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We could do the same and just translate this function straight to CNF, however there is a much better and more declarative solution than that in our case. Instead, we can just assume values which we know are fixed in the output of the hash. This will restrict the search space to discard any execution paths where the assumptions would not be true anymore. Because we are not in a brute force setting, but a constraint solving setting this is very simple to express.
We assume the following: Only compute hashes which have N bytes [N depends on the target] of leading zeros. It might seem unintuitive to "fix" output variables to certain values, however remember that the code is not executed in a regular fashion but translated as a big formula of constraints. Assumptions on the outputs will result in restrictions of the input -- in our case this means only valid nonces will be considered.
This serves three purposes: Again, in comparison, brute force just blindly computes hashes with no way of specifying what we are looking for. The SAT-based solution only computes hashes that comply with the mining specification of a valid hash. The most important part is defining the assertion, or the property P as it is called in the section above. The key idea here is that the counterexample produced by the model checker will contain a valid nonce given a clever enough assertion.
A bounded model checker is primarily a bug finding tool. You specify the invariant of your system, which should always hold, and the model checker will try to find an execution where this invariant is violated i.
That is why the P above is negated in the formula. Thus, the invariant, our P, is set to "No valid nonce exists". This is naturally expressed as the assertion. Which the model checker will encode to its negation as "a valid nonce does exist", i. If a satisfiable solution is found, we will get an execution path to a valid nonce value. In reality, this is encoded more elegantly. Since the leading zeros of a hash are already assumed to be true, all that remains to be asserted is that the value of the first non-zero byte in the valid hash will be below the target at that position.
Again, we know the position of the non-zero byte for certain because of the target. For example, if our current target is the following:. Then the following assertion states that a certain byte in state of the hash has to be above 0x As the assertion is negated, the SAT solver will be instructed to find a way to make the flag equal to 0. The only way this can be done is by playing with the only free variable in the model -- the nonce.
In that way, we just translated the bitcoin mining problem into SAT solving land. Combining the ideas from the above sections results in a conceptual SAT-based bitcoin mining framework. In pseudo C code this looks as follows:. The advantage of using the built-in solver is that, in case of satisfiability, the model checker can easily retrieve a counterexample from the solution which consists of all variable assignments in the solution.
A violation of the assertion implies a hash below the target is found. Let us inspect a counterexample when run on the genesis block as input. At state below, the flag was found to be 0 which violates the assertion. Moving upwards in the execution trace we find a valid hash in state Finally, the value of the non-deterministically chosen nonce is recovered in state The implementation of the above program generates a large CNF formula with about ' variables and ' clauses.
In order to evaluate its performance I generated two benchmark files where one has a satisfiable solution and the other does not. I restricted the nonce range the possible values to be chosen to values for each file. The files are available on the following github project.
Unsurprisingly, the solvers are not capable of solving this problem efficiently as of now. However, it is interesting to see the differences in runtime. This is interesting as Cryptominisat has been specifically tuned towards cryptographic problems as it is able to detect and treat xor clauses differently to normal clauses .
This feature is extensively used in this case, in the above run the solver found over non-binary xor clauses. The crypto-focused optimisations of Cryptominisat could potentially have helped in solving this more efficiently than the other solvers.
However, it is very surprising that ZChaff wins the SAT challenge with a good margin to the next solver. ZChaff is the oldest of all solvers presented here, the version I am using is 9 years old.
This could indicate that the heuristics applied by modern SAT solvers do not help in this particular instance. Generally, it is not known what makes a SAT instance hard or easy, which leaves only speculation or analysis of the stats provided by the SAT solvers to come to useful conclusions. I could speculate that the avalanche effect of the hash function produces a very structured CNF formula with high dependencies between clauses and variables.
Perhaps a higher degree of randomisation applied by heuristics performs less well than straight-forward DPLL. I leave this to someone with more SAT solving knowledge to decide. While the performance numbers are not great compared to GPU mining we have to keep in mind that this is entirely unoptimised and there are many ways of how this can be sped up. To give an idea of the performance gains that can be achieved with little effort I am going to use a combination of features:.
In this experiment, I am going to use Cryptominisat as it performed well in the UNSAT challenge and has a large number of parameters with parameter tuning and slicing. The restrict parameter is a way to only branch on the 32 most active variables which is intended for cryptography key search -- 32 was picked arbitrarily. In the second row, I tried running it with the plain parameter which deactivates all simplification heuristics, in order to see if the speculations around the ZChaff-speed improvement could also apply to Cryptominisat.
For the last row, I increased the nonce range to 10' values which leads to an interesting result. The unoptimised run of this file is only 3m39s; this is half the expected time when we take the 42s benchmark on nonces and assume that the search time increases linearly with the nonce range. This does not seem to be the case.
Apart from parameter tuning there's quite a few things that should have an even larger impact on performance. Here are a couple of examples:.
A very intriguing, and perhaps unintuitive property of the algorithm proposed is that with increasing bitcoin difficulty, or equally lower target, the search could become more efficient , at least in theory.
This is because we can assume more about the structure of a valid hash -- a lower target means more leading zeros which are assumed to be zero in the SAT-based algorithm.
If this is true and is a substantial effect then this is an important issue since the rate of the money supply is regulated with the difficulty. The higher the difficulty, the less likely it is that an individual block has a valid nonce.