After looking at the paper, I feel that I have a limited understanding of what is presented. It presents a formal methodology for adapting mathematical problems to proof-of-work and presents one such problem as an example. Whether or not solutions to the problem presented are actually useful is not clear to me. Other than my criticism below, the paper looks solid, but it is outside of my area of expertise, so I'll let someone else do the real analysis.
My general criticism of PoUW relates to the potential economic flaws in the concept, which is something that I don't see addressed in this paper.
First, PoW is necessary to the security of the Bitcoin protocol through the rule that the "true" chain is the one that required the most work. In Bitcoin, that work is not actually measured in hashes or joules. It is measured in time. The difficulty adjustment mechanism ensures that no matter how much effort is spent on mining, a block always takes 10 minutes of work on average. Thus, the difficulty adjustment mechanism is also a necessary component of the protocol, at least for Bitcoin.
Second, miners are incentivized to increase (or decrease) their mining capacities until the cost of mining approaches the revenue.
Now, I believe that the idea that converting wasted work into useful work would reduce the overall net cost of mining (or increase the overall net benefit) is flawed. If the result of mining provides additional value from useful work, then miners would be incentivized to increase their mining capacities until the cost of mining again approaches the revenue
plus the additional value of the useful work. In the end, the overall cost of mining would be equal to the cost of the wasted work plus the value of the useful work. Thus, nothing has been gained.
Furthermore, if the value of the useful work is greater than the cost of wasted work, then the mining incentives would be dominated by the value of the useful work. Though I haven't thoroughly examined the result of this scenario, I suspect that it could also result in the failure of the protocol.
There is a potential way to resolve these flaws. If the useful work is a public good, then the miners would gain no value from it and there would be no incentive to increase mining capacity based on the value of the useful work. The problem is that it is not clear that a true public good can exist. The protocol would be a failure if miners discovered a way to exploit the value of the useful work that was originally thought to be a public good.
One of the most commonly used PoWs, such as in Bitcoin, is simply to find a value s so that hashing it together with the given challenge (e.g. with SHA-256) maps to anything with a certain amount of leading 0’s.
I like to point out this misconception whenever it comes up. That is not how Bitcoin (or any other PoW coin that I am aware of) actually works.