COMP90088: Cryptocurrencies and decentralised ledgers Semester 1 2022
Tutorial Sheet, Week 5 Week of March 28, 2022
1. Bitcoin block arrival time. Recall that the arrival of Bitcoin blocks is a Poisson process with rate r=0.1 minutes-1. This means that the average time to find a block is 10 minutes at any instant, regardless of how long it has been since the last block was found.
Note: it is recommended to solve the following problems using a spreadsheet or statistics library a. What is the probability that at least one block is found in the next 10 minutes?
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b. What is the probability that exactly 6 blocks are found in the next 60 minutes?
c. What is the probability that 6 or more are found in the next 60 minutes?
d. What is the median time to wait to find the next block? To find 6 blocks? Another way of thinking about this is: for what value of t is the probability that you find at least one block in t minutes (or at least 6 blocks in t minutes) equal to 50%?
2. Soft forks and hard forks. Which of the following changes to Bitcoin could be made with a soft fork? Which would require a hard fork?
a. Requiring each block to include a government-issued certificate asserting that it was mined using renewable electricity sources
b. Changing the block reward schedule to be fixed at 6.25 BTC forever
c. Requiring each miner to donate 1 BTC in each block to a Bitcoin Development Fund
d. Banning ¡°empty¡± blocks with no transaction besides the coinbase
e. Adding support for the SHA-3 hash function in scripts (replacing an unused opcode)
f. Removing the limit of 20,000 signatures per block
g. Changing the block frequency to be once every 5 minutes
3. Mining pool sabotage: Recall that mining pools enable individual miners to share risk and reward, lowering the variance of their earnings while keeping the same expected value. Participants repeatedly submit shares (blocks that are valid at a lower difficulty) to prove how much work they are doing. Whenever the pool finds a block, the coinbase from that block is split among the participants in proportion to the number of shares submitted. One risk is sabotage, in which a participant submits shares, but withholds full solutions if they are found.
a. Consider two pools, P1 and P2 with mining power ¦Á1 and ¦Á2, respectively. What will P1¡¯s expected share of the total earnings be in the long run if it dedicates ¦Â<¦Á1 power towards sabotaging P2? Note that when P1 finds a block it gets the entire coinbase. When P2 finds a block, P1 receives a fraction of the coinbase proportional to the number of shares P1 generated while mining for P2.
Hint: P2¡¯s total mining power is now ¦Á2+¦Â, but only ¦Á2 is used for finding a new block. Because ¦Â power is no longer used to find blocks, P2¡¯s useful mining power, as a fraction of the entire network, is now ¦Á2/(1-¦Â). The same reasoning also applies to P1. You may assume that the block discovery rate is always 10 minutes.
b. Provide concrete values for ¦Á1, ¦Á2, ¦Â in which this attack is advantageous for P1.
c. Suppose P2 wants to protect itself by kicking out participants observed to be reporting a suspiciously low rate of valid blocks compared to how many shares they report. Explain
why this might inadvertently punish honest participants.
d. Suppose two pools, each with power ¦Á, sabotage each other with power ¦Â<¦Á. For what
range of ¦Â will the two pools lose revenue by attacking each other? How much will they lose? What classic game from game theory is this situation an instance of?
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