Detecting Eve through the error rate

The BB84 protocol does not merely share a key: it makes it possible to know whether an eavesdropper has listened in. This is what radically sets it apart from classical cryptography.

Why Eve introduces errors

An eavesdropper, Eve, does not know the basis chosen by Alice (it is announced only afterward). If she wants to read a photon, she must measure it in a basis she chooses at random.

• One time in two, she guesses the correct basis: she reads the right bit and can send on a correct photon.
• One time in two, she chooses the wrong basis: her measurement disturbs the state, and the photon she sends on to Bob is now in the wrong state.

When Bob then measures in the correct basis (Alice's), he will sometimes get a wrong bit because of Eve's disturbance. These disagreements are errors where there should not be any.

The error rate (QBER)

After basis reconciliation, Alice and Bob sacrifice a sample of their raw key: they publicly compare a subset of bits and count the disagreements. This is the QBER (Quantum Bit Error Rate).

                   nombre de bits qui diffèrent
   QBER  =  ─────────────────────────────────────────
              nombre de bits comparés (échantillon)

   Sans espion (canal idéal) :  QBER ≈ 0 %
   Ève intercepte tout        :  QBER ≈ 25 %

Why 25%? Eve picks the wrong basis one time in two (1/2). When she got it wrong, the photon sent on gives Bob a wrong bit one time in two (1/2). Over the kept positions: 1/2 × 1/2 = 1/4 of errors.

The decision

   QBER estimé
      │
      ├─ faible (sous un seuil)  ──►  pas d'espion : on garde la clé
      │
      └─ trop élevé              ──►  espion probable : on JETTE la clé

The bits sacrificed for the estimate are then discarded (they have been revealed). If the QBER is acceptable, the rest of the key is kept and purified (error correction and privacy amplification).

A security guaranteed by physics

The eavesdropper cannot get around this test: measuring disturbs (uncertainty) and she cannot clone to avoid measuring (no-cloning). Her presence necessarily translates into errors. The detection depends on no computational assumption.

In summary

By intercepting the photons, Eve chooses the wrong basis half the time and introduces errors. Alice and Bob sacrifice a sample of their key to estimate the error rate (QBER): a QBER that is too high (up to ~25% if Eve listens to everything) betrays the eavesdropper and causes the key to be rejected. The detection is guaranteed by physics, not by computational hardness.