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Is this usable? Or does the generation of the modulus outweigh any gains in the difference in signing time? No, it does not work. The generation cost itself isn't too important, because signing any one nonce doesn't help the client sign any other nonce, so you can just use a single key all the time (if you had to create a new keypair for every puzzle, ...

Edit: I just realized that the scheme that was proposed has serious security problems, and should not be used. The question is rather confused and ambiguous, but there seems to be a serious security problem. If I understand the proposal correctly, you are giving the client the private exponent $d$ and the modulus $n$ and the iteration count $k$ (say, ...

A good PRNG can't be exhausted, so your argument against random bytes doesn't matter. firstly I'd like the server to be completely stateless. Sounds impossible to me. a PRNG needs state. A clock needs state. An open connection is state too You probably want to store open challenges too. This part isn't strictly necessary, but very convenient. ...

It is hard to tell from this question, but I suspect that you may have managed to confuse yourself about what problem you have: i.e., that you've gotten caught up in a particular mechanism, and would benefit from taking a moment to step back and look at the actual requirements. I found it hard to extract the actual requirements from your question. In these ...

$\quad$ chosen-prefix partial collisions for scrypt This does somewhat better at #3 than your second option because working requires storing previously obtained scrypt outputs. (I think it would be better to require a at most a given Hamming distance between the outputs rather than equality of a specific set of bits of the output. $\:$ The first of those ...

One obvious approach would be look at a discrete log problem. That is, you pick an appropriate group of order $n$ (where $n$ is prime) and a generator $G$; for a challenge, you pick a random element $H$, and you ask for the value $0 \le k < n$ such that $H = kG$. This meets your requirements (1) the value produced is effectively random, requirement (2) ...

Using time of day is predictable. Challenges should be unpredictable to an adversary. If they were not (as you sort of say), the adversary could solve a set of challenges ahead of time and then use them in rapid succession to cause a DOS. You want to meter access to the service, which is about not only ensuring clients do some work, but that work is done at ...

Here is one simple solution. After a new user installs the Pokemon software, before the new user can use that software (for tournament-eligible activities), they must register it with a central tournament server. The tournament server gives the user's software a random 128-bit seed $S$, and remembers this seed. The Pokemon software now creates a ...

Fundamentally, all you can do is make it "hard" to roll a character. For example, if it takes about five seconds of computation to roll a character, that will annoy ordinary users as they wait 5 seconds, but it will limit cheaters to 12 rolls per minute. Of course, cheaters can always get their hands on more computing power, and the delays will annoy ...