The BB84 quantum key distribution protocol begins with Alice choosing two uniformly random bit strings $X=x_{1} x_{2} \ldots x_{m}$ and $Y=y_{1} y_{2} \ldots y_{m}$.

(a) In terms of these strings, describe Alice's process of conjugate coding for the BB84 protocol.

(b) Suppose Alice and Bob are distantly separated in space and have available a noiseless quantum channel on which there is no eavesdropping. They can also communicate classically publicly. For this idealised situation, describe the steps of the BB84 protocol that results in Alice and Bob sharing a secret key of expected length $m / 2$.

(c) Suppose now that an eavesdropper Eve taps into the channel and carries out the following action on each passing qubit. With probability $1-p$, Eve lets it pass undisturbed, and with probability $p$ she chooses a bit $w \in\{0,1\}$ uniformly at random and measures the qubit in basis $B_{w}$ where $B_{0}=\{|0\rangle,|1\rangle\}$ and $B_{1}=\{(|0\rangle+|1\rangle) / \sqrt{2},(|0\rangle-|1\rangle) / \sqrt{2}\}$. After measurement Eve sends the post-measurement state on to Bob. Calculate the bit error rate for Alice and Bob's final key in part (b) that results from Eve's action.