Cryptography Reference
In-Depth Information
until she learns Alice's and Bob's basis choices by eavesdropping on their
classical channel. She then measures the stored photons in the correct basis
to determine unambiguously the value of the bit (this is sometimes called a
“photon number splitting” (PNS) attack). Finally, she can make a
combined
attack by using the two strategies in some combination. In Ref. [7] it is shown
that the optimum attack is always either a direct or an indirect attack, depend-
ing on the value of a parameter
y
, which depends on channel and detector
characteristics and the technological capabilities attributed to Eve [7]. For the
case of a fiber optic channel, it is possible in principle for Eve to replace
the cable with a lossless medium, so that those pulses whose polarizations
she can measure are guaranteed to reach Bob. In this case we take
y
. For
the free-space case, such an attack may not be feasible, but she can achieve a
similar effect by using entanglement. In this version of the indirect attack, Eve
and an accomplice located near Bob prepare pairs of entangled photons in ad-
vance. Eve then entangles one of these pairs with a photon emitted by Alice.
Her accomplice can now make measurements on the entangled state, gaining
information about the photons at Eve's location without losing photons to the
attenuation in the channel. If we allow for such attacks, we still have
y
=
η
.
If we do not attribute this level of technology to Eve, it is appropriate to take
y
=
η
=
ηα
. Note also that Eve can perform direct attacks using classical optical
equipment, but that the indirect attacks require the use of a quantum memory.
There are three regions of interest. If
y
1
√
2
,y
∼
, the indi-
rect attack is stronger, and the maximum information that Eve can obtain is
>
1
−
(
i
.
e
.
0
.
293
)
m
2
[[
)
−
1
max
ν
=
ψ
≥
2
(µ)
−
(
1
−
y
e
−
y
µ
−
e
−
µ
[1
·{
+
µ(
1
−
y
)
]
}
]
.
(7.16)
1
3
√
2
(
,y
∼
If
y
<
1
−
i
.
e
.
0
.
206
)
, the direct attack is stronger, and Eve's informa-
tion is
m
2
max
ν
=
ψ
2
(µ)
y
+
1
e
−
µ
√
2 sinh
√
2
+
1
2 cosh
√
2
−
−
.
(7.17)
Finally, if
y
lies between these two regions, the relative strength of the attacks
depends on the number of photons in the pulse. The information leaked to
Eve is
e
−
µ
sinh
√
2 sinh
√
2
m
2
max
ν
=
ψ
(µ)
+
µ
−
y
2
∞
−
2
k
1
]
+
2
ψ
(µ)
θ(σ
(
k, y
)
−
1
)
[1
−
(
1
−
y
)
2
k
e
k
=
,
2
1
−
k
+
[1
−
θ(σ
(
k, y
)
−
1
)
]
(
1
−
)
(7.18)
e
