In a BB84 key exchange, Alice and Bob sift bits after comparing bases

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum information theory and communication · Quantum information theory and communication

Quantum Information and Technology, Ch Quantum key distribution and quantum networks

Problem

In a BB84 key exchange, Alice and Bob sift

N = 2\times10^5

bits after comparing bases. Unknown to them, an eavesdropper intercepts every photon on a randomly chosen fraction

f = 0.30

of the rounds, measuring each intercepted photon in a basis (

Z

X

) chosen uniformly at random and resending the state she obtains; she leaves the remaining rounds untouched. ((a)) Find the quantum bit error rate

Q

this attack produces in the sifted key. ((b)) Alice and Bob reveal a random sample of

m = 2\times10^4

sifted bits and find a sample error rate equal to the value from part (a). Using an upper bound at

z = 5

standard errors, compute

Q_{\rm UB}

, the value used in the security analysis, and state how many sifted bits

N'

remain undisclosed for the key. ((c)) The remaining

N'

bits go through information reconciliation at efficiency

f_{\rm EC} = 1.10

and privacy amplification with security parameter

\epsilon = 10^{-10}

. Using

\ell = N'\left[1 - (1+f_{\rm EC})\,h(Q_{\rm UB})\right] - 2\log_2(1/\epsilon),

with

h

the binary entropy in bits, find the length

\ell

of the final secret key. ((d)) Keeping

m=2\times10^4

z=5

and

f_{\rm EC}=1.10

fixed, but letting the intercepted fraction

f

vary, find the largest

f

for which the bracket

1-(1+f_{\rm EC})h(Q_{\rm UB})

in part (c) stays positive, that is, the largest attack this protocol can tolerate before the asymptotic key length is driven to zero.

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