Detecting an eavesdropper

Let us see how using the Q-box allows us to detect eavesdropping.

Suppose that Alice and Bob are using a Q-box to share random strings, as we just described. An eavesdropper, Eve, wants to also learn the random string that Alice and Bob are sharing. To do so she tries to take measurements from the Q-boxes before they reach Bob.

For every Q-box she sees, Eve randomly performs either the T/B or the L/R measurement: remember that it’s impossible to perform both of these measurements, and that performing one of these measurements destroys the state of the box. To try to hide her actions from Bob she re-initialises the Q-box using the outcome she found, and sends the Q-box on to Bob. For instance, if Eve makes the T/B measurement and gets the output Top, she re-initialises the box in the state Top.

Sometimes Eve can get away with her interception, but in other cases Eve’s action causes a noticeable disturbance.

Suppose that Alice chooses the T/B encoding and sends Bottom, and that the true internal state of the Q-box (which nobody can know) is BL. Eve doesn’t know which encoding Alice chose, so now has to choose whether to make a T/B or L/R measurement:

• If Eve chooses to make the T/B measurement, she will always get the outcome Bottom; she will then re-initialise the box as Bottom and hence when Bob receives the box it is as if the interception was never made. (It could be that the internal state of the box is different before and after Eve’s action, e.g. the state could change from BL to BR, but since neither Alice, Bob nor Eve can know the precise internal state, this difference is immaterial.)
• If Eve chooses to make the L/R measurement then she gets the outcome Left. She will then re-initialise the box as Left, making the internal state either TL or BL each with probability (1 /2). If Bob makes the T/B measurement he could get an outcome of Top, which would disagree with Alice. However, as we saw before, without the eavesdropper if Alice uses the T/B encoding and Bob reads out using the T/B measurement, they should always get the same strings.

Thus, the attempt to eavesdrop can cause a difference that Alice and Bob can detect! By performing more checks between values on which they know they should agree, Alice and Bob can gain more confidence that there is no-one trying to intercept their string.

Your turn!

Work through what happens if Alice uses the L/R encoding and sends Left with Bob using the L/R measurement, when Eve makes:

1. a L/R measurement
2. a T/B measurement.

For each case, decide whether Eve causes a disturbance or not. Use the comments to report how you’re getting on.

Summary

Using these Q-boxes it is possible for Alice and Bob to set up a shared random string and to know whether or not it has been eavesdropped; this string can function as a secret key for later sending secret messages. You might like to read about the one-time pad (or Vernam cipher) to see how to do this with perfect secrecy (in essence it is a Vigenère cipher where the codeword is random and as long as the message). The following video gives a nice introduction.