
Seminar Report On
QUANTUM CRYPTOGRAPHY
Submitted by
SANTHIMOL A. K.
In the partial fulfillment of requirements in degree of
Master of Technology in Computer and Information Science
DEPARTMENT OF COMPUTER SCIENCE
COCHIN UNIVERSITY OF SCIENCE AND TECHNOLOGY
KOCHI682022
2008 Page 2
ACKNOWLEDGEMENT
I thank GOD almighty for guiding me throughout the seminar. I would like to
thank all those who have contributed to the completion of the seminar and helped me
with valuable suggestions for improvement.
I am extremely grateful to Prof. Dr. K Poulose Jacob, Director, Department of
Computer Science, for providing me with best facilities and atmosphere for the creative
work guidance and encouragement. I would like to thank my coordinator,
Mr.G.Santhosh Kumar, Lecturer, Department of Computer Science, for all help and
support extend to me. I thank all staff members of my college and friends for extending
their cooperation during my seminar.
Above all I would like to thank my parents without whose blessings, I would not
have been able to accomplish my goal.
SANTHIMOL AK Page 3
ABSTRACT
Quantum cryptography uses quantum mechanics to guarantee secure
communication. It enables two parties to produce a shared random bit string known only
to them, which can be used as a key to encrypt and decrypt messages.
An important and unique property of quantum cryptography is the ability of the
two communicating users to detect the presence of any third party trying to gain
knowledge of the key. This results from a fundamental part of quantum mechanics: the
process of measuring a quantum system in general disturbs the system. A third party
trying to eavesdrop on the key must in some way measure it, thus introducing detectable
anomalies. By using quantum superpositions or quantum entanglement and transmitting
information in quantum states, a communication system can be implemented which
detects eavesdropping. If the level of eavesdropping is below a certain threshold a key
can be produced which is guaranteed as secure, otherwise no secure key is possible and
communication is aborted.
The security of quantum cryptography relies on the foundations of quantum
mechanics, in contrast to traditional public key cryptography which relies on the
computational difficulty of certain mathematical functions, and cannot provide any
indication of eavesdropping or guarantee of key security.
Quantum cryptography is only used to produce and distribute a key, not to
transmit any message data. This key can then be used with any chosen encryption
algorithm to encrypt and decrypt a message, which can then be transmitted over a
standard communication channel. The algorithm most commonly associated with QKD is
the onetime pad, as it is provably secure when used with a secret, random key.
KEY WORDS:
qubit, uncertainty, entanglement, bit commitment, BB84 protocol, Ekert
protocol, key distribution, onetimepadPage 4
CONTENTS
1. INTRODUCTIONÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦..1
2. CLASSICAL CRYPTOGRAPHYÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦2
2.1. DEFINITIONÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦2
2.2. ONETIMEPAD Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦..Â¦Â¦3
2.3. LIMITATIONSÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.....4
3. QUBITÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.............4
3.1. QUBIT REPRESENTATIONÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.5
4. QUATUM CRYPTOGRAPHYÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦6
4.1. UNCERTAINITYÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.........Â¦7
4.2. ENTANGLEMENTÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.Â¦.........7
4.3. BB84 PROTOCOLÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.Â¦..........8
4.4. EKERT PROTOCOLÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.........10
5. QUANTUM BIT COMMITMENT PROTOCOLÂ¦Â¦Â¦.............................11
5.1. BB84 QUANTUM BIT COMMITMENT PROTOCOLÂ¦Â¦Â¦Â¦Â¦Â¦Â¦.Â¦12
5.1.1.
THE COMMIT PROCEDUREÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.Â¦Â¦..12
5.1.2.
THE UNVEIL PROCEDUREÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.Â¦Â¦13
6. OUTLOOKSÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.Â¦.13
6.1. OTHER QUANTUM KEY DISTRIBUTION PROTOCOLS..Â¦Â¦Â¦.Â¦Â¦..14
6.2. EXPERIMENTAL STATUSÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.Â¦Â¦..14
6.3. CURRENT CHALLENGESÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦.Â¦...15
7. CONCLUSION & FUTURE SCOPEÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦..Â¦17
8.
REFERENCESÂ¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦Â¦...18Page 5
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
1
1. INTRODUCTION
Cryptography is the science of keeping private information from unauthorized
access, of ensuring data integrity and authentication, and other tasks. In this survey, we
will focus on quantumcryptographic key distribution and bit commitment protocols and
we in particular will discuss their security. Before turning to quantum cryptography, let
me give a brief review of classical cryptography, its current challenges and its historical
development.
Two parties, Alice and Bob, wish to exchange messages via some insecure
channel in a way that protects their messages from eavesdropping. An algorithm, which
is called a cipher in this context, scrambles Aliceâ„¢s message via some rule such that
restoring the original message is hardâ€if not impossibleâ€without knowledge of the
secret key. This scrambled message is called the ciphertext. On the other hand, Bob
(who possesses the secret key) can easily decipher Aliceâ„¢s ciphertext and obtains her
original plaintext. Figure 1 in this section presents this basic cryptographic scenario.
Fig. 1. Communication between Alice and Bob, with Eve listening. Page 6
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
2
2. CLASSICAL CRYPTOGRAPHY
Overviews of classical cryptography can be found in various text books (see, e.g.,
Rothe [2005] and Stinson [2005]). Here, we present just the basic definition of a
cryptosystem and give one example of a classical encryption method, the onetime pad.
Definition 2.1. A (deterministic, symmetric) cryptosystem is a fivetuple (P, C, K, E, D)
satisfying the following conditions:
1. P is a finite set of possible plaintexts.
2. C is a finite set of possible ciphertexts.
3. K is a finite set of possible keys.
4. For each k ? K, there are an encryption rule e
k
? E and a corresponding decryption
rule d
k
? D, where e
k
: P? C and d
k
: C? P are functions satisfying d
k
(e
k
(x)) = x
for each plaintext element x ? P.
In the basic scenario in cryptography, we have two parties who wish to communicate
over an insecure channel, such as a phone line or a computer network. Usually, these
parties are referred to as Alice and Bob. Since the communication channel is insecure, an
eavesdropper, called Eve, may intercept the messages that are sent over this channel. By
agreeing on a secret key k via a secure communication method, Alice and Bob can make
use of a cryptosystem to keep their information secret, even when sent over the insecure
channel. This situation is illustrated in Figure 1.
The method of encryption works as follows. For her secret message m, Alice uses
the key k and the encryption rule e
k
to obtain the ciphertext c = e
k
(m). She sends Bob the
ciphertext c over the insecure channel. Knowing the key k, Bob can easily decrypt the
ciphertext by the decryption rule d
k
:
d
k
© = d
k
(e
k
(m)) = m.
Knowing the ciphertext c but missing the key k, there is no easy way for Eve to determine
the original message m.Page 7
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
3
There exist many cryptosystems in modern cryptography to transmit secret
messages. An early wellknown system is the onetime pad, which is also known as the
Vernam cipher. The onetime pad is a substitution cipher. Despite its advantageous
properties, which we will discuss later on, the onetime padâ„¢s drawback is the costly
effort needed to transmit and store the secret keys.
Fig.2. Letters and punctuation marks encoded by numbers from 0 to 29.
Fig. 3. Encryption and decryption example for the onetime pad.
Example 2.2 (OneTime Pad). For plaintext elements in P , we use capital letters and
some punctuation marks, which we encode as numbers ranging from 0 to 29, see Figure2.
As is the case with most cryptosystems, the ciphertext space equals the plaintext space.
Furthermore, the key space K also equals P , and we have P =C= K={0, 1, . . . , 29}.
Next, we describe how Alice and Bob use the onetime pad to transmit their
messages. A concrete example is shown in Figure 3. Suppose Alice and Bob share a joint
secret key k of length n = 12, where each key symbol k
i
? {0, 1, . . . , 29} is chosen
uniformly at random. Let m = m
1
m
2
. . . m
n
be a given message of length n, which Alice
wishes to encrypt. For each plaintext letter m
i
, where 1 = i = n, Alice adds the plaintext
numbers to the key numbers. The result is taken modulo 30. For example, the last letter
of the plaintext from Figure 3, D, is encoded by m
12
=03. The corresponding key is
m
12
= 28, so we have c
12
= 3 + 28 = 31. Since 31 = 1 mod 30, our plaintext letter D is
encrypted as B. Decryption works similarly by subtracting, character by character, the
key letters from the corresponding ciphertext letters. So the encryption and decryption
can be written as respectively c
i
= (m
i
+ k
i
) mod 30 and m
i
=(c
i
 k
i
) mod 30, 1 = i = n. Page 8
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
4
2.3. Limitations
Cryptographic technology in use today relies on the hardness of certain
mathematical problems. Classical cryptography faces the following two problems. First,
the security of many classical cryptosystems is based on the hardness of problems such as
integer factoring or the discrete logarithm problem. But since these problems typically
are not provably hard, the corresponding cryptosystems are potentially insecure. For
example, the famous and widely used RSA publickey cryptosystem [Rivest et al. 1978]
could easily be broken if large integers were easy to factor. The hardness of integer
factoring, however, is not a proven fact but rather a hypothesis.1.We mention in passing
that computing the RSA secret key from the corresponding public key is polynomialtime
equivalent to integer factoring [May 2004].
Second, the theory of quantum computation has yielded new methods to tackle
these mathematical problems in a much more efficient way. Although there are still
numerous challenges to overcome before a working quantum computer of sufficient
power can be built, in theory many classical ciphers (in particular publickey
cryptosystems such as RSA) might be broken by such a powerful machine. However,
while quantum computation seems to be a severe challenge to classical cryptography in a
possibly not so distant future, at the same time it offers new possibilities to build
encryption methods that are safe even against attacks performed by means of a quantum
computer. Quantum cryptography extends the power of classical cryptography by
protecting the secrecy of messages using the physical laws of quantum mechanics.
3. QUBITS
The most important unit of information in computer science is the bit. There are
two possible values that can be stored by a bit: the bit is either equal to 0 or equal to
1. These two different states can be represented in various ways, for example by a
simple switch or by a capacitor: if not charged, the capacitor holds the value zero; if
charged, it holds the value one. Page 9
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
5
There exist many possibilities to physically represent a qubit in practice, as every
quantum system with at least two states can serve as a qubit. For example, the spin of an
atom or the polarization5 of a light particle can represent the state of a qubit. Even a cat
with its two basic states dead and alive, introduced by SchrÃƒÂ¶dinger [1935] to
visualize fundamental concepts of quantum mechanics, might serve as a representation.
The catâ„¢s problemâ€or fortune from the animalâ„¢s point of viewâ€when being used as a
quantum system is its sheer size compared to that of an atom or light particle. There is no
way to protect such a big quantum instance from interaction with its environment, which
in turn will result in decoherence of the superposition of the cat.
3.1. Qubit Representation
In general, a quantum state ?) is an element of a finitedimensional complex
vector space (or Hilbert space) H. We denote the scalar product of two states ?) and f)
by (?f), where (? = ?)
T
is the conjugate transpose of ?). It is convenient to deal with
normalized states, so we require (??) = 1 for all states ?) that have a physical meaning.
The quantum analog of the bit is called qubit, which is derived from quantum bit.
A qubit ?) is an element of a twodimensional Hilbert space, in which we can introduce
an orthonormal basis, consisting of the two states 0) and 1). Unlike its classical
counterpart, the quantum state can be in any coherent superposition of the basis states:
?) = a0) + ÃƒÅ¸1),
(1)
where a and ÃƒÅ¸ are, in general, complex coefficients. This is due to the fact that the
quantum mechanical equation of motion, the SchrÃƒÂ¶dinger equation, is linear: Any linear
superposition of its solutions (the quantum states) is also a solution. Since we require
quantum states to be normalized, we find that the coefficients in (1) have to fulfill
a
2
+ ÃƒÅ¸
2
= 1, where  Ã‚Â·  denotes the absolute value. Page 10
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
6
4. QUANTUM CRYPTOGRAPHY
Quantum Cryptography, or Quantum Key Distribution (QKD), uses quantum
mechanics to guarantee secure communication. It enables two parties to produce a shared
random bit string known only to them, which can be used as a key to encrypt and decrypt
messages. An important and unique property of quantum cryptography is the ability of
the two communicating users to detect the presence of any third party trying to gain
knowledge of the key. This results from a fundamental part of quantum mechanics: the
process of measuring a quantum system in general disturbs the system. A third party
trying to eavesdrop on the key must in some way measure it, thus introducing detectable
anomalies. By using quantum superpositions or quantum entanglement and transmitting
information in quantum states, a communication system can be implemented which
detects eavesdropping. If the level of eavesdropping is below a certain threshold a key
can be produced which is guaranteed as secure (i.e. the eavesdropper has no information
about), otherwise no secure key is possible and communication is aborted.
The security of quantum cryptography relies on the foundations of quantum
mechanics, in contrast to traditional public key cryptography which relies on the
computational difficulty of certain mathematical functions, and cannot provide any
indication of eavesdropping or guarantee of key security.
Quantum cryptography is only used to produce and distribute a key, not to
transmit any message data. This key can then be used with any chosen encryption
algorithm to encrypt (and decrypt) a message, which can then be transmitted over a
standard communication channel. The algorithm most commonly associated with QKD is
the onetime pad, as it is provably secure when used with a secret, random key.
Quantum cryptography exploits the quantum mechanical property that a qubit
cannot be copied or amplified without disturbing its original state. This is the statement
of the NoCloning Theorem [Wootters and Zurek 1982], which is easily proven: Assume
there exists a unitary transformation U that can copy two states ?
1
) and ?
2
): Page 11
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
7
where 0) is an arbitrary input state. If we equate the scalar products of the lefthand and
righthand sides, it follows by the unitarity of U that (?
1
?
2
) = (?
1
?
2
)
2
, which implies that
(?
1
?
2
) equals 0 or 1. This means that we can copy only orthogonal or identical states. In
contrast, arbitrary unknown states cannot be perfectly cloned. (Note that orthogonal or
identical states are not viewed as unknown states, since we do know they are
orthogonal, for example.)
The essence of this theorem is the main ingredient of quantum key distribution,
where Alice and Bob use a quantum channel to exchange a sequence of qubits, which
will then be used to create a key for the onetime pad in order to communicate over an
insecure channel. Any disturbance of the qubits, for example caused by Eve trying to
measure the qubitsâ„¢ state, can be detected with high probability.
Quantum cryptographic devices typically employ individual photons of light and
take advantage of either the Heisenberg Uncertainity principle or Quantum
Entanglement.
4.1. Uncertainity
Unlike in classical physics, the act of measurement is an integral part of quantum
mechanics. So it is possible to encode information into quantum properties of a photon in
such a way that any effort to monitor them disturbs them in some detectable way. The
effect arises because in quantum theory, certain pairs of physical properties are
complementary in the sense that measuring one property necessarily disturbs the other.
This statement is known as the Heisenberg uncertainty principle. The two complementary
properties that are often used in quantum cryptography, are two types of photon's
polarization, e.g. rectilinear (vertical and horizontal) and diagonal (at 45Ã‚Â° and 135Ã‚Â°).
4.2. Entanglement
It is a state of two or more quantum particles, e.g. photons, in which many of their
physical properties are strongly correlated. The entangled particles cannot be described
by specifying the states of individual particles and they may together share information in
a form which cannot be accessed in any experiment performed on either of the particles
alone. This happens no matter how far apart the particles may be at the time. Page 12
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
8
4.3. BB84 Protocol
The BB84 protocol was proposed by Charles H.Bennett and Gilles Brassard
[1984]. This is the first protocol designed to employ quantum mechanics for two parties
to agree on a joint secret key. In this protocol, Alice and Bob use a quantum channel to
send qubits. They are also connected by a classical channel, which is insecure against an
eavesdropper but unjammable. Alice and Bob use four possible quantum states in two
conjugate bases (say, the rectilinear basis+and the diagonal basisÃƒâ€”).We use 0)
+
and 0)
Ãƒâ€”
=
(0)
+
+1)
+
)/v2 for the classical signal 0, and we use 1)
+
and 1)
Ãƒâ€”
= (0)
+
 1)
+
)/v2 for
the classical signal 1. Note that the two bases are connected by the socalled Hadamard
transformation
in the following way: We have H0)
+
= 0)
Ãƒâ€”
and H1)
+
= 1)
Ãƒâ€”
, and viceversa, since
H
2
= 1.
The protocol works as follows (see also Table I for illustration):
(1) Alice randomly prepares 2n qubits, each in one of the four states 0)
+
, 0)
Ãƒâ€”
, 1)
+
, or1)
Ãƒâ€”
,
and sends them to Bob.
(2) For each qubit that Bob receives, he chooses at random one of the two bases (+ or Ãƒâ€”)
and measures the qubit with respect to that basis. In the case of a perfectly noiseless
channel, if Bob chooses the same basis as Alice, his measurement result is the same
as the classical bit that Alice prepared. If the bases differ, Bobâ„¢s result is completely
random. Page 13
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
9
(3) Alice tells Bob via the classical channel which basis she used for each qubit. They
keep the bits where Bob has used the same basis for his measurement as Alice. This
happens in about half the cases, so they will have approximately n bits left. These are
forming the socalled sifted key.
(4) Alice and Bob choose a subset of the sifted key to estimate the errorrate. They do so
by announcing publicly the bit values of the subset. If they differ in too many cases,
they abort the protocol, since its security cannot be guaranteed.
(5) Finally, Alice and Bob obtain a joint secret key from the remaining bits by performing
error correction and privacy amplification.
Eveâ„¢s goal is to learn at least some part of the key. Thus, an obvious strategy for her
is to intercept the qubits being transmitted from Alice to Bob. She cannot simply copy the
qubits, since this would contradict the NoCloning Theorem. In order to extract some
information, she is forced to measure (and thus destroy) them. But since she does not
know the basis in which they were prepared (Alice announces this information only after
Bob received all signals), she can only guess or just flip a coin for the selection of the
measurement basis. In about half the cases, she will happen to choose the same basis as
Alice and get completely correlated bit values. In the other half, her results will be
random and uncorrelated. Bob certainly expects to receive something from Alice, so Eve
needs to send some qubits to him. However, she still has no idea which basis Alice used,
so she prepares each qubit in the same basis as she measured it (or she chooses a basis at
random). These newly created qubits again match Aliceâ„¢s bases in only half of the cases.
After Bob receives Eveâ„¢s qubits, he measures them, and Alice and Bob apply the sifting.
Because of Eveâ„¢s disturbance, about half of Bobâ„¢s key was measured in a different basis
than it was prepared by Alice. Since Bobâ„¢s result is random in those cases, his sifted key
will contain about 25% errors. In the errorestimation stage, if Alice and Bob obtain such
a high error rate, it would be wise for them to abort the protocol.
If the error rate is below an agreed threshold value, Alice and Bob can eliminate
errors with (classical) error correction. A simple method for error correction works as
follows: Alice chooses two bits at random and tells Bob the XORvalue of the two bits.
Bob tells Alice if he has the same value. In this case, they keep the first bit and discard Page 14
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
10
the second bit. If their values differ, they discard both bits. The remaining bits form the
key.
The last stage of the protocol is privacy amplification [Maurer 1993; Bennett et al.
1995]â€a procedure in which Alice and Bob eliminate (or, at least, drastically reduce)
Eveâ„¢s knowledge about the key. They do so by choosing random pairs of bits of the sifted
key and replacing them by their corresponding XORvalues. Thus, they halve the length
of the key, in order to amplify their privacy. Note that Eve has less knowledge about
the XORvalue, even if she knew the values of the single bits with high probability (but
not with certainty).
Note that these simple methods for error correction and privacy amplification do not
always work. For the general case, there exist more sophisticated strategies.
4.4. Ekert Protocol
In 1991 Artur Ekert proposed a new QKD protocol whose security relies on the
entangled pairs of photons. These can be created by Alice, by Bob, or by some source
separate from both of them, including eavesdropper Eve. The photons are distributed so
that Alice and Bob each end up with one photon from each pair.
The scheme relies on two properties of entanglement. First, the entangled states are
perfectly correlated in the sense that if Alice and Bob both measure whether their
particles have vertical or horizontal polarizations, they will always get the same answer
with 100% probability. The same is true if they both measure any other pair of
complementary (orthogonal) polarizations. However, the particular results are completely
random; it is impossible for Alice to predict if she (and thus Bob) will get vertical
polarization or horizontal polarization.
Second, any attempt at eavesdropping by Eve will destroy these correlations in a way
that Alice and Bob can detect. Page 15
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
11
5. QUANTUM BIT COMMITMENT PROTOCOL
When talking about quantum cryptography, everyone is thinking about key
distribution. There are, however, other cryptographic applications as well, such as bit
commitment. A bit commitment protocol based on quantum mechanics was introduced
by Brassard et al. [1993]. The unconditional security of the protocol (which means that
the security of the protocol is independent of the computational resources, such as
computing time, amount of memory used, and computer technology of the cheater) has
been accepted without proof [Yao 1995]. Two years after it had been proposed, the
protocol turned out to be insecure [Mayers 1995].
A commitment protocol is a procedure in which one party, say Alice, deposits a
message such that no one (and in particular not Alice) can read it nor change it. At some
point in the future, Alice can announce her message, and with high certainty it can be
proven that the revealed message is the same as the one Alice had deposited originally.
To illustrate this situation, suppose Bob wants to auction off a diamond ring, subject to
the condition that each person wishing to participate in the auction can bid only one
single amount of money. After each person has chosen a specific amount, the highest
bidder gets the ring. So everyone writes their own bid on a piece of paper, puts it into a
personal safe, which is then locked and given to Bob. Until all bids have been submitted
to Bob, each bidder keeps the key matching the lock of his or her safe. In this way Bob
cannot see any of the bids, which in turn cannot be changed once they have been
submitted, since only Bob has access to the committed safes. All keys are handed over to
Bob after he has received all safes from the people participating in the auction. The
different offers are compared in public, so that everybody can be sure that only the
highest bidder walks away with the diamond and an empty wallet.
We can describe this commitment protocol mathematically as follows: The
protocol has two stages, the commit phase and the unveil phase. Alice commits herself to
the data m by computing c = f (m), and she sends c to Bob. Alice unveils the commitment
by showing Bob the preimage m of c. In classical cryptography, and in particular in
publickey cryptography, oneway functions are used for commitment. In quantum
cryptography, we want to make use of the laws of quantum mechanics to create a fair Page 16
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
12
protocol for both sides. Bit commitment is a special case of a commitment protocol,
where the data m consists of only one single bit.
It is widely believed that it is impossible to create a perfectly secure classical bit
commitment protocol. Regarding the extension to the quantum world, it was shown that
unconditionally secure quantum bit commitment is also impossible [Mayers 1997; Lo and
Chau 1997]. However, when relaxing the security constraints, quantum bit commitment
becomes possible in slightly modified frameworks. One example is Kentâ„¢s quantum bit
commitment protocol, which is based on special relativity theory [Kent 1999]. Another
example is due to Damgard et al. [2005] who proposed a quantum bit commitment
protocol that is secure in the bounded storage model.
5.1. BB84 Quantum bit commitment protocol
A quantum bit commitment protocol can be created from the BB84 quantum key
distribution protocol with a few minor changes in the BB84 protocol[Bennett and
Brassard 1984]. Just as in the classical bit commitment protocol, the quantum protocol
starts with the commit phase and ends with the unveil phase.
5.1.1. The commit procedure:
(1) Alice chooses a bit b ? {0, 1}.
(2) Alice creates a random binary string w = w
1
Ã‚Â· Ã‚Â· Ã‚Â·w
n
with n bits.
(3) If Alice wants to commit to 0, she does a quantum encoding of each bit w
i
in the two
basis states of the rectilinear basis +. If she wants to commit to 1, she encodes the bits
in the two basis states of the diagonal basis Ãƒâ€”. Let ?
i
denote the basis chosen for w
i
.
(4) Alice sends the sequence of n encoded quantum states to Bob.
(5) Bob chooses a random measurement basis (rectilinear or diagonal) for each of the
received quantum states, i.e., he chooses a string of random bases =
1
Ã‚Â· Ã‚Â·
n
? {+,
Ãƒâ€”}
n
. He measures the i
th
state in the basis
i
, and denotes the outcome by .
If we take a look at the two density matrices for the n states corresponding to b = 0
and b = 1, respectively, it is easy to see that they are the same, and equal to the identity
matrix. Thus, Bob has no chance to get any information about the bit b. Page 17
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
13
5.1.2. The unveil procedure:
(1) Alice publishes b (i.e., the basis that she used for encoding) and the string w.
(2) For about half of the n states, Bob used the same basis for his measurement as Alice
used for encoding. In these cases Bob can verify that Aliceâ„¢s revealed bits are
matching his measurement results.
How could a dishonest party cheat in this protocol? For example, Alice could choose
the bit b = 1 for the commit phase, so she encodes the states with the diagonal basis Ãƒâ€”.
Later during the unveil phase, she changes her mind and tells Bob that she committed to
the bit b = 0, so Bob assumes that Alice has used the rectilinear basis +. In approximately
n/2 cases, Bob measures the states with the rectilinear basis +, and in these cases Alice
has to guess the bits Bob measured. Since Aliceâ„¢s success to make a right guess for one
bit is 1/2, her overall cheating will not be detected with a probability of (1/2)
n/2
. Once n is
chosen large enough, Alice has practically no chance to manipulate the protocol by this
probabilistic method.
But what if Alice uses specially entangled states? Alice could create n pairs of
entangled states and send one part of each pair to Bob. She doesnâ„¢t have to commit to a
bit in the beginning, because she can perform a measurement right before the unveil
phase. If, for example, she chooses bit b = 0, she measures the states that she has kept in
the rectilinear basis +. Bobâ„¢s measurement results will be perfectly correlated, due to the
shape of the entangled state. If Alice wants to choose bit b = 1 instead, she measures the
states that she has kept in the diagonal basis Ãƒâ€”. The state is forminvariant under a basis
rotation by 45
?
, Aliceâ„¢s announced encoded states will again match Bobâ„¢s measurement
results. Thus, Bob has no chance to notice the attack.
6. OUTLOOKS
The security of quantum key distribution relies on the inviolable laws of quantum
mechanics: nonorthogonal quantum states are used as signal states in the BB84 protocol.
The impossibility of perfect cloning of nonorthogonal states implies the security of this
protocol. Page 18
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
14
In the security proof for the BB84 protocol, we have employed an equivalent
entanglementbased protocol. The main idea is that local measurements on a maximally
entangled state, shared by Alice and Bob, have perfectly correlated outcomes that can be
used as the key. A maximally entangled state is necessarily pure, and a pure state cannot
be entangled with an eavesdropperâ„¢s stateâ€thus Eve cannot learn anything about the key.
The idea for quantum cryptography with entangled states goes back to Ekert [1991], who
suggested to confirm the existence of quantum correlations in the state of Alice and Bob
by a Bell inequality test.
6.1. Other quantum key distribution protocols
A variety of quantum key distribution protocols can be found in the literature. All
known prepareandmeasure schemes can be seen as variations of the BB84 protocol,
which are obtained by changing the number and/or dimension of the quantum states.
Bennett [1992] proposed a protocolâ€which now is named after him the B92 protocolâ€
in which only two nonorthogonal states are used. In the socalled sixstate protocol [BruÃƒÅ¸
1998; BechmannPasquinucci and Gisin 1999], the six eigenstates of the three Pauli
operators are used. In this protocol, it is more difficult for Eve to retrieve any
information, thus the security is enhanced.
A recently suggested protocol [Scarani et al. 2004] introduces a new sifting
method: rather than announcing the basis, Alice gives Bob a list of two nonorthogonal
states from which the signal state was taken. This protocol has certain security
advantages that are connected with experimental implementations of quantum
cryptography.
6.2. Experimental status
In recent years, much effort has been devoted to experiments on quantum
cryptography, and much progress has been made. In most experiments, polarized photons
are representing the qubits: photons are polarized if their electromagnetic field oscillates
in a fixed direction of space.
Polarizationbased encoding works best for freespace communication systems
rather than fiberoptic lines. Data are transmitted faster in freespace systems, but theyPage 19
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
15
cannot traverse the longer distances of fiberoptic links. In March 2004, NEC scientists in
Japan sent a single photon over a 150km fiberoptic link, breaking the transmission
distance record for quantum cryptography.
The most commercially viable QKD systems rely on fiberoptic links limited to
100 to 120 km. At longer distances, random noise degrades the photon stream. Quantum
keys cannot travel far over fiber optic lines, and, thus, they can work only between
computers directly connected to each other.
As of March 2007 the longest distance over which quantum key distribution has
been demonstrated using optic fiber is 148.7 km, achieved by Los Alamos/NIST using
the BB84 protocol. Significantly, this distance is long enough for almost all the spans
found in today's fiber networks. The distance record for free space QKD is 144km
between two of the Canary Islands, achieved by a European collaboration using
entangled photons (the Ekert scheme) in 2006, and using BB84 enhanced with decoy
states[8] in 2007. The experiments suggest transmission to satellites is possible, due to
the lower atmospheric density at higher altitudes. For example although the minimum
distance from the International Space Station to the ESA Space Debris Telescope is about
400 km, the atmospheric thickness is about an order of magnitude less than in the
European experiment, thus yielding less attenuation compared to this experiment.
6.3. Current Challenges
Like everything in the world of information security, quantum cryptography is not
panacea. The main drawbacks of quantum cryptography are due to the following two
reasons:
Man in the middle attack
Quantum cryptography is vulnerable to a maninthemiddle attack when used
without authentication to the same extent as any classical protocol, since no principle of
quantum mechanics can distinguish friend from foe. As in the classical case, Alice and
Bob cannot authenticate each other and establish a secure connection without some
means of verifying each other's identities (such as an initial shared secret). If Alice andPage 20
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
16
Bob have an initial shared secret then they can use an unconditionally secure
authentication scheme (such as CarterWegman,) along with quantum key distribution to
exponentially expand this key, using a small amount of the new key to authenticate the
next session. Several methods to create this initial shared secret have been proposed, for
example using a 3rd party or chaos theory.
Photon number splitting attack
In the BB84 protocol Alice sends quantum states to Bob using single photons. In
practice many implementations use laser pulses attenuated to a very low level to send the
quantum states. These laser pulses contain a very small amount of photons, for example
0.2 photons per pulse, which are distributed according to a Poissonian distribution. This
means most pulses actually contain no photons (no pulse is sent), some pulses contain 1
photon (which is desired) and a few pulses contain 2 or more photons. If the pulse
contains more than one photon, then Eve can split of the extra photons and transmit the
remaining single photon to Bob. This is the basis of the photon number splitting attack,
where Eve stores these extra photons in a quantum memory until Bob detects the
remaining single photon and Alice reveals the encoding basis. Eve can then measure her
photons in the correct basis and obtain information on the key without introducing
detectable errors.
There are several solutions to this problem. The most obvious is to use a true
single photon source instead of an attenuated laser. While such sources are still at a
developmental stage QKD has been carried out successfully with them. However as
current sources operate at a low efficiency and frequency key rates and transmission
distances are limited. Another solution is to modify the BB84 protocol, as is done for
example in the SARG04 protocol, in which the secure key rate scales as t3 / 2. The most
promising solution is the decoy state idea, in which Alice randomly sends some of her
laser pulses with a lower average photon number. These decoy states can be used to
detect a PNS attack, as Eve has no way to tell which pulses are signal and which decoy.
Using this idea the secure key rate scales as t, the same as for a single photon source. This
idea has been implemented successfully in several QKD experiments, allowing for high
key rates secure against all known attacks.Page 21
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
17
7. CONCLUSION AND FUTURE SCOPE
Quantum cryptography promises to revolutionize secure communication by
providing security based on the fundamental laws of physics, instead of the current state
of mathematical algorithms or computing technology. The devices for implementing such
methods exist and the performance of demonstration systems is being continuously
improved. Within the next few years, if not months, such systems could start encrypting
some of the most valuable secrets of government and industry.
Future developments will focus on faster photon detectors, a major factor limiting
the development of practical systems for widespread commercial use. Chip Elliott, BBN's
principal engineer, says the company is working with the University of Rochester and
NIST's Boulder Laboratories in Colorado to develop practical superconducting photon
detectors based on niobium nitride, which would operate at 4 K and 10 GHz.
The ultimate goal is to make QKD more reliable, integrate it with today's
telecommunications infrastructure, and increase the transmission distance and rate of key
generation. Thus the Longterm goals of quantum key distribution are the realistic
implementation via fibers, for example, for different buildings of a bank or company ,
and free space key exchange via satellites. Quantum cryptography already provides the
most advanced technology of quantum information science, and is on the way to achieve
the (quantum) jump from university laboratories to the real world. Page 22
QUANTUM CRYPTOGRAPHY
DEPARTMENT OF COMPUTER SCIENCE, CUSAT
18
8. REFERENCES
1. Quantum Cryptography: A Survey
DAGMAR BRUSS, GÃƒÂBOR ERDÃƒâ€° LYI, TIM MEYER, TOBIAS RIEGE, and JÃƒâ€“ RG ROTHE
ACM Computing Surveys, Vol. 39, No.2, Article 6, Publication date: June 2007
2. http://en.wikipedia.org/wiki/Quantum_cryptography
3. http://www.aip.org/tip/INPHFA/vol10/iss6/p22.html
4. http://www.perimeterinstitute.ca/persona...n/QKD.html
5. http://www.cs.brandeis.edu/pablo/qbc/node4.htm 
