Question

In a system using a blockchain for distributed consensus, in order to add a block to a chain, a participating node must solve the following puzzle: it must find a value x such that its hash, H (x || seed), is less than T. The hash function is such that a given value of x can uniformly map to any integer in [0, 2²⁵⁶ — 1].

Assume T is set to 2²²⁶.
(a) What is the probability that a given value of x, randomly chosen by the participating node, is a winning solution to the puzzle (i.e. H (x || seed) < T)?

Answer 1

The probability of a given value of x being a winning solution to the puzzle in a blockchain system is 1/2^30, based on the uniform distribution of a hash function that evenly maps values in the specified range.

Step-by-step explanation:

The probability that a given value of x, randomly chosen by the participating node, is a winning solution to the puzzle (i.e., H (x || seed) < T) can be calculated using the principles of uniform distribution. Since the hash function maps any value of x to an integer in the range [0, 2²²²²²² — 1] uniformly, the probability P of finding a hash that is less than T (where T is set to 2²²²) is simply the ratio of the number of successful outcomes to the total number of possible outcomes. Therefore, the probability P is given by P = T / 2²²²²²². Plugging in the values, we get P = 2²²² / 2²²²²²², which simplifies to P = 1 / 2³0. Hence, the probability of a given x being a winning solution is 1 divided by 2 raised to the power of 30.