 16-round truncated differential that predicts that the 2nd byte of the ciphertext difference will be 0 (under the same input difference). Their characteristic holds with probability 2
^{-57.3} (they say). Again, this is not very useful.
Analogously to the first result, one might conjecture that the 2nd byte of the ciphertext difference will be 0 with probability 2^{-8} + 2^{-57.3} for Twofish, but probability 2^{-8} for an ideal cipher. If this were true, one might be able to mount a distinguishing attack with 2^{100} chosen plaintexts or so (another very rough estimate). But I have no idea whether the conjecture is true.

- They also show a 5-round truncated differential characteristic that predicts that the input difference that is non-zero everywhere except in its 9th byte will lead to an output difference of the same form. This characteristic has probability 2
^{-119.988896}, they say (but they also say that they have made some approximations, and the actual probabilities can be a little smaller or a little larger). Compared to an ideal cipher, where one would expect this to happen by chance with probability 2^{-120}, this isn’t very interesting. It’s hard to imagine how this could be useful in a distinguishing attack.

The paper theorizes that all of these characteristics might be useful in an attack, but I would be very careful about drawing any conclusions. It can be very tricky to go from single-path characteristics whose probability is much smaller than the chances of it happening by chance in an ideal cipher, to a real attack. The problem is in the part where you say “let’s just assume all other paths behave randomly.” Often the other paths do not behave randomly, and attacks that look promising fall flat on their faces.

We simply don’t know whether these truncated differentials would be useful in a distinguishing attack. But what we do know is that even if everything works out perfectly to the cryptanalyst’s benefit, and if an attack is possible, then such an attack is likely to require a totally unrealistic number of chosen plaintexts. 2^{100} plaintexts is something like a billion billion DVDs’ worth of data, or a T1 line running for a million times the age of the universe. (Note that these numbers might be off by a factor of 1,000 or so. But honestly, who cares? The numbers are so huge as to be irrelevent.) And even with all that data, a distinguishing attack is not the same as a key recovery attack.

Again, I am not trying to belittle the results. Moriai and Yin did some great work here, and they deserve all kinds of credit for it. But even from a theoretical perspective, Twofish isn’t even remotely broken. There have been no extensions to these results since they were published five years ago. The best Twofish cryptanalysis is still the work we did during the design process: available on the Twofish home page.

Posted on November 23, 2005 at 12:15 PM •
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