<<

. 2
( 9)



>>


R(X |Y ) ¤ H (X |Y ). (2.45)


References
1. Carter, L., Wegman, M.N.: Universal classes of hash functions. J. Comput. Syst. Sci. 18(2),
143“154 (1979) 15
2. Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley-Interscience (1991). URL
http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09“20&amp;path=ASIN/04710625
96 19
3. National Institute of Standards and Technology: FIPS PUB 186“2: Digital Signature Standard
(DSS). National Institute for Standards and Technology, Gaithersburg, MD, USA (2000). URL
http://www.itl.nist.gov/¬pspubs/¬p186“2.pdf 14
4. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge
University Press, Cambridge (2000). URL http://www.amazon.ca/exec/obidos/redirect?tag=
citeulike09“20%&amp;path=ASIN/0521635039 3, 5, 9, 10
5. R´ nyi, A.: On measures of information and entropy. Proceedings of the 4th Berkeley Sympo-
e
sium on Mathematics, Statistics and Probability, pp. 547“561 (1961) 20
6. Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-
key cryptosystems. Commun. ACM 21(2), 120“126 (1978) 14
7. Shannon, C.E.: A mathematical theory of communication. The Bell System Technical Jour-
nal 27, 379“423, 623“656 (1948). URL http://cm.bell-labs.com/cm/ms/what/shannonday/
shannon1948.pdf 19
8. Stinson, D.R.: Universal hashing and authentication codes. In: J. Feigenbaum (ed.) CRYPTO,
Lecture Notes in Computer Science, Vol. 576, pp. 74“85. Springer, Heidelberg (1991) 15
9. Wegman, M.N., Carter, L.: New hash functions and their use in authentication and set equality.
J. Comput. Syst. Sci. 22(3), 265“279 (1981) 14, 15, 16, 17, 18, 19
10. Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299(5886), 802“803
(1982). DOI 10.1038/299802a0. URL http://dx.doi.org/10.1038/299802a0 13
Chapter 3
Quantum Key Distribution

M. Pivk




In this chapter a complete QKD protocol is presented, starting from the transmission
via the quantum channel up to the communication over the public channel. The
protocol described here is the BB84 protocol, named after Bennett and Brassard
[5]. There are other protocols like the B92 protocol [3], the six-state protocol [8], the
SARG protocol [19] and the Ekert protocol [10], which are not discussed here. We
are focusing on BB84, the most known QKD protocol. In Fig. 3.1 you see an abstract
sequence diagram of BB84. ka is the pre-shared secret needed for authentication and
K is the ¬nal key generated after BB84 is executed. We begin with the ¬rst stage,
the transmission of the photons, which is the physical representation of the qubits,
from Alice to Bob. This phase of the protocol is discussed in detail in Sect. 3.1.
Afterward the communication switches to the public channel (Sect. 3.2).
There, the ¬rst phase is the sifting phase, where Alice and Bob negotiate which
bits are used and which bits are discarded. To avoid a man-in-the-middle attack by
Eve, this message exchange must be authenticated. After agreeing on the bits and
being sure that Eve has not modi¬ed messages by using an authentication scheme,
Alice and Bob go on to the reconciliation phase or error correction phase. Because
the quantum channel is not a noiseless channel, Alice and Bob do not share the
same identical string. There is a small portion of errors in Bob™s string, which is
corrected in this phase. Again Eve has the possibility to modify messages during
this phase to her interest. Therefore, Alice and Bob must authenticate this phase.
Passing reconciliation, Alice and Bob share a string, which is identical with very
high probability. But this string cannot be used as a key yet. Eve™s information
about the string must be considered. She has gained information during the error
correction and maybe also during the quantum transmission. Hence, Alice and Bob
must map their string via a function to a smaller subset, so that Eve™s knowledge
decreases almost to zero. This stage is called privacy ampli¬cation and afterward
Alice and Bob share a secret key only known by them.


M. Pivk (B)
Safety & Security Department, Quantum Technologies, AIT Austrian Institute of Tech-
nology GmbH, Lakeside B01A, 9020 Klagenfurt, Austria, mario.pivk@ait.ac.at:
http://www.ait.ac.at




Pivk, M.: Quantum Key Distribution. Lect. Notes Phys. 797, 23“47 (2010)
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-04831-9 3
24 M. Pivk

Fig. 3.1 BB84. ka is the k a Alice Bob ka
pre-shared key. K is the
generated key after protocol
Quantum channel
execution


sifting

authka(sifting)


error correction

authka(error correction)


privacy amplification


K K


3.1 Quantum Channel

As mentioned, the ¬rst stage of the BB84 protocol is the quantum transmission.
Therefore, we take a closer look at the physical realization: How can qubits be
represented and transmitted? In the second part we discuss what Alice and Bob have
to do, when they perform the BB84 protocol. Additionally, an equation is given,
which represents the theoretical throughput of the quantum channel.


3.1.1 Physical Realization

One way to represent a qubit is by using single photons. They are very suitable
for QKD, because photons hardly interact with each other and they can overcome
long distances with low loss in optical ¬bers. Polarization is one of the photon™s
attributes which can serve as a qubit property. There exists rectilinear polarization
(horizontal/0—¦ and vertical/90—¦ ) and diagonal polarization (+45—¦ and ’45—¦ shifted).
We can map horizontal and vertical with the qubits states |0 and |1 , and +45—¦
and ’45—¦ shifted with the states |+ and |’ , respectively. Thus, we have the same
statistical probabilities as discussed in Sect. 2.1.3, which we will use next.
There are many different systems for implementing QKD. In Chap. 6 seven sys-
tems are presented which are developed for SECOQC.
One possibility to generate single photons is by attenuating the output of a
laser. This attenuator is characterized by the mean photon number, which de¬nes
the rate of photons passing the attenuator. Usually, the mean photon number is
0.1, meaning that 90% of the time no photon passes the attenuator but in 10%
of cases, the probability that a single photon passes is better than 95% (for more
information on the computation of the probability see [17]). Nevertheless, the
3 Quantum Key Distribution 25

output of the laser may consist of several path; in BB84 due to the amount of
different polarization states four are needed. But after a pulse passes the attenuator
it has an arbitrary polarization; therefore, linear polarizers and half-plates are used
to create the desired polarization. These elements can be seen as linear operators
(Pauli matrices) known from Sect. 2.1.2 performing the bit and phase ¬‚ip on the
qubits.
After the photons have the right polarization they pass through a beam splitter,
where they are combined. Now the pulse is transmitted over the quantum channel
(optical ¬ber or free space) and arrives at the receiver. There, a random choice is
made by the receiver as to which polarization ¬lter to use (e.g., horizontal or +45—¦ ).
If the receiver chooses differently to the sender we know from Sect. 2.1.3 that there
is a chance of 50% for the photon to pass the ¬lter (and change the polarization
to horizontal) or to get absorbed (this holds for both +45—¦ and ’45—¦ polarized
photons). If the receiver chooses the same basis as the sender the photon passes
with 100% probability if it was polarized horizontal and with 0% probability if it
was polarized vertical. Behind the ¬lter a detector clicks every time a photon passes.
(this sending“receiving architecture is taken from [22]).


3.1.2 Photon Transmission and Throughput
Back to our scenario, Alice is on the transmitter and Bob on the receiver site of
the above-described quantum channel. Alice chooses now randomly two strings
independent of each other with length m. The ¬rst string represents the basis for
the quantum transmission and the second the proper value of the speci¬c bit. Alice
and Bob have the same mapping scheme in common. A sample mapping is de¬ned
in Tables 3.1 and 3.2.

Table 3.1 Base mapping
Base Representation bit
Rectilinear 0
Diagonal 1



Table 3.2 Value mapping
Rectilinear Diagonal value bit
Horizontal (0—¦ ) +45—¦ 0
Vertical (90—¦ ) ’45—¦ 1


Alice starts now to transmit. Therefore, she takes the ¬rst bit of the ¬rst string,
indicating which base to use (e.g., a 0-bit denotes to use the rectilinear basis), and
the ¬rst bit of the second string indicating which value to take (e.g. a 1 denotes in
the rectilinear base to polarize the photon vertical and in the diagonal base to shift
the polarization by ’45—¦ ). She applies this procedure on all m-bits of both strings
26 M. Pivk

and sends them as photons via the quantum channel to Bob. Bob on his part chooses
also a random string with length m, for his base choices. With these bits he measures
the incoming photons with the corresponding ¬lter. Note that it makes no sense to
measure with both ¬lters, because after measurement the original polarization is lost
as mentioned in the previous section or in Sect. 2.1.3. Bob keeps the measurement
results to himself. Consider that Bob™s measurements do not completely match with
Alice™s ones, due to optical misalignment, disturbance on the quantum channel,
noise in Bob™s detectors, or the presence of an eavesdropper Eve, although both
choose the same basis.
Before describing the next step, we continue by analyzing the throughput of the
quantum channel. In [24] a theoretical function is given by which the several losses
on the quantum channel can be computed:


gq = μ · ± f · ±e · ·det · kdead . (3.1)


The gain of the quantum channel gq consists now of the mean photon number μ,
which we discussed in the previous section. The next factor ± f represents the ¬ber
loss. This loss is distance dependent and increases with higher distances. For the
receiver™s detector ·det as detection ef¬ciency and kdead as factor accounting for the
reduction of the photon detection rate due to the dead time is given. The dead time
is the hold-off time following each detection event; during this time the bias voltage
of the device is below a certain level such that no photon can be detected. ±e is the
additional loss of the system.
To compute the number of Bob™s measurement results the data rate f data is
required, which is the number of photons Alice™s laser can send. This rate is given
in hertz and can have range from MHz to GHz. These different rates have impact
on the gain of the quantum channel: with higher rates the ef¬ciency shrinks. So the
key material Bob receives per second can be computed by f data · gq . The Eq. 3.1
conforms with experimental test done in [24].
As long as the strings which Alice and Bob need for the choice of bases are
randomly chosen (the probability to send a qubit in the rectilinear base or diagonal
base is 50%), half of Bob™s results, i.e., the raw key, must be discarded because of
the independence of both strings (see later in Sect. 3.2.1). To minimize this loss of
raw key bits a concept was designed in [13], which increases the ef¬ciency. Instead
of having the same probability for 0™s and 1™s in the string, the authors propose to
increase the probability for one of them. In fact the photons will be transmitted more
often in one base than in the other. Thus, the rate in which Alice and Bob select the
same base will increase. We will discuss this more precisely in Sect. 3.2.1.
Finally, the communication between Alice and Bob on the quantum channel is
¬nished. They now switch to the public channel. So far, Alice holds the two strings
of length m containing her choice of base and value and Bob a string of length m
containing his choice of bases and his measurement results of length gq · m. The
smaller size is due to the loss on the quantum channel.
3 Quantum Key Distribution 27

3.2 Public Channel
The communication via the public channel is necessary for Alice and Bob, since
they must negotiate on which bits they perform the next steps. Because they choose
their bases for the quantum transmission randomly they must know in which cases
they used the same basis. After agreeing on them, the errors must be corrected,
since the communication via the quantum channel is noisy. And as a last step Alice
and Bob must reduce Eve™s knowledge, which she gained during the protocol. By
intercepting and resending photons Eve gains knowledge of the raw key (this issue
is re¬‚ected by the error rate) and also eavesdropping on the messages on the public
channel increases her knowledge. If Eve has the possibility to modify messages,
she can start a man-in-the-middle attack and then share keys with both disguised as
one™s peer.


3.2.1 Sifting

The ¬rst phase on the public channel is sifting. After Alice has sent random bits
mapped into randomly chosen quantum bases via photons, further steps are required
such that she shares the same bit string with Bob. The ¬rst thing Alice must know
is which photons Bob has measured. By reason of the fact that only a small fraction
of pulses contain photons, and several losses on the quantum channel and at Bob™s
detector, Alice does not know which bit Bob received. Therefore, Bob sends a mes-
sage on the public channel to inform Alice which photons he has measured. The
easiest way is to send a string as long as the string chosen at the beginning for the
bases, which we indicate with length m. Bob sets now the position where he was
able to make a measurement to 1 and the other positions to 0. Since we want to save
communication traf¬c, we can bene¬t from the fact that the gain of the quantum
channel gq (Eq. 3.1) is a very small factor (<0.01). Bob must only tell Alice which
position he received. To represent a position log2 m-bits are necessary. Let 2n be the
length of the raw key, which is equal to gq · m, then 2n log2 m-bits are necessary
to tell Alice the measured positions. This would be more ef¬cient if the following
equation is ful¬lled:

2n log2 m = gq · m log2 m < m.

We know that the gain of the quantum channel is less than 0.01, so it is suf¬cient
that m < 2100 . The sending rate is maximal in the range of GHz (109 ≈ 230 ), hence
adequate for this sifting transmission mode.
With this message Alice knows only which position Bob has measured but not
if he has used the same basis, because they have randomly and independently cho-
sen the bases. Thus, Bob sends a second message, in which he publishes the bases
he used. An optimized approach sends only those positions where he has success-
fully measured a photon. Thus, the length of the second message is 2n. Note that
Bob sends only his sequence bases (see Sect. 3.1.2) reduced by the bits where no
28 M. Pivk

measurement was possible, and not the measurement results. After Alice receives
both messages she can reduce her secret bit value string by canceling out those
positions which Bob did not receive (Bob™s ¬rst message) and those positions where
she uses different bases at transmission (comparison by her base string and Bob™s
second message). To make sure Bob shares the same string with her. Alice sends
a message to Bob containing her choice of bases for those positions where Bob
received a photon. Bob himself performs the same procedure as Alice and cancels
out those positions with different bases. Finally, the sifting phase is over and Alice
and Bob share a secret key: the so-called sifted key. However, error remains due to
the noisy quantum channel.

Alice Bo b
1011011000101010 1101110010011000
0010010110110101


010 1010 00101
2,3,4,6,7,8,9,12,13,14,15,16
0010010110110101

_101_1001__11000
010 1011 10101


_011_1100__01010 010 1010 00101


011011 011011

quantum public
base string key string canceled
channel channel

Fig. 3.2 QKD protocol until the end of the sifting phase

But what is the length of this key? Since Alice and Bob have chosen the two
bases for transmission and measuring randomly and independently, the probability
that they use the same basis is 1 · 1 + 1 · 1 = 1 . The length of the sifted key reduced
22 22 2
by mismatching bases from the raw key is 2n = n. The complete process until
2
the output of the sifted key is demonstrated in an example in Fig. 3.2. The sending
key length is m = 16. Assuming that the gain of the quantum channel is given by
gq = 0.75 (which is not a realistic value), we use Tables 3.1 and 3.2 for mapping
bits to photons. So the sifted key length is n = m · g p · 1 = 6.
2
As mentioned in Sect. 3.1.2 it is suggested [13] to choose the strings representing
the photon bases with different probability. So far, we chose the rectilinear or the
diagonal basis with probability 1 . Assume now that Alice and Bob choose a basis
2
with probability δ ∈ [0, 1], whereby the other basis is used with complementary
probability 1’δ. The protocol gain for the normal BB84 is 0.5 as we have computed
before; the protocol gain g p for the case that the probability of base usages can be
changed is

g p = δ · δ + (1 ’ δ) · (1 ’ δ) = 2δ 2 ’ 2δ + 1. (3.2)
3 Quantum Key Distribution 29

1


0.9


0.8
protocol gain




0.7


0.6


0.5


0.4
0 0.2 0.4 0.6 0.8 1
delta

Fig. 3.3 Protocol gain g p when delta δ is variable (Eq. 3.2)




The fact that δ is variable gives us the probability to increase the protocol gain.
As you can see in Fig. 3.3 the gain g p rises to 1 if δ goes to 1 (to 0 “ analog for the
other basis). So the case δ = 0.5 would be the worst case.
But what happens with the security if δ goes to 1. In Sect. 3.2.3.1 we will
discuss this in detail. To say it right away, the chance for Eve to eavesdrop the
photons without being detected increases because Alice and Bob must negotiate at
least which bases to use with higher probability, and Eve can use this information.
The errors Eve produces decrease in the set where the basis is chosen with higher
probability and increase in the set of the other basis. Using the naive error esti-
mation, which calculates the error probability of both sets together, Alice and Bob
will fail to detect Eve, because the proportion of the sets changes. Finally, when
δ reaches 1, Eve will not introduce any errors. Unless δ = 1, if we use a re¬ned
error analysis, which re¬‚ects the error probability of both sets (detailed explanation
in Sect. 3.2.3.1), Eve can still be detected. In [13] it is described that the set of
bits after sifting must have enough samples to make an accurate error estimation.
Therefore, some constraints on δ are de¬ned. Assume that m t photons are trans-
mitted from Alice to Bob, which Bob was able to measure. Then, on average, only
m t (1 ’ δ)2 photons belong to the case where the basis with the smaller probability
is used. The size of this set was analyzed in [13] with the information we allow
Eve to obtain on the ¬nal key. If we except a ¬xed but arbitrarily small amount
of information on the ¬nal key, the number of this set is required only to scale
logarithmically with the length of the ¬nal key k. Thus, δ must ful¬ll the following
equation:
30 M. Pivk

m t (1 ’ δ)2 ≥ O (log k) ,
log k
δ ¤1’O ,
mt
if m t ’ ∞ then δ ’ 1 and g p ’ 1. (3.3)

Note that δ never reaches 1 and the fact that the protocol gain goes to 1 can easily
be seen in Fig. 3.3.


3.2.2 Authentication of Sifting

Due to the possibility of a man-in-the-middle attack (combination of intercept/resend
attack, see Sect. 5.2.1, and modi¬cation of messages on the public channel) by
Eve we must ensure that the sifting phase is authenticated by Alice and Bob. We
know this phase consists of three messages exchanged between Alice and Bob. Two
messages sent by Bob, where he tells Alice which photons he was able to measure
and which bases he used for the measurement. The last message completing sifting
phase is sent by Alice, where she tells Bob which bases she used for those photons
Bob measured. To authenticate those messages we use Wegman“Carter authentica-
tion, as explained in Sect. 2.2.2. For Bob™s two messages and for Alice™s message
we generate authentication tags and append them to the respective messages.
Because Wegman“Carter authentication is symmetric, we have to use a secret
key shared by Alice and Bob. Since key material is generated in every iteration a
small part of it can be used for the authentication of the next round. Unfortunately,
when we start the ¬rst round a pre-shared key must be available, which is exchanged
previously through a secret channel (face-to-face or in another way). In order that
key material remains after withdrawal by authentication, the choice of the algorithm
is a major concern. Following Eq. 2.35 the authentication cost for Wegman“Carter
is 4 · ((b + log2 log2 a) · log2 a) for an input size of a and an authentication tag size
of b. The key size grows only in the logarithmic scale, so it would be useful for our
intended purpose.
Similar to [11] we compute now the authentication cost for the sifting phase. In
the previous section we derived that the length of Bob™s ¬rst message is 2n log2 m
and the length of the second message is 2n. The sum is 2n(1 + log2 m) and thus the
authentication key length w1 needed for the tag is

w1 = 4 · gauth + log2 log2 2n 1 + log2 m · log2 2n 1 + log2 m , (3.4)

where gauth is the length of the resulting tag. Alice and Bob compute their tag using
the hash function indexed by the authentication key and Alice compares if the tags
match.
If Alice veri¬es that the message is from Bob she sends her message with her
choice of bases, which has length 2n. We denote the authentication key length for
Alice™s message as w2 and the length is computed as
3 Quantum Key Distribution 31

w2 = 4 · gauth + log2 log2 (2n) · log2 (2n). (3.5)

If Bob determines that the generated tag with cost w2 matches with Alice™s tag
both can be sure that the probability that Eve modi¬ed the message is at most µ,
which is the property of the class of hash function (µ-ASU2 ).
The required key material to authenticate the sifting phase is the sum of all costs
w1 , w2 (Eqs. 3.4 and 3.5) and results in the total sifting authentication costs


t S = 4 · gauth + log2 log2 2n 1 + log2 m · log2 2n 1 + log2 m
+ 4 · gauth + log2 log2 (2n) · log2 (2n). (3.6)




3.2.3 Reconciliation
The reconciliation stage is split into two parts. The ¬rst major part is the error
estimation. Its aim is to ¬nd the correct error rate. To guarantee an optimal error
correction, part two of reconciliation. The error correction has the task of correcting
all errors in Bob™s string via public discussion. Here some information about the
string must be disclosed and exchanged between Alice and Bob.


3.2.3.1 Error Estimation
The error estimation is an important step in the QKD protocol to determine the
proper error rate of the sifted key. Usually, in BB84, the error rate p is estimated by
picking a small random subset of bits with length r from those given in the sifted
key. This test string is publicly compared by Alice and Bob and yields in a certain
number of errors e. If the length of the test string is chosen adequate to the length of
the sifted key n, the error probability is

e
p= . (3.7)
r

Of course, since a part of the sifted key has been announced, those bits must be
deleted to avoid information leakage to Eve. This elimination has little effect on
the length of the ¬nal key if the length of the sifted key is large and if the error
estimation is not executed every round. The error rate found for a round can also be
used in the following rounds.
If the error rate p turns out to be very large, then either eavesdropping has
occurred or the channel is somehow unusually noisy. However, the sifted key is dis-
carded and Alice and Bob may re-start the whole protocol again (on another quan-
tum channel). This threshold pmax , for which the rate should not exceed p ¤ pmax ,
can be set to, e.g., 11% because at the moment the best error correction code
32 M. Pivk

approaches a maximal tolerated error rate of 12.9% [21]. If the error rate p is
reasonably small ( p ¤ pmax ), Alice and Bob can continue with error correction.
An improved error estimation was presented in [1] and [13]. In the re¬ned error
analysis there does not compute a single error rate but two values are estimated.
The idea is not to merge the measurements during the quantum transmission, where
Alice and Bob have used the same basis, into one set (more precisely the sifted key)
and choose the random test subset out of it. They choose two random test subsets.
The ¬rst subset consists of all measurements where Alice and Bob have used the
¬rst basis (e.g., rectilinear) and the second one where they used the second basis
(e.g., diagonal). Let r1 be the length of the ¬rst subset and r2 be the length of the
second subset just before Alice and Bob publicly compare both strings and estimate
the number of errors e1 and e2 , respectively, which leads to the error rates

e1
p1 = , (3.8)
r1

e2
p2 = , (3.9)
r2

where p1 is the error rate for the photons measured with the ¬rst basis and p2 those
with the second basis and
p1 + p2
p= . (3.10)
2

The advantage compared to the single error rates is as follows: If Eve starts a
speci¬c attack like the biased eavesdropping strategy (a specialization of intercept
and resend, see Sect. 5.2.1). Then she chooses the probability q1 for measuring
each photon sent from Alice to Bob in the ¬rst basis (e.g., rectilinear) and a second
probability q2 for measuring each photon in the second basis (e.g., diagonal). Hence,
with probability 1 ’ q1 ’ q2 she does not measure the photon. When Alice and Bob
use the same basis errors occur only if Eve uses a different basis. Regarding Bob™s
side, the photons measured by Eve are randomized and yield an incorrect bit in half
of these cases. This introduces an error rate of p1 = q22 for the ¬rst basis. Similarly,
for the second basis an error rate p1 = q22 is obtained (without respect to the normal
noise on the quantum channel). If we apply these two error rates and Eq. 3.10 to the
requirement p ¤ pmax it results in (q1 + q2 ) ¤ 4 pmax . Eve has the possibility to vary
her probability q1 in a big range. In contrast if we use the constraint p1 , p2 ¤ pmax
which has the same property as the single error rate constraint in a random noisy
channel, Eve™s possibility to choose the probabilities is shortened with q1 , q2 ¤
2 pmax . With the re¬ned error analysis the biased eavesdropping strategy is avoided
(e.g., q1 = 3 pmax , q2 = 0 is accepted when we measure only one error rate, but
rejected by the detailed error measurement).
In Sect. 3.2.1 we have heard about a strategy to vary the probability of used
bases at sending and receiving, to boost the protocol gain g p (in standard BB84
protocol the probability for sending in one of the bases is 1 as described in Eq. 3.2).
2
3 Quantum Key Distribution 33

The dif¬culty now is that with increasing or decreasing δ, a biased eavesdropping
strategy where Eve aware of δ is more dif¬cult to detect using the simple error
estimation rate. In this case the simple error rate is

δ 2 p1 + (1 ’ δ)2 p2 δ 2 q1 + (1 ’ δ)2 q2
= .
p= (3.11)
δ 2 + (1 ’ δ)2 2(δ 2 + (1 ’ δ)2 )

If Eve chooses the strategy q1 = δ and q2 = 1 ’ δ we see in Fig. 3.4 that when
δ ’ 1 or δ ’ 0 the error probability p ’ 0 unlike the re¬ned error analysis where
δ ’ 1 then p1 ’ 0 but p2 ’ 0.5. Regardless of how δ is chosen (0 < δ < 1) if
the measurement set is quite big (as de¬ned in Sect. 3.2.1) then there would be no
problem to detect Eve, which is not possible with the simple error analysis.

0.5
complete set
subset base 1
subset base 2
0.45

0.4

0.35

0.3
error rate




0.25

0.2

0.15

0.1

0.05

0
0 0.2 0.4 0.6 0.8 1
delta

Fig. 3.4 Error rates (variable δ) after a biased eavesdropping attack with q1 = δ




3.2.3.2 Error Correction
Henceforth we will model the quantum channel as binary symmetric channel (BSC).
In a BSC (p) we have an alphabet of two symbols (0, 1) which are transmitted over
the channel from a sender to a receiver. With probability p noise is added to the
transmitted symbol, meaning the receiver gets a 1 instead of a 0 and a 0 instead of a
1, respectively. For every symbol we have a probability 1’ p that it will be correctly
received (see Fig. 3.5).
After the sifting phase Alice holds a random string ska and Bob skb with equal
length |ska | = |skb | = n. The difference between these two strings depends on
the quantum bit error rate (QBER) p of the channel. Starting the QKD protocol
34 M. Pivk

1“p
0 0
p
sender receiver
p
1
1
1“p

Fig. 3.5 Binary symmetric channel (BSC): symbols are exposed to noise with probability p


this probability p is unknown or rests on empirical values. Further on, the error
estimation phase as presented in Sect. 3.2.3.1 gives a good approximation of the
error rate. Observe that dist (ska , skb ), the amount of places in which both strings
differ (Hamming distance), is nearly np. Setting ska = A and skb = B (A and B
are random variables) the conditional entropy (Eq. 2.38) of A given B is

H ( A|B) = H (A • B) = nh( p),

where h( p) is now short for Shannon entropy H (X ) and the random variable X is a
Bernoulli trial with parameter p (for each x ∈ X , p(x) is the same).
The task is to correct Bob™s string B without disclosing enough information that
gives Eve chances to reconstruct the very same string. Therefore, a reconciliation
protocol R p is de¬ned, which runs on both strings A, B and results in the string
S by exchanging some information Q on the public channel. We write S = ⊥ if
the protocol fails to produce S. Since Q must be exchanged on the public channel,
an eavesdropper Eve can gain some information on S, which can be expressed by
I E (S|Q). This is the expected amount of bits that an eavesdropper Eve can get on S
given Q.
Brassard and Salvail [7] have formulated some de¬nitions to characterize such
reconciliation protocols. The ¬rst de¬nition deals with the robustness.

De¬nition 3.1 A reconciliation protocol R p is µ-robust if

(∃N0 (µ))(∀n ≥ N0 (µ)) pr ob(A = ±, B = β) pr ob(R p (±, β) = [⊥, ·]) ¤ µ
±,β∈{0,1}n


where 0 ¤ µ ¤ 1.

If a protocol is µ-robust the probability to fail is maximal µ. The next theorem is
a direct consequence of the noiseless coding theorem.

Theorem 3.1 (∀ p ¤ 1 ) (∀reconciliation protocol R p ) If there exists 0 ¤ µ ¤ 1 such
2
that R p = [S, Q] is µ-robust then

I E (S|Q)
≥ 1,
lim
nh( p)
n’∞
3 Quantum Key Distribution 35

where n is the length of the transmitted string.


In other words, Eve™s information of S is greater than or equal to the information
of A given B. Eve cannot have less than this information if the reconciliation proto-
col R p is successful. But if these two pieces of information are equal the protocol is
optimal. Let us de¬ne it by the next de¬nition:


De¬nition 3.2 A protocol R p is optimal if

(∀µ ≥ 0)[R p = [S, Q] is µ-r obust]

and
I E (S|Q)
= 1,
lim
nh( p)
n’∞


where the public channel is a BSC(p).
In [7] a construction of a optimal protocol is presented. The basic idea is that
Alice creates a random label of her string A with length approximately nh( p), when
the length of string A is n. If Alice sends this label over the channel, Eve would
only have this information and so De¬nition 3.2 holds, the protocol is optimal.
When Bob obtains the label f ( A), he computes all possible inputs, resulting in a
set S{B | f (B ) = f ( A)}. The string B ∈ S with minimal Hamming distance from
B is the desired S. Brassard and Salvail [7] described this protocol as unpractical,
n
because Alice and Bob require 2m2 functions. But if the function f (x) is chosen
from a univer sal2 class of hash functions (see De¬nition 2.5) the protocol remains
optimal. The speci¬cation of a univer sal2 class of hash functions can be done in a
short and ef¬cient way. The only problem which remains is that there are no known
ef¬cient algorithms to compute the set S{B |h(B ) = h(A)}, h ∈ H, if the hash
value of A is known.
The goal is to create an ef¬cient reconciliation protocol, which is de¬ned as
follows:

De¬nition 3.3 A reconciliation protocol R p is ef¬cient if there is a polynomial t(n)
¯p
such that T R (n) ¤ t(n) for n suf¬ciently large, where n is the length of the strings
¯p
transmitted over the secret channel and T R (n) represent the expected running time
of R p , given an n-bit long input string.


De¬nition 3.4 A reconciliation protocol R p is ideal if it is both optimal and
ef¬cient.
Brassard and Salvail determined when their optimal protocol becomes ideal,
using the univer sal2 class of hash function H3 [9]. They proved that their protocol
is ideal if and only if NP ⊆ BPP, which is a hypothesis.
36 M. Pivk

With the fact that a ideal reconciliation protocol depends on a open question in
complexity, which is unlikely to be true, we have to ¬nd another solution. In Sect.
2.2.1 (universal hashing) we had a similar problem. There we de¬ne a str ongly
univer sal2 class of hash functions, where the size of this class becomes too large
and unpractical. But just a small change of the probability on the theoretical bound
makes these classes useful. Here again if we do not demand optimality and allow
the reconciliation protocol to transmit a small amount of leaked information above
the theoretical bound the protocol becomes ef¬cient and useful. In [7] this charac-
terization was de¬ned as in De¬nition 3.5.
p
De¬nition 3.5 A reconciliation protocol Rζ is almost ideal if for all ζ ≥ 0 we have
p
1. (∃µ ≥ 0)[Rζ = [S, Q] is µ-robust]
2. limn’∞ I E (S|Q) ¤ 1 + ζ
nh( p)
p
3. (∃ polynomial t)(∃N0 (ζ ))(∀n ≥ N0 (ζ ))[T Rζ ¤ t(n)]
¯

for n the length of the string transmitted over the B SC( p).


An almost-ideal protocol has an error probability bounded by µ approaching 0 for
increasing n (see De¬nition 3.51). The amount of leaked information is allowed to
be slightly greater than the theoretical bound (see De¬nition 3.52), but the parameter
ζ indicating the excess of information is chosen by Alice and Bob before the start
p
¯
of the protocol. After their choice the expected runtime T Rζ of the reconciliation
protocol must be bounded by a polynomial (see De¬nition 3.53).
Such an almost-ideal protocol was presented in [7] and an earlier version in [4].
This simple protocol called CASCADE leaks an amount of information close to the
theoretical bound, when the error probability is below 15%.
The appropriate part of CASCADE for the error correction is BINARY. When
Bob™s string contains an odd number of errors, Alice and Bob can perform an inter-
active binary search on the strings to ¬nd and correct one error, respectively. At the
beginning Alice sends Bob the parity of the entire string and Bob checks if his parity
bit differs from Alice™s bit. If not, there is an even number of errors (possibly zero)
in the string and nothing is done. If the parity bits differ the following steps are run
through to ¬nd the error:

1. Alice sends Bob the parity of the ¬rst half of the string.
2. Bob determines whether an odd number of errors occurred in the ¬rst or in the
second half by testing the parity of his string and comparing it to the parity sent
by Alice.
3. With the half determined in step 2 we start again at step 1, until the erroneous bit
is found.

For a better understanding see the visualization in Fig. 3.6. The leaked informa-
tion for a string with an even number of errors is one bit and for a string with an odd
number log n + 1 bits, whereas one error is corrected.
3 Quantum Key Distribution 37




skip bit
transmitted
by Alice
erroneous bit parity bit

Fig. 3.6 BINARY corrects exact one error of a block with an odd number of errors




CASCADE proceeds in several passes, which are de¬ned by Alice and Bob
before execution relative to the error probability p. Alice and Bob hold string A =
A1 , ..., An and B = B1 , ..., Bn ( Ai , Bi ∈ {0, 1}), respectively. For the ¬rst pass they
choose k1 (determination of parameter k1 is shown later) and split their strings into
blocks of length k1 . Block v for pass 1 is de¬ned by K v = {l|(v ’ 1)k1 < l ¤ vk1 },
1

v = 1... kn1 . On each block BINARY is performed. This can be done parallel for
each block to minimize the communication effort. For passes i > 1, Alice and Bob
choose a ki and a random function f i : [1..n] ’ 1.. kn1 . Now the block j in
pass i has the form K ij = {l| f i (l) = j}. Again Alice and Bob perform BINARY on
these blocks to correct errors. In the previous version of CASCADE [4] the leaked
information about the string is eliminated during execution of BINARY by removing
the last bit of each subset for which a parity bit is computed. In [7] an improvement
is introduced where the removed bits are kept. This allows us to correct more errors
in later passes. Because if in pass i > 1 an error Bl = Al of block K ij is corrected,
each block K v from previous passes containing this bit l ∈ K v , 1 ¤ u < i, has now
u u

an odd number of errors and can be corrected with less effort. Therefore, a set K of
these blocks is created. The smallest block in K is chosen and BINARY is executed
on it. After correcting the error Bl , we create again a new set B with the previous
blocks containing bit Bl from each pass from 1 to i. Now set K = B ∪ K\B © K is
determined. If K = … then by choosing again the smallest block further errors can
be eliminated. Pass i ends when all blocks K ij are checked and when there are no
more blocks in K .
38 M. Pivk

For determining the parameter ki let δi ( j) be the probability that after the pass
i ≥ 1, 2 j errors remain in block K v . Let E i be the expected number of errors in K v
1 1

after pass i. So that after the ¬rst pass we have

δ1 ( j) = prob(X = 2 j) + prob(X = 2 j + 1), (X ≈ Bin(k1 , p)),


k1
2

E1 = 2 jδ1 ( j).
j=1


Suppose now that k1 is chosen such that

k1
2
1
δ1 (l) ¤ δ1 ( j) (3.12)
4
l= j+1


and

ln 1
E1 ¤ ’ 2 (3.13)
2

are satis¬ed and for the following passes we simply de¬ne ki = 2ki’1 for i > 1.
Brassard and Salvail have shown in [7] that if k1 satis¬es Eqs. 3.12 and 3.13 the
amount of information I (ω) per block of length k1 leaked after ω passes can be
bounded as follows:
k1
ω 2
1 ’ (1 ’ 2 p)k1 jδ1 ( j)
I (ω) ¤ 2 + log k1 + 2 log k1 . (3.14)
2l’1
2 l=2 j=1


For different error probabilities p ∈ {0.01, 0.05, 0.10, 0.15} in four passes the
theoretical bound and the leaked information I (4) are computed with Eq. 3.14. In
addition empirical tests were performed to ¬nd the average amount of leaked infor-
mation I (4) (10 tests with n = 10, 000). The results of [7] are present in Table 3.3.
ˆ
I (4)
The column f ( p) kh( p) gives the percentage of additional leaked information relative
to the theoretical bound.

Table 3.3 CASCADE benchmark
f ( p) = I (4)
ˆ
p k1 I (4) kh( p) I (4) kh( p)
0.01 73 6.47 5.89 6.81 1.16
0.05 14 4.60 4.01 4.64 1.16
0.10 7 3.81 3.28 3.99 1.22
0.15 5 3.80 3.05 4.12 1.35
3 Quantum Key Distribution 39

Finally, the reconciliation phase is ¬nished. After getting random strings for
Alice and Bob from the sifting phase with length n and error probability p the
described reconciliation protocol (CASCADE) has corrected Bob™s string and only a
very low probability remains that Bob™s string has still an error. The leaked informa-
tion was kept down near the theoretical bound with an additional leakage of 16“35%
depending on the error probability. In Chap. 4 a advancement of this protocol is
presented, called Adaptive Cascade.


3.2.4 Con¬rmation/Authentication of Error Correction

The next step in the protocol is on the one hand to con¬rm the strings in Alice™s
and Bob™s possession and on the other hand to authenticate the reconciliation phase.
Fortunately, this can be done in one step based on the fact that Eve could mount
a man-in-the-middle attack during the reconciliation phase. Without the authenti-
cation step, Alice and Bob could believe that their strings, on which they operate,
are identical when they are not. Only one different bit is necessary to produce com-
pletely uncorrelated strings at the privacy ampli¬cation phase (see Sect. 3.2.5). The
best solution for Alice and Bob is to verify that the outputs of the error correction
phase are the same. This will authenticate the complete communication during the
reconciliation phase, because an intervention of Eve will introduce errors.
The veri¬cation can be done by hashing the corrected random strings and com-
pare the resulting tags. We also should keep the size of the tag small compared to
the input (corrected shifted key) size. Every additional bit will increase the leaked
information and accordingly Eve™s knowledge.
As in Sect. 3.2.2 we have to sacri¬ce a part of already shared key for authen-
tication. Again the best choice to authenticate the corrected shifted key is to use
Wegman and Carter authentication as presented in Sect. 2.2.2. Gilbert and Hamrick
acknowledge that this is the best choice and analyzed the total authentication
costs for the error correction phase [11]. After Eq. 2.35 the authentication cost
is 4 · ((b + log2 log2 a) · log2 a) for an input size of a and an authentication tag
size of b.
The size of the corrected sifted key at the end of the error correction phase is n.
The required authentication key length w1 for the generation of an authentication
tag is
w1 = 4 · (g EC + log2 log2 n) · log2 n, (3.15)

where g EC is the length of the resulting tag. Alice and Bob compute the tag using
the hash function indexed by the authentication key and compare if the tags match.
The main problem is if Bob sends this tag to Alice, Eve can modify the tag with
arbitrary bits to convince Alice that the strings do not match when, in fact, they do.
To avoid this denial-of-service attack Bob must authenticate his message. Therefore,
he computes another tag, which authenticates the previous tag as his tag and sends
it to Alice. Again he needs a part of the shared key to generate the tag with length
gauth which costs
40 M. Pivk

w2 = 4 · (gauth + log2 log2 g EC ) · log2 g EC . (3.16)

If Alice determines that her generated tag with cost w1 matches with Bob™s tag
and the authentication tag with cost w2 does as well, she can be sure that the strings
are equal. If the tag with cost w1 does not match but the second does, there must be
at least an error in the string of Bob. In the case that both tags do not match, she can
assume that either Eve manipulates the messages or the shared keys of Alice and
Bob are different.
To indicate Bob that the tags match Alice sends a piece of the key with length

w3 = g EC
˜ (3.17)

to Bob. Again to avoid the man-in-the-middle attack, Alice must authenticate her
message. With the same tag length gauth as Bob™s before, yielding an authentication
cost of

w4 = 4 · (gauth + log2 log2 g EC ) · log2 g EC .
˜ ˜ (3.18)

After Bob has compared the tags of Alice with his ones they agree the authenti-
cation step for the error correction is complete. Alice and Bob can now be sure that
with probability of 1 ’ µ the strings are equal. Because they use an µ- ASU2 class
of hash functions the collision probability for two distinct values is at most µ as
the De¬nition 2.5 for universal hash functions declare. Also the probability for Eve
to modify one of this authentication messages is at most µ, even if she suppresses
modi¬cations during the error correction phase or make Alice and Bob believe they
have different strings when, in fact, they have not.
The required key material to authenticate the reconciliation phase is the sum of
all costs w1 , w2 , w3 , w4 (see Eqs. 3.15, 3.16, 3.17, and 3.18), which results in the
total error correction authentication costs

t EC = 4(g EC + log2 log2 n) log2 n + 4(gauth + log2 log2 g EC ) log2 g EC
+ g EC + 4(gauth + log2 log2 g EC ) log2 g EC .
˜ ˜ ˜ (3.19)


3.2.5 Privacy Ampli¬cation

The last step to the secret key is the privacy ampli¬cation phase. Bennett, Brassard,
Cr´ peau, and Maurer [6] give a good explanation what privacy ampli¬cation is:
e
Privacy ampli¬cation is the art of distilling highly secret shared information,
perhaps for use as a cryptographic key, from a larger body of shared information
that is only partially secret. Let Alice and Bob be given a random variable W , as a
random n-bit string, while an eavesdropper Eve learns a correlated random variable
V , providing at most t < n bits of information about W , i.e., H (W |V ) ≥ n ’ t.
The details of the distribution PV W are generally unknown to Alice and Bob, except
3 Quantum Key Distribution 41

that it satis¬es this constraint as well as possibly some further constraints. They
may or may not know PW . Alice and Bob wish to publicly choose a compression
function g : {0, 1}n ’ {0, 1}r such that Eve™s partial information on W and her
complete information on g gives her arbitrarily little information about K = g(W ),
except with negligible probability (over possible choices for g). The resulting K is
virtually uniformly distributed, given all Eve™s information; it can hence be used
safely as a cryptographic key.
The size r of the secret that Alice and Bob can distill depends on the kind as well
as the amount of information available to Eve. Eve can obtain

• t arbitrary bits of W ,
• t arbitrary parity checks of W ,
• the result of an arbitrary function mapping n-bit strings to t-bit strings,
• the string W transmitted through a binary symmetric channel with bit error prob-
ability p satisfying h( p) = 1 ’ n , and hence with capacity n where h denotes
t t

the binary entropy function.

Assume that we have the scenario where Alice and Bob are connected by an inse-
cure channel to which Eve has passive perfect access. Note that with authentication
of the communication active perfect access has the same characteristics, which will
not be discussed here for reasons of simplicity. So after Alice™s transmission on
the quantum channel each of them has knowledge of a correlated random variable.
Alice X , Bob Y , and Eve Z . Assume that these variables are distributed according
to some joint probability function p X Y Z , whereby Eve has partially control over this
distribution. Based on the fact that neither Alice nor Bob has an advantage compared
to Eve concerning information (I (X ; Y ) > I (X ; Z ), I (X ; Y ) > I (Y ; Z )), after the
sifting phase which can be denoted as random variable C, Alice can compute a string
W from X and C such that Alice™s uncertainty about W is 0 and Bob™s uncertainty
about W is smaller than Eve™s one:

H (W |XC) = 0,
H (W |Y C) < H (W |ZC). (3.20)

The reconciliation phase helps to remove Bob™s uncertainty about W ; there-
fore, Alice sends a bit string D with length l is slightly larger than H (W |Y C)
such that H (W |Y C D) ≈ 0. Eve™s uncertainty H (W |ZC D) is lower bounded by
H (W |ZC) ’ l, which can be substantially positive, due to Eq. 3.20. We summarize
Eve™s knowledge ZC D about W to the random variable V .
The aim of privacy ampli¬cation is now to generate a secret key K , of which
Eve has negligible amount of information. Therefore, Alice and Bob agree on a
function g (known by Eve) to generate K = g(W ). This g is chosen randomly from
a set G of function to avoid that Eve can decide beforehand which strategy to use.
Thus, this function is also a random variable G on the set G. The output length of
G : W’ {0, 1}r should decrease Eve™s information about K to a minimum. Assume
I (W ; V ) ¤ t then I (K ; GV ) ≈ 0 and H (K |GV ) ≈ r . But how must r be chosen? In
42 M. Pivk

[6] they state r = n ’ t ’ s and the security parameter s decreases Eve™s information
’s
rapidly, because I (K ; GV ) ¤ 2 2 .
ln
The theorem and the corollaries of [6] are presented now, which yields an upper
bound to Eve™s information. It emerges that for the function set G the universal
classes of hash functions as described in Sect. 2.2.1 become useful, due to their
properties.

Theorem 3.2 Let X be a random variable over the alphabet X with probability
distribution p X and R´ nyi entropy R(X ), let Q = G(X ). Then
e

2r’R(X )
r ’R(X )
H (Q|G) ≥ R(Q|G) ≥ r ’ log2 1 + 2 ≥r’ ,
ln 2

where G is the random variable corresponding to the random choice (with uniform
distribution) of a member of a universal class of hash functions X ’ {0, 1}r .

The ¬rst inequality follows from Eq. 2.45. The second inequality can be proven
by using Jensen™s inequality and using the fact that we use universal hash function.
For the complete proof we refer to [6]. The last inequality follows from log2 (1+y) ¤
y
. The next corollary is derived from the above theorem.
ln 2

Corollary 3.1 Let PV W be an arbitrary probability distribution and let v be a par-
ticular value of V observed by Eve. If Eve™s R´ nyi entropy R(W |V = v) about W
e
is known to be at least c and Alice and Bob choose K = G(W ) as their secret key,
then

2r ’c
H (K |G, V = v) ≥ r ’ log2 1 + 2 ≥r’ ,
r’c
ln 2

where G is chosen at random from a universal class of hash functions W ’ {0, 1}r .

Corollary 3.2 Let W be a random n-bit string with uniform distribution over
{0, 1}n , let V = e(W ) for an arbitrary eavesdropping function e : {0, 1}n ’ {0, 1}t
for some t < n, let s < n ’ t be a positive safety parameter, and let r = n ’ t ’ s.
If Alice and Bob choose K = G(W ) as their secret key, then Eve™s expected infor-
mation about the secret key K , given G and V , satis¬es

2’s
I (K ; GV ) ¤ ,
ln 2

where G is chosen at random from a universal class of hash functions {0, 1}n ’
{0, 1}r .

Finally, corollary 3.2 can be proven with usage of Corollary 3.1 (see [6]). Note
that the security parameter does not depend on the error probability p or else vari-
ables, if the upper bound for Eve™s knowledge t is known.
3 Quantum Key Distribution 43

Let us recapture the entire protocol to identify the knowledge t Eve has obtained.
The easiest knowledge to identify is in the reconciliation phase and the authenti-
cation of the reconciliation. We know that during error correction at least nh( p)
bits ( p is the error probability) are necessary to correct all errors. In Sect. 3.2.3.2 a
reconciliation protocol has been presented which reaches nearly this limit. It differs
only by a factor f ( p) given as in Table 3.3. Thus, the knowledge Eve accumulates
during this phase is n f ( p)h( p). If we formulate this as a fraction „1 by which the
reconciled key is to shorten then

n f ( p)h( p)
„1 ( p) = = f ( p)h( p) = ’ f ( p) p log2 p + (1 ’ p) log2 (1 ’ p) .
n
(3.21)
In the con¬rmation/authentication of the error correction phase a hash value of
the key is transmitted. This tag has length g EC , giving Eve additional information.
The last phase where Eve can gain knowledge is the sifting phase, if she eaves-
drops several photons during quantum transmission. How much information Eve
gets, depends on the error probability p. L¨ tkenhaus derived in [14] and [15] a frac-
u
tion „2 , by which we need to shorten the reconciliation key in addition to the other
parameters to get a secure key. Therefore, he used a trivial extension of Corollary
3.1:

I (K ; GV ) = H (K ) ’ H (K |GV ) = r ’ H (K |GV ) ¤ r ’ R(K |GV ). (3.22)

This inequality can be applied due to Eq. 2.45. Furthermore, in [14] it is shown
that the collision probability of a key k is bounded above by the collision probability
of his sifted key for averaged g and v, hence

I (K ; GV ) ¤ r ’ R(K |GV ) ¤ r ’ R(W |V ). (3.23)

We set the information I (K ; GV ) to zero and thus the shortening is

n ’r R(W |V ) w
„2 = =1’ = 1 + log2 pc (v) v , (3.24)
n n
w
means averaged with respect to v and pc (v) is the collision probability
where v
2
w∈W p(w|y) . We can express the collision probability for the sifted key as the
w
product of the collision probability for single bits of this sifted key pc (v) v =
( pc ( p))n with respect to the error rate of the quantum channel p. In [14] this colli-
(1)

sion probability was derived as

1
pc ( p) ¤ + 2 p ’ 2 p2 for p ¤ 1/2,
(1)
(3.25)
2
which gives, ¬nally,

„2 ( p) ¤ log2 1 + 4 p ’ 4 p 2 for p ¤ 1/2. (3.26)
44 M. Pivk

The gain for the privacy ampli¬cation g pa is composed of „1 from Eq. 3.21, „2
from Eq. 3.26, n (security parameter length) and g n (authentication tag length)
s EC




s g EC
g pa = 1 ’ „1 ( p) ’ „2 ( p) ’ ’
n n
= 1 + f ( p)( p log2 p + (1 ’ p) log2 (1 ’ p))
s g EC
’ log2 1 + 4 p ’ 4 p 2 ’ ’ for p ¤ 1/2. (3.27)
n n

Note that the two fractions n and g n are not crucial factors for the ¬nal key,
s EC


because they are almost ¬xed values and if the length of the key increases they
are comparatively small. Figure 3.7 shows that the maximal acceptable error rate
is around 10% for CASCADE and around 11% if we reach Shannon™s limit. The
factors n and g n are not considered.
s EC




1 Cascade
Shannon™s limit



0.8
gain of privacy amplification




0.6



0.4



0.2



0
0 0.02 0.04 0.06 0.08 0.1 0.12
error rate

Fig. 3.7 The gain of privacy ampli¬cation g pa for increasing quantum error rate p

Concluding the QKD protocol we give a universal class of hash function, which
can be used to reduce the reconciliation key of length n to a ¬nal key of length k.
Therefore, the multiplication in ¬nite ¬elds is de¬ned as

De¬nition 3.6 Let A = GF (2n ) and B = {0, 1}k . Let h c (x) be de¬ned as the last k
bits of the product c · x in a polynomial representation of GF (2n ). The set

HGF (2n )’{0,1}k = {h c : c ∈ GF (2n )}

is a universal family of hash functions [23].
3 Quantum Key Distribution 45

This family requires only n bits to identify the particular function (the value of c).
In particular it is chosen for privacy ampli¬cation because the multiplication of
large blocks can be done ef¬ciently. We know that by using the traditional shift-
and-add algorithm, the multiplication of two elements of GF (2n ) can be achieved
in quadratic time. In [2] a simple algorithm is presented, which performs a multipli-
cation in a · l log l steps for some small constant a.
The value c, which chooses a function of the family of hash functions, must be
random. Alice or Bob could randomly choose such a value c and announces it pub-
lictly to its peer. But we must consider that this data exchange must be authenticated,
not to be exposed to an attack by Eve. Recalling the sifting phase, where Bob has
sent Alice an authenticated message, in which he told her, which photons he was
able to measure and what bases he has used. A convention was that Bob chooses
his bases randomly, so Alice and Bob share a 2n authenticated random string (the
length is 2n because normal sifting has a gain of 0.5). They can now extract the ¬rst
n bits of this string and use it as c. For this reason the communication during the
privacy ampli¬cation phase and thus an additional authentication is not necessary.
As an example let the string for the random choice of base, which Bob sent
authenticated to Alice, be 0010011101011000101. After performing sifting and rec-
onciliation, the reconciliation key, shared by Alice and Bob, is 1101001110. Due to
the error rate and the information leaked during reconciliation we have computed
that Eve has 4 bits of information, so we must shorten the key to the length of 6 bits.
We extract the value of c from Bob™s random choice of bases, which is 0010011101
and multiply c with the reconciliation key.

0010011101 · 1101001110 = 100000011011010110

The ¬nal key consists now of the last six bits of the result (010110).


3.3 QKD Gain

In Sect. 3.1 and 3.2 discussing the communication over the classical and the quan-
tum channel we heard how a QKD system works and some equations are given to
compute the output of the several stages. The gain of the quantum channel gq in
Eq. (3.1), the protocol gain g p in Eq. (3.2), and the gain of privacy ampli¬cation g pa
in Eq. (3.27) form the complete gain. Additionally we must subtract the required
authentication key (t S (3.6) and t EC (3.19)) which are needed for the next round. So
the length k of the ¬nal authenticated key for one round is

k = m · gq · g p · g pa ’ t S ’ t EC , (3.28)

where m is the length of the random sequence Alice generates at the beginning and
sends via the quantum channel to Bob. The major problem with this protocol is that
for the ¬rst round we need a pre-shared secret between Alice and Bob. Thus, the key
46 M. Pivk

output length would be limited to the length of the secret. After a certain “warm-up”
phase the protocol would reach the expected output on average.



3.4 Finite Resources

All assumptions regarding the postprocessing (sifting, error estimation and correc-
tion, privacy ampli¬cation) and hence the security of quantum key distribution pro-
tocols have been proven in the asymptotic limit. However, the actual implementa-
tions of the protocols can only use ¬nite resources, such as limited computational
power and keys of ¬nite length. This fact has already been addressed for special
protocols like the BB84 protocol [12] or the six-state protocol [16]. In both scenarios
it has been shown that the values for the sifted key rate in the ¬nite scenario differ
signi¬cantly from the expected values from the asymptotic limit.
As described in [20] also the de¬nition of security has to be altered when consid-
ering ¬nite resources. In the asymptotic limit a key of length l is said to be secure if
its deviation µ from a perfect key tends to zero as l increases. Another property most
security de¬nitions lack is composability. Composability assures that a key coming
from a quantum cryptographic protocol can safely be used in classical applications
(e.g., for encryption with a one-time pad). For the nonasymptotic scenario Scarani
and Renner presented a security de¬nition, which satis¬es also composability [20].
Here a key is considered to be secure if the difference between the generated key
and a perfect key is smaller than µ, which means that µ is the maximum probability
that the generated key differs from a perfect key.
The main goal Scarani and Renner pursue in their paper is to obtain a sifted
key rate r based on a certain number of signals, a security parameter µ, and some
losses from the error correction. As starting point they take the relation from the
asymptotic limit, where the sifted key rate r is

r = S X |E ’ H X |Y , (3.29)

where S is the von Neumann and H the Shannon entropy. The goal is achieved using
tools from Renner™s PhD thesis [18]. The exact formulas and proofs are described
in [20] and will not be further discussed here. The major result is that the key rate
becomes positive, e.g., for a little more than 104 signals and an error rate of 0.5% or
at least 105 signals and an error rate of 5%.



References
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3 Quantum Key Distribution 47

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09027 30, 39
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distribution. Eur. Phys. J. D 41(3), 599“627 (2007) 46
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its unconditional security. Journal of Cryptology 18, 133 (2005). URL http://www.citebase.
org/abstract?id=oai:arXiv.org:quant-ph/0011056 26, 28, 29, 32
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18. Renner, R.: Security of Quantum Key Distribution. Ph.D. thesis, Swiss Federal Institute of
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Chapter 4
Adaptive Cascade

S. Rass and C. Kollmitzer




4.1 Introduction
Quantum cryptographic key exchange is a promising technology for future secret
transmission, which avoids computational infeasibility assumptions, while (almost)
not presuming pre-shared secrets to be available in each peer™s machine. Never-
theless, a modest amount of pre-shared secret information is required in adjacent
link devices, but this information is only needed for authentication purposes. So
quantum key distribution cannot create keys from nothing, rather it is a method of
key expansion. The remarkable feature of quantum cryptography is its ability to
detect eavesdropping by the incident of an unnaturally high quantum bit error rate.
On the other hand, it has no defense against person-in-the-middle attacks by itself,
which is why authentication is of crucial importance.
We are particularly interested in the ¬nal steps of the BB84 protocol, in which
Alice and Bob correct errors in the bit strings they created from their measurements.
Key reconciliation involves Alice and Bob publicly exchanging parity bits in order
to correct errors and distill identical keys.


4.2 Error Correction and the Cascade Protocol
Coming to the core topic of this chapter, let us pay closer attention to the error
correction mechanism which has been proposed along with the experimental imple-
mentation of BB84 [2]. Errors in physical transmission media often exhibit burst
structures, that is, a sequence of consecutive errors is more likely to occur than

S. Rass (B)
System Security Research Group, Institute of Applied Informatics, Universitaet Klagenfurt,
Universitaetsstrasse 65-67, 9020 Klagenfurt, Austria, stefan.rass@uni-klu.ac.at
C. Kollmitzer
Safety & Security Department, Quantum Technologies, AIT Austrian Institute of Technology
GmbH, Lakeside B01A 9020 Klagenfurt, Austria, christian.kollmitzer@ait.ac.at;
http://www.ait.ac.at




Rass, S., Kollmitzer, C.: Adaptive Cascade. Lect. Notes Phys. 797, 49“69 (2010)
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-04831-9 4
50 S. Rass and C. Kollmitzer

sparse scattering. Consequently, a popular trick is to permute the bits prior to any
error correction in order to chop down long bursts into small pieces, ideally leaving
an almost uniform pattern of erroneous bits in the result. This is the ¬rst step in
the Cascade protocol that has been summarized in Chap. 3. After having agreed
on a publicly known permutation of bits, Alice and Bob take their shuf¬‚ed strings
and partition it into blocks of size k, such that a single block is believed to con-
tain no more than one error with high probability. In [2], a theoretical treatment
of the optimal choice of block size is missing, and this is precisely the gap we
intend to close here. The problem of how errors are scattered across the raw key
has been tackled on statistical grounds before [17]. For this analysis, a binomial
distribution of errors within the blocks is assumed, which is later approximated by
a Poisson distribution. We shall take a different route here, considering the process
that induces the errors to be Poissonian, as well as adapting the initial block-size

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