Cryptography Reference
In-Depth Information
qubits were received and in what measurement basis, and Alice has indicated
to Bob which basis choices correspond to her own. We consider here the
important special case where the number of photons in the pulses sent by
Alice follow a Poisson distribution with parameter
. This is an appropriate
description when the source is a pulsed laser that has been attenuated to
produce weak coherent pulses. In this case, the length of the sifted string may
be expressed as [7]
µ
r
d
,
m
2
n
=
ψ
≥
1
(ηµα)(
1
−
r
d
)
+
(7.5)
where
is the transmission probability
in the quantum channel, and
r
d
is the probability of obtaining a dark count
in Bob's detector during a single pulse period.
η
is the efficiency of Bob's detector,
α
is the probability of
encountering
k
or more photons in a pulse selected at random from a stream
of Poisson pulses having a mean of X photons per pulse:
ψ
≥
k
(
X
)
∞
∞
e
−
X
X
l
l
!
ψ
≥
k
(
X
)
≡
k
ψ
l
(
X
)
=
,
(7.6)
l
=
l
=
k
Other types of photon sources may be treated by appropriate modifications of
Equations (7.5) and (7.6). A comprehensive treatment of this subject, including
an extensive analysis of factors contributing to
, is found in Ref. [7].
The next terms represent information that is either in error or that may
be leaked to Eve during the rest of the protocol. This information is removed
from the sifted string by the algorithm used for privacy amplification, and so
the corresponding number of bits must be subtracted from the length of the
sifted string to obtain the size of the final key that results.
The first such term,
e
T
, represents the errors in the sifted string. This may
be expressed in terms of the parameters already defined and the intrinsic
channel error probability
r
c
:
α
,
m
2
r
d
2
e
T
=
ψ
≥
1
(ηµα)
r
c
(
1
−
r
d
)
+
(7.7)
where the intrinsic channel errors are due to relative misalignment of Alice's
and Bob's polarization axes and, in the case of fiber optics, the dispersion
characteristics of the transmission medium. These errors are removed by an
error correction protocol that results in additional
q
bits of information about
the key being transmitted over the classical channel. We express this as
Q
x,
n
e
T
e
T
q
≡
xh
(
e
T
/
n
)
=
e
T
(7.8)
e
T
/
n
where
h
is the binary entropy function for a bit whose
a priori
probability
of being 1 is
p
. The factor
x
is introduced as a measure of the ratio by which a
particular error correction protocol exceeds the theoretical minimum amount
(
p
)
