Restricted Adversaries

In this section, we assume the following about the adversary Eve:

- Eve is limited to single photon measurements, but can perform any such measurement without errors.
- Eve can determine the number of photons in a pulse, and can separate out single photons from a pulse without disturbing their state.
- Eve has no quantum memory, all her measurements are completed before the bases are announced.
- Eve can replace the fiber between Alice and
Bob by a more transparent one. For simplicity, we in fact assume
that she can replace it by a perfect fiber.
^{2} - Eve cannot influence the physical setup of Alice or Bob's apparatus. For instance in our experiment, there is a non negligible probability that a photon injected into Bob's apparatus will be lost before it reaches the detectors, or will be measured incorrectly - we assume that Eve cannot remove these losses and errors.

Once one has chosen not to assume a completely unbounded adversary -
as we do here - it is always debatable *which* restrictions one
should place on Eve, and we do not claim to have the only reasonable
answer to this. We have tried to be quite generous to Eve compared to
today's technology, but at the same time we have tried to avoid
assumptions that seem to us to be unnatural or unnecessarily generous.
In particular, the first two assumptions are long shots given today's
state-of-the-art cavity QED experiments, where quantum non-demolition
measurements on single photons are carried out. However by development
of the existing technology they may become feasible in the near
future. We also do not expect Eve to perform collective attacks, which
require a quantum computer of some sort. The third assumption denies
Eve' ability to store photons in an unknown quantum state. Even though
proposals for quantum memory exist in the literature, none of them
seems to assume memory time longer than a fraction of a second. It is
always possible to wait with the announcement of the basis for that
long time and therefore we discard the quantum memory feature. Our
last assumption is the main difference between our analysis and the
less optimistic one from Brassard et al. [3], where it is
assumed that Eve can somehow make Bob detect photons perfectly.

In the following, we analyze which attacks Eve may launch on various types of pulses sent by Alice, under the constraints we assumed. Then we put this together, to see what the maximal damage is that Eve can do. Of course, the goal of Eve will all the time be to obtain as much information as possible about the bits sent, while causing as little disturbance on the signal as possible. In particular, Eve must make sure that the error rate and size of the raw key is close to what Alice and Bob expects, since otherwise they will abandon this run of the protocol. For simplicity, we will focus on what happens asymptotically for long keys.

By the *raw key*, we mean in the following the bit string as sent
by Alice after she has removed the bits for which Bob did not detect
the corresponding pulses or measured in the wrong basis.