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 $2$

$2561$

following the standard conversion from bit strings to integers.

b$255$b$254...$ b$1$b$0->$

$255$

X

i$=0$

bi$2$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

$12$d$4$b$6$df$6233$

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.