The Background
Bitcoin tracks chain work by counting the (anticipated) minimal variety of hashes it will take to create a series of the identical variety of blocks and identical issue steps[1]. Naturally, gathered work measured on this vogue (e.g. anticipated variety of hashes) will increase with every new block that’s added to the chain.
For a given goal t, the minimal anticipated variety of hash makes an attempt essential to discover a block which meets the proof of labor standards (hash lower than or equal to t) is geometrically distributed[2] with parameter p = t / 2^256.
We will calculate the entropy, w, of such a geometrically distributed random variable[3] by the equation w(p) = (-(1-p)log2(1-p) – plog2(p)) / p, and utilizing p = t / 2^256 we will trivially rewrite this right into a operate of the goal as an alternative, w(t).
Utilizing this technique we will calculate the minimal work (e.g. entropy) w(t), denominated in bits, obligatory to explain the proof of labor sign. Different elements of the blockchain (similar to what knowledge is within the blocks in addition to the proof of labor sign) should not included on this evaluation so that’s the reason it’s an estimate of the minimal encoding.
Plugging in some numbers:
- for block zero (and any block with the bottom attainable issue, therefore
p = 1 / 2^32), we getroughly 33.44 bitsof labor per block. - for a more moderen block, say block
#602784we getroughly 76.97 bitsof labor per block.
The Query
Can we full the above evaluation for each issue interval and sum the entropies to acquire a decrease sure on the minimal complexity, in bits, of the (proof of labor elements of the) Bitcoin blockchain? Or is there some assumption (similar to having the ability to sum the entropies on this context?) that’s unsafe/flawed?
An Tried Reply & Some Motivation
Performing such an evaluation[4] appears to provide a consequence, as of block 602784, of roughly 36.9 million bits or 4.6 megabytes and rising.
Nevertheless, one cause why the strategy appears fascinating is that, whether it is cheap, then plainly all the maths which the bitcoin protocol at the moment does with “time” (within the unix timestamp sense, similar to issue changes, and block timestamp should advance median of final n block’s timestamps) might as an alternative be performed in “bits.” As such, this could possibly be useful for when/if the protocol is prolonged into function throughout huge distances of spacetime?
Then once more, there’s most likely one thing terribly flawed with this evaluation which is why the query is being requested right here.
References
[1] Strongest vs Longest chain and orphaned blocks
[2] How is it that concurrent miners don’t subvert one another’s work?
[3] https://math.stackexchange.com/questions/490559/entropy-of-geometric-random-variable
[4] https://github.com/philbertw4/mathematicalbitcoin/blob/grasp/scala/SatsPerBit.sc