Editing Quantum key distribution (section)

=== Intercept and resend ===
The simplest type of possible attack is the intercept-resend attack, where Eve measures the quantum states (photons) sent by Alice and then sends replacement states to Bob, prepared in the state she measures. In the BB84 protocol, this produces errors in the key Alice and Bob share. As Eve has no knowledge of the basis a state sent by Alice is encoded in, she can only guess which basis to measure in, in the same way as Bob. If she chooses correctly, she measures the correct photon polarization state as sent by Alice, and resends the correct state to Bob. However, if she chooses incorrectly, the state she measures is random, and the state sent to Bob cannot be the same as the state sent by Alice. If Bob then measures this state in the same basis Alice sent, he too gets a random result—as Eve has sent him a state in the opposite basis—with a 50% chance of an erroneous result (instead of the correct result he would get without the presence of Eve). The table below shows an example of this type of attack.

{| class="wikitable" style="text-align: center; margin: 1em auto 1em auto"
|-
! Alice's random bit
| style="width:40pt;"| 0 || style="width:40pt;"| 1 || style="width:40pt;"| 1 || style="width:40pt;"| 0 || style="width:40pt;"| 1 || style="width:40pt;"| 0 || style="width:40pt;"| 0 || style="width:40pt;"| 1
|-
! Alice's random sending basis
| [[File:PlusCM128.svg|15x15px]] || [[File:PlusCM128.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:PlusCM128.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:PlusCM128.svg|15x15px]]
|-
! Photon polarization Alice sends
| [[File:Arrow north.svg|20x20px]] || [[File:Arrow east.svg|20x20px]] || [[File:Arrow southeast.svg|15x15px]] || [[File:Arrow north.svg|20x20px]] || [[File:Arrow southeast.svg|15x15px]] || [[File:Arrow northeast.svg|15x15px]] || [[File:Arrow northeast.svg|15x15px]] || [[File:Arrow east.svg|20x20px]]
|-
! Eve's random measuring basis
| [[File:PlusCM128.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:PlusCM128.svg|15x15px]] || [[File:PlusCM128.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:PlusCM128.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:PlusCM128.svg|15x15px]]
|-
! Polarization Eve measures and sends
| [[File:Arrow north.svg|20x20px]] || [[File:Arrow northeast.svg|15x15px]] || [[File:Arrow east.svg|20x20px]] || [[File:Arrow north.svg|20x20px]] || [[File:Arrow southeast.svg|15x15px]] || [[File:Arrow east.svg|20x20px]] || [[File:Arrow northeast.svg|15x15px]] || [[File:Arrow east.svg|20x20px]] 
|-
! Bob's random measuring basis
| [[File:PlusCM128.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:PlusCM128.svg|15x15px]] || [[File:Multiplication Sign.svg|15x15px]] || [[File:PlusCM128.svg|15x15px]] || [[File:PlusCM128.svg|15x15px]]
|-
! Photon polarization Bob measures
| [[File:Arrow north.svg|20x20px]] || [[File:Arrow northeast.svg|15x15px]] || [[File:Arrow northeast.svg|15x15px]] || [[File:Arrow southeast.svg|15x15px]] || [[File:Arrow east.svg|20x20px]] || [[File:Arrow northeast.svg|15x15px]] || [[File:Arrow north.svg|20x20px]] || [[File:Arrow east.svg|20x20px]]
|-
! PUBLIC DISCUSSION OF BASIS
| colspan=8 |
|-
! Shared secret key
| 0 ||  || 0 ||  ||  || 0 ||  || 1
|-
! Errors in key
| ✓ ||  || ✘ ||  ||  || ✓ ||  || ✓
|}

The probability Eve chooses the incorrect basis is 50% (assuming Alice chooses randomly), and if Bob measures this intercepted photon in the basis Alice sent he gets a random result, i.e., an incorrect result with probability of 50%. The probability an intercepted photon generates an error in the key string is then 50% × 50% = 25%. If Alice and Bob publicly compare <math>n</math> of their key bits (thus discarding them as key bits, as they are no longer secret) the probability they find disagreement and identify the presence of Eve is

{{center|<math>P_d = 1 - \left(\frac{3}{4}\right)^n</math>}}

So to detect an eavesdropper with probability <math>P_d = 0.999999999</math> Alice and Bob need to compare <math>n = 72</math> key bits.