We're going to be deploying a new crypto scheme in Tahoe-LAFS next year -- the year 2010. Tahoe-LAFS is used for long-term storage, and I won't be surprised if people store files on Tahoe-LAFS in 2010 and then rely on the confidentiality and integrity of those files for many years or even decades to come. (People started storing files on Tahoe-LAFS in 2008 and so far they show no signs of losing interest in the integrity and confidentiality of those files.)

This long-term focus makes Tahoe-LAFS's job harder than the job of protecting transient network packets. If someone figures out in 2020 or 2030 how to spoof a network transaction that you sent in 2010 (see [1]), it'll be far too late to do you any harm, but if they figure out in 2030 how to alter a file that you uploaded to a Tahoe-LAFS grid in 2010, that might harm you.

Therefore I've been thinking about how to make Tahoe-LAFS robust against the possibility that SHA-256 will turn out to be insecure.

A very good way to do this is to design Tahoe-LAFS so that it relies as little as possible on SHA-256's security properties. The property that seems to be the hardest for a secure hash function to provide is collision-resistance. We are analyzing new crypto schemes to see how many security properties of Tahoe-LAFS we can continue to guarantee even if the collision-resistance of the underlying secure hash function fails, and similarly for the other properties of the secure hash function which might fail [2].

This note is not about that design process, though, but about how to maximize the chance that the underlying hash function does provide the desired security properties.

We could use a different hash function than SHA-256 -- there are many alternatives. SHA-512 would probably be safer, but it is extremely expensive on the cheap, low-power 32-bit ARM CPUs that are one of our design targets [3], and the output size of 512 bits is too large to fit into Tahoe-LAFS capabilities. There are fourteen candidates left in the SHA-3 contest at the moment. Several of them have conservative designs and good performance, but there is always the risk that they will be found to have catastrophic design flaws or that a great advance in hash function cryptanalysis will suddenly show how to crack them. Of course, a similar risk applies to SHA-256!

So I turn to the question of how to combine multiple hash functions to build a hash function which is secure even if one or more of the underlying hash functions turns out to be weak.

I've read several interesting papers on the subject -- such as [4, 5] and especially "Robust Multi-Property Combiners for Hash Functions Revisited" by Marc Fischlin, Anja Lehmann, and Krzysztof Pietrzak [6]. The good news is that it turns out to be doable! The latter two papers show nice strong theoretical results -- ways to combine hash functions so that the resulting combination is as strong or stronger than the two underlying hash functions. The bad news is that the proposal in [6] builds a combined function whose output is twice the size of the output of a single hash function. There is a good theoretical reason for this [4], but it won't work for our practical engineering requirements -- we need hash function outputs as small as possible (partially due to usability issues)

The other bad news is that the construction proposed in [6] is complicated, underspecified, and for the strongest version of it, it imposes a limit on the length of the inputs that you can feed to your hash function. It grows to such complexity and incurs such limitations because it is, if I may call it this, "too theoretical". It is designed to guarantee certain output properties predicated on minimal theoretical assumptions about the properties of the underlying hash functions. This is a fine goal, but in practice we don't want to pay such a high cost in complexity and performance in order to gain such abstract improvement. We should be able to "hedge our bets" and achieve a comfortable margin of safety with a very simple and efficient scheme by making stronger, less formal, but very plausible assumptions about the underlying hash functions. Read on.

I propose the following combined hash function C, built out of two hash functions H1 and H2:

C(x) = H1(H1(x) || H2(x))

The first observation is that if H1 is collision-resistant then so is C. In practice I would expect to use SHA-256 for H1, so the resulting combiner C[SHA-256, H2] will be at least as strong as SHA-256. (One could even think of this combiner C as just being a tricky way to strengthen SHA-256 by using the output of H2(x) as a randomized salt -- see [7].)

The next observation is that finding a pair of inputs x1, x2 which collide in *both* H1 and in H2 is likely to be much harder than finding a pair of inputs that collide in H1 and finding a pair of inputs that collide in H2 (see [5]).

Now the reason that a combiner like this one is not published in theoretical crypto literature is that it obviously could fail if the outer hash function H1 fails. For example, even if H2 is collision- resistant, if H1 turns out to be susceptible to collisions, then theoretically speaking C[H1, H2] might be susceptible to collisions. However, in real life C[H1, H2] would most likely still be collision resistant!

All practical attacks on real hash functions so far (if I understand correctly) are multi-block attacks in which the attacker is able to feed a sufficiently long and unconstrained input to the hash functions that the effects of the later parts of his inputs are able to manipulate the state generated by the earlier parts of his inputs. My combiner C uses H1 in its outer invocation on a single- block-sized input, which means no such multi-block attacks are possible on the outer invocation. In addition, the inputs that the attacker gets to feed to the outer invocation of H1 are highly constrained. Basically, he would already have to be very good at manipulating the inner invocations H1 and H2 in ways that he isn't supposed to before he can even begin to manipulate the outer invocation of H1.

A measure of the practical security of a combiner like this one would be "how safe would it be if it were instantiated using broken practical hash functions such as MD5 and SHA1?". It appears to me (from an admittedly cursory analysis) that there is no realistic way to find collisions in C[MD5, SHA1] even though there are realistic ways to find collisions in MD5 and in SHA1. Of course, I'm not proposing to use C[MD5, SHA1]! I'm proposing to use C[SHA-256, _] where _ is some other hash function which is believed to be strong. The example of instantiating C with MD5 and SHA1 just goes to show that C is a hash function which is stronger than either of its two underlying hash functions.

The other desirable security properties such as second-preimage resistance and pre-image resistance seem to follow the same pattern as collision-resistance -- C[H1, H2] seems to be much stronger than H1 or H2 alone.



[4] Krzysztof Pietrzak: "Non-Trivial Black-Box Combiners for Collision-Resistant Hash-Functions don't Exist" [5] Jonathan J. Hoch, Adi Shamir: "On the Strength of the Concatenated Hash Combiner when All the Hash Functions are Weak" [6] Marc Fischlin, Anja Lehmann, Krzysztof Pietrzak: "Robust Multi- Property Combiners for Hash Functions Revisited"

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