1. Cryptography - an Overview
I can't speak without an interception.
This is private; please get off my line.
Please tell me when I can have my privacy. - Ray and Dave Davies
The purpose of cryptography is to transmit information in such a way
that access to it is restricted entirely to the intended recipient, even
if the transmission itself is received by others. This science is of
increasing importance with the advent of broadcast and network
communication, such as electronic transactions, the Internet, e-mail, and
cell phones, where sensitive monetary, business, political, and personal
communications are transmitted over public channels.
Cryptography operates by a sender scrambling or encrypting the original message or plaintext in a systematic way that obscures its
meaning. The encrypted message or cryptotext is transmitted, and the receiver
recovers the message by unscrambling or decrypting the transmission.
Originally, the security of a cryptogram depended on the secrecy of the
entire encrypting and decrypting procedures. Today, however, we use ciphers in which the algorithm for encrypting and decrypting could
be revealed to anybody without compromising the security of a particular
message. In such ciphers a set of specific parameters, called a key, is used together with the plaintext as an
input to the encrypting algorithm, and together with the cryptotext as an
input to the decrypting algorithm. The encrypting and decrypting
algorithms are publicly announced; the security of the cryptogram depends
entirely on the secrecy of the key. To prevent this being discovered by
accident or systematic search, the key is chosen as a very large number.
Once the key is established, subsequent secure communication can take
place by sending cryptotext, even over a public channel that is vulnerable
to total passive eavesdropping, such as public announcements in mass
media. However, to establish the key, two users, who may not be in contact
or share any secret information initially, will have to discuss it, using
some other reliable and secure channel. But since interception is a set of
measurements performed by an eavesdropper on a channel, however difficult
this might be from a technological point of view, any classical key
distribution can in principle be passively monitored, without the
legitimate users realizing that any eavesdropping has taken place.
Cryptographers have tried hard to solve this key distribution problem. The 1970s
brought a clever mathematical discovery in the form of public key
cryptography (PKC) [1, 2]. The idea of PKC is for each user to randomly
choose a pair of mutually inverse transformations -- a scrambling
transformation and an unscrambling transformation -- and to publish the
directions for performing the former but not the latter. The
transformation is designed so that the unscrambling operation cannot be
deduced easily from the scrambling operation, enabling only the user to
read scrambled messages. In these systems users do not need to agree on a
secret key before they send a message. They work similarly to a drop
mailbox with two locks. The owner of the mailbox provides everybody with a
key for dropping mail into his box, but only he has the key to open it and
read the messages inside. PKC was introduced in 1976 .
PKC systems exploit the fact that certain mathematical operations are
easier to do in one direction than the other. The systems avoid the key
distribution problem, but unfortunately their security depends on unproven
mathematical assumptions about the intrinsic difficulty of certain
operations. The most popular public key cryptosystem, RSA
(Rivest-Shamin-Adleman), gets its security from the difficulty of
factoring large numbers . This means that if
ever mathematicians or computer scientists come up with fast and clever
procedures for factoring large numbers, then the whole privacy and
discretion of widespread cryptosystems could vanish overnight. Indeed,
recent work in quantum computation
suggests that in principle quantum
computers might factorize huge integers in practical times, which
could jeopardize the secrecy of many modern cryptography techniques .
But quantum technology promises to revolutionize secure communication
at an even more fundamental level. While classical cryptography relies on
the limitations of various mathematical techniques or computing technology
to restrict eavesdroppers from learning the contents of encrypted
messages, in quantum cryptography the information is protected by the laws
of physics. This Hot Topic will discuss some of the basics of how this can
2. Classical Cryptography
Gentlemen do not read each other's mail - Henry Stimson, U.S.
Secretary of State
Cryptography is the art of devising codes and ciphers, and cryptoanalysis is the art of
breaking them. Cryptology is the combination of the two. In the literature
of cryptology, information to be encrypted is known as plaintext, and the parameters of the encryption
algorithm that transforms the plaintext are collectively called a key. The keys used to encrypt most messages,
such as those used to exchange credit-card information over the Internet,
are themselves encrypted before being sent .
The schemes used to disguise keys are thought to be secure, because
discovering them would take too long for even the fastest computers.
Existing cryptographic techniques are usually identified as
"traditional" or "modern." Traditional techniques date back for centuries,
and use operations of coding (use of alternative words or phrases),
transposition (reordering of plaintext), and substitution (alteration of
plaintext characters). Traditional techniques were designed to be simple,
for hand encoding and decoding. By contrast, modern techniques use
computers, and rely on extremely long keys, convoluted algorithms, and
intractable problems to achieve assurances of security.
There are two branches of modern cryptographic techniques: public key
encryption and secret key encryption. In PKC, as mentioned above, messages
are exchanged using an encryption method so convoluted that even full
disclosure of the scrambling operation provides no useful information for
how it can be undone. Each participant has a "public key" and a "private
key"; the former is used by others to encrypt messages, and the latter is
used by the participant to decrypt them.
The widely used RSA algorithm is one example of PKC. Anyone wanting to
receive a message publishes a key, which contains two numbers. A sender
converts a message into a series of digits, and performs a simple
mathematical calculation on the series using the publicly available
numbers. Messages are deciphered by the recipient by performing another
operation, known only to him . In principle,
an eavesdropper could deduce the decryption method by factoring one of the
published numbers, but this is chosen to typically exceed 100 digits and
to be the product of only two large prime
numbers, so that there is no known way to accomplish this
factorization in a practical time.
In secret key encryption, a k-bit "secret key" is shared by two
users, who use it to transform plaintext inputs to cryptotext for
transmission and back to plaintext upon receipt. To make unauthorized
decipherment more difficult, the transformation algorithm can be carefully
designed to make each bit of output depend on every bit of the input. With
such an arrangement, a key of 128 bits used for encoding results in a
choice of about 1038 numbers. The encrypted message should be
secure; assuming that brute force and massive parallelism are employed, a
billion computers doing a billion operations per second would require a
trillion years to decrypt it. In practice, analysis of the encryption
algorithm might make it more vulnerable, but increases in the size of the
key can be used to offset this.
The main practical problem with secret key encryption is exchanging a
secret key. In principle any two users who wished to communicate could
first meet to agree on a key in advance, but in practice this could be
inconvenient. Other methods for establishing a key, such as the use of
secure courier or private knowledge, could be impractical for routine
communication between many users. But any discussion of how the key is to
be chosen that takes place on a public communication channel could in
principle be intercepted and used by an eavesdropper.
One proposed method for solving this key distribution problem is the
appointment of a central key distribution server. Every potential
communicating party registers with the server and establishes a secret
key. The server then relays secure communications between users, but the
server itself is vulnerable to attack. Another method is a protocol for
agreeing on a secret key based on publicly exchanged large prime numbers,
as in the Diffie Hellman key exchange. Its security is based on the
assumed difficulty of finding the power of a base that will generate a
specified remainder when divided by a very large prime number, but this
suffers from the uncertainty that such problems will remain intractable.
Quantum encryption, which will be discussed later, provides a way of
agreeing on a secret key without making this assumption.
Communication at the quantum level changes many of the conventions of
both classical secret key and public key communication described above.
For example, it is not necessarily possible for messages to be perfectly
copied by anyone with access to them, nor for messages to be relayed
without changing them in some respect, nor for an eavesdropper to
passively monitor communications without being detected . To understand these ideas, we must first
discuss some underlying physics.
3. Quantum Cryptography Fundamentals
Nobody understands quantum theory. - Richard Feynman, Nobel
Electromagnetic waves such as light waves can exhibit the phenomenon
of polarization, in which the direction of the electric field vibrations
is constant or varies in some definite way. A polarization filter is a
material that allows only light of a specified polarization direction to
pass. If the light is randomly polarized, only half of it will pass a
According to quantum theory, light
waves are propagated as discrete particles known as photons. A photon is a
massless particle, the quantum of the
electromagnetic field, carrying energy, momentum, and angular momentum. The polarization of the
light is carried by the direction of the angular momentum or spin of the
photons. A photon either will or will not pass through a polarization
filter, but if it emerges it will be aligned with the filter regardless of
its inital state; there are no partial photons. Information about the
photon's polarization can be determined by using a photon detector to
determine whether it passed through a filter.
"Entangled pairs" are pairs of photons generated by certain particle
reactions. Each pair contains two photons of different but related
polarization. Entanglement affects the randomness of measurements. If we
measure a beam of photons E1 with a polarization filter, one-half of the
incident photons will pass the filter, regardless of its orientation.
Whether a particular photon will pass the filter is random. However, if we
measure a beam of photons E2 consisting of entangled companions of the E1
beam with a filter oriented at 90 degrees (deg) to the first filter, then
if an E1 photon passes its filter, its E2 companion will also pass its
filter. Similarly, if an E1 photon does not pass its filter then its E2
companion will not.
The foundation of quantum cryptography lies in the Heisenberg
uncertainty principle, which states that certain pairs of physical
properties are related in such a way that measuring one property prevents
the observer from simultaneously knowing the value of the other. In
particular, when measuring the polarization of a photon, the choice of
what direction to measure affects all subsequent measurements. For
instance, if one measures the polarization of a photon by noting that it
passes through a vertically oriented filter, the photon emerges as
vertically polarized regardless of its initial direction of polarization.
If one places a second filter oriented at some angle q to the vertical, there is a
certain probability that the photon will pass through the second filter as
well, and this probability depends on the angle q. As q increases, the probability of the
photon passing through the second filter decreases until it reaches 0 at
q = 90 deg (i.e., the
second filter is horizontal). When q = 45 deg, the chance of the photon
passing through the second filter is precisely 1/2. This is the same
result as a stream of randomly polarized photons impinging on the second
filter, so the first filter is said to randomize the measurements of the
Polarization by a filter: Unpolarized light enters a
vertically aligned filter, which
absorbs some of the light and polarizes the remainder in the vertical
direction. A second filter tilted at some angle q absorbs some of the
polarized light and transmits the rest, giving it a new polarization.|
(From "Quantum Cryptography" by Charles H. Bennett, Gilles Brassard,
and Artur K. Ekert,
A pair of orthogonal (perpendicular) polarization states
used to describe the polarization of photons, such as horizontal/vertical,
is referred to as a basis. A pair of bases
are said to be conjugate bases if the
measurement of the polarization in the first basis completely randomizes
the measurement in the second basis , as in
the above example with q =
45 deg. It is a fundamental consequence of the Heisenberg uncerty
principle that such conjugate pairs of states must exist for a quantum
If a sender, typically designated Alice in the literature,
uses a filter in the 0-deg/90-deg basis to give the photon an initial
polarization (either horizontal or vertical, but she doesn't reveal
which), a receiver Bob can determine this by using a filter aligned to the
same basis. However if Bob uses a filter in the 45-deg/135-deg basis to
measure the photon, he cannot determine any information about the initial
polarization of the photon .
These characteristics provide the principles behind quantum
cryptography. If an eavesdropper Eve uses a filter aligned with Alice's
filter, she can recover the original polarization of the photon. But if
she uses a misaligned filter she will not only receive no information, but
will have influenced the original photon so that she will be unable to
reliably retransmit one with the original polarization. Bob will either
receive no message or a garbled one, and in either case will be able to
deduce Eve's presence.
4. Quantum Cryptography Application
And I would send a message
To find out if she's talked,
But the post office has been stolen,
And the mailbox is locked. - Bob Dylan
Sending a message using photons is straightforward in principle, since
one of their quantum properties, namely polarization, can be used to
represent a 0 or a 1. Each photon therefore carries one bit of quantum
information, which physicists call a qubit.
To receive such a qubit, the recipient must determine the photon's
polarization, for example by passing it through a filter, a measurement
that inevitably alters the photon's properties. This is bad news for
eavesdroppers, since the sender and receiver can easily spot the
alterations these measurements cause. Cryptographers cannot exploit this
idea to send private messages, but they can determine whether its security
was compromised in retrospect.
The genius of quantum cryptography is that it solves the problem of
key distribution. A user can suggest a key by sending a series of photons
with random polarizations. This sequence can then be used to generate a
sequence of numbers. The process is known as quantum key distribution. If
the key is intercepted by an eavesdropper, this can be detected and it is
of no consequence, since it is only a set of random bits and can be
discarded. The sender can then transmit another key. Once a key has been
securely received, it can be used to encrypt a message that can be
transmitted by conventional means: telephone, e-mail, or regular postal
The first published paper to describe a cryptographic protocol using
these ideas to solve the key distribution problem was written in 1984 by
Charles Bennett and Gilles Brassard . In
it, Bennett and Brassard described an unconditionally secure quantum key
distribution system. The system is called the BB84 system (after Bennett
and Brassard, 1984), and its operation is as follows .
Alice and Bob are equipped with two polarizers each, one aligned with
the rectilinear 0-deg/90-deg (or +) basis that will emit - or | polarized
photons and one aligned with the diagonal 45-deg/135-deg (or X) basis that
will emit \ or / polarized photons. Alice and Bob can communicate via a
quantum channel over which Alice can send photons, and a public channel
over which they can discuss results. An eavesdropper Eve is assumed to
have unlimited computing power and access to both these channels, though
she cannot alter messages on the public channel (see below for discussion
Alice begins to send photons to Bob, each one polarized at random in
one of the four directions: 0, 45, 90, or 135 deg. As Bob receives each
photon, he measures it with one of his polarizers chosen at random. Since
he does not know which direction Alice chose for her polarizer, his choice
may not match hers. If it does match the basis, Bob will measure the same
polarization as Alice sent, but if it doesn't match, Bob's measurement
will be completely random. For instance, if Alice sends a photon | and Bob
measures with his + polarizer oriented either - or |, he will correctly
deduce Alice sent a | photon, but if he measures with his X polarizer, he
will deduce (with equal probability) either \ or /, neither of which is
what Alice actually sent. Furthermore, his measurement will have destroyed
the original polarization.
To eliminate the false measurements from the sequence, Alice and Bob
begin a public discussion after the entire sequence of photons has been
sent. Bob tells Alice which basis he used to measure each photon, and
Alice tells him whether or not it was the correct one. Neither Alice nor
Bob announces the actual measurements, only the bases in which they were
made. They discard all data for which their polarizers didn't match,
leaving (in theory) two perfectly matching strings. They can then convert
these into bit strings by agreeing on which photon directions should be 0
and which should be 1.
This provides a way for Alice and Bob to arrive at a shared key
without publicly announcing any of the bits. If an eavesdropper Eve tries
to gain information about the key by intercepting the photons as they are
transmitted from Alice to Bob, measuring their polarization, and then
resending them so Bob does receive a message, then since Eve, like Bob,
has no idea which basis Alice uses to transmit each photon, she too must
choose bases at random for her measurements. If she chooses the correct
basis, and then sends Bob a photon matching the one she measures, all is
well. However, if she chooses the wrong basis, she will then see a photon
in one of the two directions she is measuring, and send it to Bob. If
Bob's basis matches Alice's (and thus is different from Eve's), he is
equally likely to measure either direction for the photon. However, if Eve
had not interfered, he would have been guaranteed the same measurement as
Alice. In fact, in this intercept/resend scenario, Eve will corrupt 25
percent of the bits . So if Alice and Bob
publicly compare some of the bits in their key that should have been
correctly measured and find no discrepancies, they can conclude that Eve
has learned nothing about the remaining bits, which can be used as the
secret key. Alternatively, Alice and Bob can agree publicly on a random
subset of their bits, and compare the parities. The parities will differ
in 50 percent of the cases if the bits have been intercepted. By doing 20
parity checks, Alice and Bob can reduce the probability of an eavesdropper
remaining undetected to less than one in a million . It is of course crucial that they do not
discuss the orientation of the polarization filters until after the
message has been sent, or Eve could use this to intercept and resend the
|An Illustration of Quantum Key
A quantum cryptography system allows two people, say Alice and Bob, to
exchange a secret key. Alice uses a transmitter to send photons in one of
four polarizations: 0, 45, 90 or 135 degrees. Bob uses a receiver to
measure each polarization in either the rectilinear basis (0 and 90) or
the diagonal basis (45 and 135); according to the laws of quantum
mechanics he cannot simultaneously make both measurements.
The key distribution requires several steps. Alice sends photons with one
of the four polarizations, which she chooses at random.
|For each photon, Bob chooses at random
the type of measurement: either the rectilinear type (+) or the diagonal
Bob records the result of his measurements but keeps it a secret.
After the transmission, Bob tells Alice the measurement types he used
(but not his results) and Alice tells him which were correct for the
photons she sent. This exchange may be overheard.
Alice and Bob keep all cases in which Bob should have measured the
correct polarization. These cases are then translated into bits (1s and
0s) to define the key.
As a check, Alice and Bob choose some bits at random to reveal. If they
agree, they can use the remaining bits with assurance that they have
not been intercepted. But if they find a substantial number of
discrepancies, it indicates unavoidable tampering due to eavesdropping,
and they should start over to transmit another key.|
(From "Quantum Cryptography" by Charles H. Bennett, Gilles
Brassard, and Artur K. Ekert,
The BB84 system is now one of several types of quantum cryptosystems
for key distribution. Another one involves cryptosystems with encoding
built upon quantum entanglement and Bell's
Theorem, proposed by Artur K. Ekert (1990) [12, 13]. The basic idea
of those cryptosystems is as follows. A sequence of correlated particle
pairs is generated, with one member of each pair being detected by each
party. An eavesdropper on this communication would have to detect a
particle to read the signal, and retransmit it in order for his presence
to remain unknown. However, the act of detection of one particle of a pair
destroys its quantum correlation with the other, and the two parties can
easily verify whether this has been done, without revealing the results of
their own measurements, by communication over an open channel.
5 . Quantum Privacy Attacks
A sekret ceases tew be a sekret if it iz once confided ... -
"Affurisms from Josh Billings: His Sayings," Henry Wheeler Shaw
Quantum cryptography obtains its fundamental security from the fact
that each qubit of information is carried by a single photon, and that
each photon will be altered as soon as it is read once. This foils
attempts to intercept message bits without being detected.
Quantum cryptographic techniques provide no protection against the
classic bucket brigade attack (also known as the "man-in-the-middle
attack"). In this scheme, an eavesdropper Eve is assumed to have the
capacity to monitor the communications channel and insert and remove
messages without inaccuracy or delay. When Alice attempts to establish a
secret key with Bob, Eve intercepts and responds to messages in both
directions, fooling both Alice and Bob into believing she is the other.
Once the keys are established, Eve receives, copies, and resends messages
so as to allow Alice and Bob to communicate. Assuming that processing time
and accuracy are not difficulties, Eve will be able to retrieve the entire
secret key, and thus the entire plaintext of every message sent between
Alice and Bob, without any detectable signs of eavesdropping.
Even if Eve does not practice interference of this kind, there are
other methods she can still attempt to use. Because of the difficulty of
using single photons for transmissions, most systems use small bursts of
coherent light instead. In theory, Eve
might be able to split single photons out of the burst, reducing its
intensity but not affecting its content. By observing these photons (if
necessary, holding them somehow until the correct basis for observation is
announced) she might gain information about the information transmitted
from Alice to Bob.
A confounding factor in detecting attacks is the presence of noise on
the quantum communication channel. Eavesdropping and noise are
indistinguishable to the communicating parties, and so either can cause a
secure quantum exchange to fail. This leads to two potential problems: a
malicious eavesdropper could prevent communication from occurring, and
attempts to operate in the expectation of noise might make eavesdropping
attempts more feasible.
6 . State of Quantum Cryptography Technologies
What hath God wrought. - first message sent by telegraph, by
Samuel F. B. Morse
Experimental implementations of quantum cryptography have existed
since 1990, and today quantum cryptography is performed over distances of
30-40 kilometers using optical fibers.
Essentially, two technologies make quantum key distribution possible:
the equipment for creating single photons and that for detecting them. The
ideal source is a so-called photon gun that fires a single photon on
demand. As yet, nobody has succeeded in building a practical photon gun,
but several research efforts are under way. Jungsang Kim at Stanford
University, California, and colleagues, for example, are working on a
light-emitting p-n junction that produces
well-spaced single photons on demand. Others are working with a
diamond-like material in which one carbon atom in the structure has been
replaced with nitrogen. That substitution creates a vacancy similar to a hole in a p-type
semiconductor, which emits single photons when excited by a laser.
Many groups are also working on ways of making single ions emit single
None of these technologies, however, is mature enough to be used in
current quantum cryptography experiments. As a result, physicists have to
rely on other techniques that are by no means perfect from a security
viewpoint. Most common is the practice of reducing the intensity of a
pulsed laser beam to such a level that, on average, each pulse contains
only a single photon. The problem here is the small but significant
probability that the pulse contains more than one photon. This extra
photon is advantageous for Eve, who can exploit the information it
contains without Alice and Bob being any the wiser.
Single-photon detection is tricky too. The most common method exploits
avalanche photodiodes. These devices
operate beyond the diode's breakdown
voltage, in what is called Geiger mode. At that point, the energy from
a single absorbed photon is enough to cause an
electron avalanche, an easily detectable
flood of current. But these devices are far from perfect. To detect
another photon, the current through the diode must be quenched and the
device reset, a time-consuming process.
Furthermore, silicon's best detection wavelength is 800 nanometers
(nm, where 1 nm = one one-billionth of a meter), and it is not sensitive
to wavelengths above 1100 nm, well short of the 1300- and 1550-nm
standards for telecommunication. At telecommunications wavelengths,
germanium (Ge) or indium-gallium-arsenide (InGaAs) detectors must be used,
even though they are far less efficient and must be cooled well below room
temperature. While commercial single-photon detectors at
telecommunications wavelengths are beginning to appear on the market, they
still lack the efficiencies useful for quantum cryptography .
The distance that the key can be transmitted is also an important
technical limitation. Most experts agree that a 67-km transmission
achieved by a group of physicists at the University of Geneva on October
2001 is close to the maximum that can be achieved with current technology.
Beyond about 80 km of cable, too few photons make it from Alice to Bob.
The range could be extended by devices that strengthen the signal as it
passes by, like those used to send telephone conversations over long
distances. However, unlike telephone repeaters, quantum versions would
have to bolster the signal without measuring the photons. Scientists have
shown that creating a repeater that doesn't measure is feasible in
principle, but the technology to building one is a long way off .
Satellites could provide an alternative means of achieving
long-distance transmission. A quantum cryptography team led by physicist
Richard Hughes at the Los Alamos National Laboratory in New Mexico is
developing a key-distribution system that sends single photons through
open air. So that the photons can be distinguished from all the others
bombarding the detector, the team uses various techniques to filter the
incoming light. In a recent paper , Hughes
and his colleagues have described how they sent keys over a distance of 10
km with rates similar to those achieved using optical fibers. Ten
kilometers is a long way short of the hundreds of kilometers between the
Earth's surface and satellites, but because air turbulence, the factor
that most disrupts the photons, occurs predominately in the lower 2 km of
the atmosphere, Hughes believes his system should be able to send signals
to satellites. The team is now trying to make the receiver light and
sturdy enough to fit in a satellite and survive a rocket launch. Combined
with optical fibers, satellites could eventually form part of a
long-distance transmission system.
In the shorter term, the technology might help to protect the security
of satellite television broadcasts. In one such breach, a hacker known as
Captain Midnight interrupted a 1986 broadcast by HBO (the Home Box Office
company) and sent over half of the company's customers a five-minute
broadcast of a message complaining about the firm's new subscription
7 . Conclusion
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 .
- W. Diffie and M. E. Hellman, IEEE Transactions on Information Theory, IT-22, pp. 644-654 (1977).
- Rivest R., Shamir A., and Adleman L., "On Digital Signatures and Public-Key Cryptosystems", MIT Laboratory for Computer Science, Technical Report, MIT/LCS/TR-212 (January 1979).
- P.W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (IEEE Computer Society, Los Alamitos, CA, 1994), p. 124.
(K. Lamond, "Credit Card Transactions: Real World and Online")
- E. Klarreich, Nature, vol. 418, 18 July 2002, pp. 270-272.
- C. H. Bennett, "Quantum Cryptography: Uncertainty in the Service of Privacy", Science, vol. 257, 7 August 1992, pp. 752-753.
- C. H. Bennett, "Quantum Cryptography: Uncertainty in the Service of Privacy", Science, vol. 257, 7 August 1992, pp. 752-753.
(S. Goldwater, "Quantum Cryptography and Privacy Amplification")
- S. K. Moore, IEEE Spectrum, May 2002.
- C. H. Bennett and G. Brassard, in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, IEEE, New York (1984).
- C. H. Bennett, G. Brassard, and A. K. Ekert, "Quantum Cryptography", Scientific American, October 1992, pp. 50-57.
- A. K. Ekert, Physical Review Letters, 67, 661 (1991).
- A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, Physical Review Letters, 69, 1293 (1992).
- R. J. Hughes, J. E. Nordholt, D. Derkacs, and C. G. Peterson, New Journal of Physics, 4, 43 (2002).