Blockchain mining $(2$pt$.)$

The output of SHA$256$ is usually shown is in hexadecimal $($each character

is $4$ bits$).$ Each output can be seen as an integer between $0$ and $2^{256}-1$

following the standard conversion from bit strings to integers.

$b_{255}{b}_{254}dots{b}_1{b}_0{\displaystyle \underset{i=0}{\overset{255}{}}}{b}_i{2}^i$

In this exercise we assume that, taking different inputs, the output of

SHA$256$ is uniformly distributed.

$($a$)$ Suppose that you take $48$ inputs, and compute SHA$256$ for each

one of them. How many outputs will be is below $2^{252},$ in average?

$($b$)$ A recent block hash was

$0000000000000000000153990$bd$58860$af$764$cc$7$a$49$c$8755416$e

$12d4b6df6233$

This number is below $2^{180}.$ How frequent is a number below $2^{180},$ in

average?

$($c$)$ Make an estimation of how much time you would need, with your

computer, to obtain a hash value below $2^{180}.$ To do so$,$ use openssl,

another application, or an estimation of a SHA$256$ execution.