# Security of BB84 QKD protocol

I am studying the BB84 protocol and I have some doubts regarding its security.

Consider the following example: Alice wants to send a message to Bob over a public quantum channel, where a third person (Eve) could try to detect the message. The first step for Alice is to compute a string of classical bits in qubits via orthogonal bases $$X$$ or $$Z$$. Then, she sends the encoded message over the channel. At this point, suppose Eve has intercepted the message.

It is known that the BB84 protocol is secure as Eve cannot intercept the message without being detected. This happens because she has to apply a set of bases in order to get Alice's message from qubits to bits; the fact is that Eve doesn't know the bases which have been used by Alice, as those are private. Based on that, when she tries to measure the qubits in her random selection of bases, if those are wrong, an altered message will be sent to Bob. We don't have to go further as my question lies in this part of the communication:

1. As Eve doesn't know Alice's bases, almost certainly she will use different bases, leading to a different encoded message for Bob which will allow to detect the interception. What happens if she select exactly the same bases as Alice's? I suppose it's not impossible; even if the probability is so small, I still think it could happen.
2. Based on 1., if Eve intercepts the whole message correctly because she used the same bases, doesn't this go against the non-cloning theorem of quantum computation, as it would imply creating the same message from unknown states.

You are correct she could guess the bases correctly however each time she has a probability of $$1/2$$ of selecting the correct basis. So if Alice sends $$n$$ qubits then the probability she selects all of the correct bases is $$1/2^n$$. For any reasonable $$n$$ this is extremely unlikely.
For some perspective, if $$n=100$$ then (based on some rough estimates) Eve is more likely to select the same grain of sand as you from all possible grains of sand on the earth (choosing uniformly) than guess all the bases correctly. If $$n=1000$$ then Eve is more likely to choose the same atom as you from all atoms in the observable universe (choosing uniformly) than guess all the bases correctly.
Typical BB84 experiments run from $$n=10^5$$ to $$n=10^9$$ and possibly even higher. So guessing many bases is out of the question.