Exercise 2.10

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 2.10}$

Analyze Eve's success in eavesdropping on the BB84 protocol if she does not even know which two bases to choose from so chooses a basis at random at each step.

a) On average, what percentage of bit values of the final key will Eve know for sure after listening to Alice and Bob's conversation on the public channel?

b) On average, what percentage of bits in her string are correct?

c) How many bits do Alice and Bob need to compare to have a $90\%$ chance of detecting Eve's presence?