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To formulate this question precisely, I will define an idealized hypothetical "perfect" hash function H(n) which has nice scalability properties, and will formulate a problem PERFECT HASHCASH in terms of that, understanding that practical considerations may end up yielding only an approximation of this ideal.

To keep it simple, we will say that our hash function H(n) takes as input a single natural number n. Then we say that H(n) is a perfect hash function iff:

  1. H(n) maps each natural number to an infinite binary sequence, of which the time complexity to compute any initial segment s is polynomial in the size of n and s, (making it a sponge function).
  2. For any initial segment of length d, the set of all natural numbers n such that H(n) shares that initial segment has natural density = 1/(2^d).

The first thing formalizes the scalability of our function, and the second thing formalizes the idea that we want all hashes to appear roughly "equally often" as an output. Other than that, our perfect hash function is a black box, and we don't care much about exactly how it works, so long as it meets the above properties, as well as the usual desiderata applying to hash functions (easy to compute, hard to invert, hard to find collisions, etc).

Predicated on the assumption that a perfect hash function exists, we can now define the problem PERFECT HASHCASH as follows: PERFECT HASHCASH takes as input a perfect hash function H, a natural number n, and an all-zeroes vector 0^d of length d, which can be thought of as a unary representation of d. A solution to PERFECT HASHCASH consists of an n and d such that H(n) starts with 0^d.

Given those inputs, it is clear that PERFECT HASHCASH is in the complexity class TFNP, since this is a function problem and a solution is guaranteed to exist.

Can we also identify PERFECT HASHCASH as a member of any complexity class finer than TFNP?

Could it perhaps be in PPP? PPA? PPAD? Something else?

For background, see Complexity class on Wikipedia.


EDIT: the above question has been overhauled, as in the way that I originally formulated it I assumed that SHA256 is what I'm now calling a perfect hash function. Many people have noted in the comments that this may not be true, so rather than place the emphasis in this question on whether SHA256 specifically has the nice scaling properties we want, I defined an idealized hash function that we hope SHA256 at least approximates nicely enough for real-world purposes, and rephrased the question in terms of that.

As a final note to clear up any potential confusion, to make PERFECT HASHCASH resemble real Hashcash, we'd have to make one more assumption: that there exists some way to start with a block of data (an email, a Bitcoin block, etc) and somehow derive a characteristic perfect hash function from that, perhaps by "salting" a different perfect hash function in a way that the result is also another perfect hash function. So in the case of a "perfect Bitcoin," all of the miners on the bitcoin network would be working with their own unique perfect hash functions H'(n) which are somehow tied to the block they're working on, and each miner would simply try H'(0), H'(1), H'(2), ... in order until they find something starting with enough 0's. Each H' would be a different input to PERFECT HASHCASH.

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+1 though not everyone on this site is a cryptographer; it may be beneficial to describe the acronyms (FNP, TFNP, etc) or to ask on crypto.stackoverflow.com –  makerofthings7 Oct 16 '13 at 14:09
    
I think it is the responsibility of the asker to make his question understandable to potential experts answering the question, giving all background information shouldn't be necessary in every question, it would inflate questions too much. –  Murch Oct 25 '13 at 10:33
    
Polynomial in what value? What's playing the role of input size here? –  Nate Eldredge Feb 1 at 20:47
    
Nate: Difficulty. –  Mike Battaglia Feb 2 at 0:21
    
The pigeonhole principle guarantees that there are an infinite number of solutions for some x. It does not guarantee an infinite number of solutions for every x - I'm not sure we can disprove that there is some x for which there is no solution. –  Meni Rosenfeld Feb 3 at 14:12
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1 Answer

There's a reason cryptographic hash functions, like the double SHA256 used for proof-of-work in Bitcoin, are not usually described using these complexity classes that classify asymptotic behavior. In fact, there are several.

  1. A technical reason is that hash functions often do not scale. For example, it is not defined how one would extend the proof-of-work to operate on 512 bits. A natural choice would be to use SHA512 then, but going from SHA256 to SHA512 takes alot of essentially arbitrary choices, like changing the number of rounds from 64 to 80, which are standardized but not in a naturally scaling way and not for arbitrarily large hash sizes.

  2. It is not relevant for the cryptographer. Even a NP-complete hash function, which would be the strongest amongst the complexity cases you listed for building a strong hash, does not guarantee all what we want from either a cryptographic hash or a proof-of-work function. To qualify for NP-completeness is merely a strong heuristic that the problem cannot be solved by an algorithm that is, asymptotically, less than exponential. But for a good hash function we want it to be, at the very limited bitcount we choose to use it at, maximally exponential in the sense that solving it is really as hard as trying every possibility in a hash function. For its corresponding proof-of-work function with such a difficulty that only a fraction x of the output range is acceptable, this would mean that we should expect to require a number of attempts of x/2 times the size of full output range to find a proof-of-work. Anything better than that would prompt an academic to call the associated hash function broken, even if it merely reduces the number of tries in half, which would still put it in an exponential complexity class and would easily be possible even with a NP-complete function.

    An impressive (but only superficially related) example of how picking something seemingly NP-complete is insufficient to get something cryptographically hard is knapsack cryptography. Of course, there the problem was that by picking special cases the complexity of the problem was reduced. The point is that even a NP-complete problem can be less difficult than indeed having to try every solution, despite it sometimes being described that way! For cryptography-grade quality, having to try every input is meant literally; for the complexity analysis, it is good enough if the asymptotic scaling remains exponential in the number of bits. So if you could reduce the problem to another NP-complete function taking only every 1000th bit as input, that would be good enough for classifying the problem as NP (and even NP-complete if a similar mapping worked in reverse), but not for being of interest for cryptographic applications.

  3. It's difficult! And I think this difficulty has already let you astray: Even your arguments for placing this problem in TFNP are, whilst really close to the truth, not true in the mathematical sense. For example, if I specify x=0, no y can produce hash(y) < x, contradicting your assertion. If all other x are fine or if there is a required minimum value for x probably depends on how you define the "strings" y you want hashcash to operate on. For Bitcoin with a limited number of bits entering the double-SHA256, I wouldn't be surprised if x=1 also does not have a solution, i.e. if no block hash can become exactly zero. Of course we'll probably never know. In practice, it is desirable that a hash function should produce total proof-of-work functions in the way you describe, but I don't think it is a proven quality. Disclaimer: I honestly don't know. You really should ask a cryptographer.

    What remains to be done to answer your question, after finding how the hash function scales and verifying that it remains polynomial for arbitrary large sizes, is just this proof that the corresponding proof-of-work function is total. If this proof can be done using the pigeon hole principle, you have shown that it is in PPP, etc.

    So where is the difficulty? For example, if y has at least as many bits as x, and if we change your hashcash to have a less-than-or-equal rather just less-than, and if we are willing to multilate it further to the point that either finding a proof-of-work or the existence of a hash collision is good enough to make "hashcash" true, then the pigeon-hole principle as explained in the wikipedia article you linked to would obviously apply.

    But anything less than that, as best as I can see, would not suffice to apply the pigeon-hole principle and hence would not answer the question if hashcash is in PPP or not. To again refer to your linked wikipedia article: Only for very few problems the answer is know, even for PPP. For the special cases of PPP, PPA and PPAD, it obviously gets even harder. If you find a solution, post it to an academic journal, not just here!

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You make a lot of good points. Some of them dig at the formulation of my question more than the spirit of it, so I've rephrased the question in terms of an idealized "perfect hash function" for which all hashes occur with equal frequency and which scales perfectly. I was assuming SHA256 is a perfect hash function, but if it isn't, then I'm interested in the situation where a perfect hash function is used. I think that addresses your technical point 1. (contd) –  Mike Battaglia Feb 5 at 7:06
    
I think my reformulation also addresses your point 2, since for some weakness like you mention to exist, it would have to affect every perfect hash function to make a sweeping statement about the entire PERFECT HASHCASH problem in general. Finally, I think my more rigorous formation also addresses your point #3. You'll note that I formulated it in terms of a number of starting bits being 0 rather than the thing being less than a target just to keep it simple, but generalizing it to the target-based approach is also straightforward if you want. –  Mike Battaglia Feb 5 at 7:12
    
@MikeBattaglia I agree you've solved many of the technicalities. There's something problematic left, though, that may be more than just a technicality: The pigeon hole principle at the root of the finer complexity classes you hope to get at is essentially a counting argument. Hence it works for problems where you can say "either I hit (on one side of) the target, or I find a collision." Since finding a collision won't solve your (perfect) hashcash problem, applying the pigeon hole principle may not be possible. At least I couldn't. –  pyramids Feb 7 at 11:32
    
pyramids: that's what I'm thinking, though I wasn't sure if there might exist some crazier and more complicated way to reduce it to something in PPP. But I guess if PPP is out, then that rules out PPAD and PPA as well, since they're in PPP. –  Mike Battaglia Feb 7 at 18:53
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