ii. quantum key distribution

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VOLUME 1, NUMBER 1 PHYSICAL REVIEW LETTERS 21 NOVEMBER 2003


1

Quantum Cryptography


Jose Miranda

Department of Physics, University of Ottawa, Ottawa, Ontario, Canada

(Received 21 November 2003, published 28 November 2003)



An application of quantum mechanics is in the solving of the key
distribution problem with cu
rrent encryption methods. This enables the use
of a system similar to a one
-
time pad which has been proven mathematically
to be secure. A quantum key distribution system uses individual photons
whose characteristics such as polarization, have been encode
d to represent a 1
or a 0. A random key containing a large number of bits can be transmitted in
this method and through a set procedure to eliminate errors, a secure key can
be obtained which guarantees security.

PACS numbers: 95.35.+d


I. INTRODUCTION



This paper will describe in general terms
how quantum mechanics can be applied in
secure communications systems.
Although this paper is titled Quantum
Cryptography, it may be more suitable to
use the title Quantum Key Distribution for
Secure Communicat
ions. Topics
discussed will be wide ranging,
commencing with a brief discussion on the
importance of secure communications in
society and the current systems that
accomplish the security. Weaknesses with
these systems will be discussed in an
attempt to p
resent the motivation for using
quantum mechanics. A general
description of a quantum key distribution
protocol will follow and then a discussion
of the future of the technology. The
intention is to present a logical sequence
of topics leading to the app
lication of
quantum mechanics in secure
communications.


In this day and age, the requirement for
secure communications is ever increasing.
This requirement is not just for
governments and big business anymore.
The average person on the street also
has a
requirement for secure communications in
their day to day activities. Consider the
following: email, internet banking, e
-
commerce and access control in services
such as satellite television. All of these
activities rely on secure communications.
Of course, government also has a
requirement to ensure that their
communications are transmitted and
received in a secure fashion. An
indication of the importance that the US
government places on having a secure
communications capability is the National
S
ecurity Agency, NSA. The NSA is the
US cryptologic organization and is the
largest employer of mathematicians in the
world. Contrary to what is portrayed in
the movies, their sole purpose is to design
cipher systems to protect US government
information w
hile concurrently taking
advantage of the weaknesses of everyone
else’s systems.


A communications system consists of
three main blocks
--

a transmitter, a
channel and a receiver. Making it secure
requires a box in the transmitter that
encrypts the data
prior to transmission
through the channel and another box in the
receiver that decrypts the data to make it
readable. Encryption simply means using
cryptography to change the data into a
form unrecognizable from its non
-
encrypted state. Cryptography is d
efined
by the Handbook of Applied Cryptography
as “mathematical techniques related to
aspects of information security”. An
example of these mathematical techniques
is one
-
way functions that easily allow a
calculation in one direction while making
it very
difficult to calculate in the reverse
direction. For example, multiplying two
very large prime numbers of several
hundred digits are easy but attempting to
factor the product is extremely difficult.
To assist in decrypting the data after it is
received,
a piece of information is required
known as a key. In the example of
factoring very large prime numbers, the
key makes the operation very simple, but
VOLUME 1, NUMBER 1 PHYSICAL REVIEW LETTERS 21 NOVEMBER 2003


2

only if the key is known. All
cryptographic systems require a key to
allow decryption.


This system of
using mathematical
functions has yet to be proven to be 100%
secure; however, for now it is not a
problem because the security of a
cryptographic algorithm is measured in the
amount of time and computer resources
required to break the algorithm. It does
n
ot have to be completely foolproof, just
difficult enough so that the information
cannot be used in a timely fashion. The
Deputy Director of NSA was once quoted
as saying “If all the personal computers in
the world
-

approximately 260 million
computers
--

were to be put to work on a
single PGP encrypted message, it would
take on average an estimated 12 million
times the age of the universe to break a
single message.” PGP encryption stands
for Pretty Good Privacy and is based on a
current standard that has

an algorithm
relying on the difficulty of factoring two
very large prime numbers. One system
that has been mathematically proven to be
100% secure is the one
-
time pad. The
one
-
time pad uses a key that is as long as
the message and is based on an alphabe
t
that has had the letters randomly
rearranged. The alphabet is used only
once and then discarded such that an
eavesdropper may never collect enough
data to notice patterns in the encrypted
messages. The disadvantage to this
system is that the sender and

receiver must
both have all of the alphabets that can be
used to encrypt the message.


There are currently two types of
cryptographic systems in use today:
secret
-
key and public
-
key systems. In a
secret
-
key system, the same key is used for
encryption a
nd decryption, whereas in
public
-
key systems, each user has a
public
-
key and a private
-
key. Encryption
is accomplished using the receiver’s
public
-
key and the message is decrypted
using a private
-
key.


In secret
-
key cryptography, because the
same key is
used for encryption and
decryption, the problem of passing the
secret key to the receiver, known as key
distribution, becomes a problem. Public
-
key systems provide a solution through the
use of the two different keys. Each
potential receiver of a message

has a
public key that is openly available to
others. When a sender wishes to transmit
a secure message to the receiver, they use
that receiver’s public key to encrypt the
message. The intended receiver then uses
their private key which is known only to
them to decrypt the message. The
weakness in this system is that the public
-
key and private
-
key are mathematically
related; therefore, it is possible that the
private key can be determined via the
public key. Recall that the security of
current systems i
s defined by the large
amount of time and computer resources to
break into the algorithms. They may be
secure now but with the advent of
quantum computers that can perform
numerous calculations simultaneously;
they may not be so secure in the future! A
s
olution is to use quantum mechanics to
solve the key management problem by
transmitting the key using a variation of a
one
-
time pad system. The main difference
is that the receiver does not require a copy
of any of the alphabets that can be used.
As a ma
tter of fact, the receiver does not
require any advance information at all.


II. QUANTUM KEY DISTRIBUTION



As stated in the beginning, this article
should be renamed to Quantum Key
Distribution as quantum cryptography does
not deal with the encryption a
lgorithms but
rather key distribution using single photon
transmission. Its strength comes from the
fact that an eavesdropper cannot obtain any
information from qubits transmitted between
the sender and the receiver without
disturbing their state. Before

continuing on
to describe quantum key distribution,
quantum bits or qubits must be introduced.
In today’s digital age, a bit is the smallest
unit to represent data. The bit can take on
one of two states, 1 or 0. In quantum
information and computation,
the analogous
“animal” is the quantum bit or qubit for
short. As a digital bit, a qubit can be in one
of two states. The notation for the two states
that a qubit can take is as follows:





0
|

and

1
|


VOLUME 1, NUMBER 1 PHYSICAL REVIEW LETTERS 21 NOVEMBER 2003


3

One importan
t difference between a qubit
and a bit is that a qubit can be in a state that
is also a linear combination of the two states
presented above:








1
|
0
|
|







In this case, the two states

0
|

and

1
|

ar
e referred to as computational basis states
and form an orthonormal basis.
1

There is
another important difference between a qubit
and a bit. A bit can be examined to
determine which state it is in, 1 or 0;
however, this is not so with qubits.
Quantum mec
hanics only allows certain
information to be determined when
examining a qubit. Using the notation
above, examining the qubit will reveal one
of only two answers: a 0 with a probability
of


2

or a 1 with probability


2
. For
example, a qubit in this state:





1
|
2
1
0
|
2
1


when measured, will give 0 or 1 50% of
the time. Whereas a bit can be represented
by two different voltage levels or two
different alignments of particles on a
magnetic tape, a qubit can be represented by
two polarizations of a ph
oton or two states of
an electron orbiting a single atom.

Returning to quantum key distribution,
how is it accomplished? There have been
several protocols that have been developed
for quantum key distribution, one of which is
the BB84 protocol. In this p
rotocol, an
encoding scheme is used to translate a binary
1 or 0 into a quantum state. For the purposes
of describing the protocol, the polarizations
of a photon will be used to represent the
states of the qubit. This protocol requires
any two incompati
ble bases; therefore, the
linear and circular polarizations will be used:

In the linearly polarized basis,

A

the
qubit’s states are translated as:


0
|
1
|








In the circularly polarized basis,

A

th
e
qubit’s states are:


0
|
1
|







Lomonaco Jr. uses the term quantum
alphabets
2

in reference to these bases. As
this term ties into the concept of the one
-
time pad and its use of differing alphabets,
this term will be used from this point
f
orward.

There are five main steps in this protocol
which will be described using three
characters, Alice, Bob and Eve. Alice is the
originator of the message, Bob is the
intended receiver and Eve is the
eavesdropper. Note that two
communications channels
are required, a
public channel and a quantum channel.


In Step 1, Alice is required to generate a
random bit sequence, a portion of which will
eventually become the key that she will use
for encryption and Bob will use for
decryption. In Step 2 Alice wil
l randomly
choose an alphabet (

A
or

A
) to represent
the state of each bit of the number sequence.
She transmits the photons individually over
the quantum communications channel to
Bob. Step 3 is the receipt of th
e photons by
Bob. Bob does not know which alphabet
Alice chose and must randomly choose an
alphabet to measure the states of each
photon as they arrive. He records his choice
of alphabet for each photon. In Step 4 Bob
contacts Alice over the public
comm
unications channel (telephone or
email) to inform her which alphabet he
chose to measure each photon. By chance,
Bob will have chosen the correct alphabet
for examining the states of some of the
photons. Bob does not inform her of the
values that he had
measured. Alice responds
by informing Bob of the photons to which he
applied the correct alphabet. Both Alice and
Bob then discard those photons to which
Bob applied the incorrect alphabet. What
remains is a sequence of random bits
referred to as a raw
key which has been
transmitted securely from Alice to Bob.
This is so because Alice has transmitted the
photons one at a time and they cannot be
split nor cloned. Additionally, any attempt
at examining the photons while in transit to
Bob will modify some

of their properties.
For example, measuring a linearly polarized
VOLUME 1, NUMBER 1 PHYSICAL REVIEW LETTERS 21 NOVEMBER 2003


4

photon with a diagonal filter will result in a
50% chance that the photon will go through,
and if it does, it will have a different
polarization when it reaches Bob. Eve has
changed the me
ssage. Measuring a
horizontally polarized photon with a vertical
filter will block that photon and will not
allow it to go through to Bob. Again, Eve
has changed the message. As a final check,
Step 5, Alice and Bob take a sample of the
raw key and over

a public communications
channel reveal the values that they have, to
determine if there were any errors introduced
by noise and Eve eavesdropping. If the error
count is below a given threshold, then what
they both have is a key that can be used in a
one
-
time pad encryption system. This key
will only be used for one message and a new
key will be developed for subsequent
messages.



III. THE FUTURE



One problem
that is keeping these
systems from general use is overcoming
noise over large distances in op
tical fibre
and through the atmosphere. In 1995,
researchers were able to transmit and
receive photons in excess of 23 kms over
an optical fibre line and through the
atmosphere over 1 km. In other research,
researchers have successfully transmitted
and r
eceived photons over distances of 1
km through the atmosphere and on going
experiments predict that distances of 10km
and 23 km are possible. On going
research in this field focuses on the
development of sources for photons and
improving the capabilities
of sensors.
There are companies already on the market
with systems such as idQuantique, a
European company that claim to have a
system that can work to a distance of 60
km.


IV. CONCLUSION



Secure communications systems are in
greater demand in today’
s digital world.
Quantum mechanics can offer greater
security in systems that are currently in
use. The advantage is to use Quantum
Key Distribution in which a random key is
transmitted from the sender to the receiver
via qubits. This provides security
by: the
use of a random key similar to a one
-
time
pad system and because of the nature of
quantum states, they cannot be examined
without changing some of their properties.




[1] Nielsen M., Chuang I.,
Quantum
Computation and Quantum Information
,
(Cambr
idge University Press, Cambridge,
UK 2001).

[2] Lomonaco Jr., S.J.,
A Quick Glance at
Quantum Cryptography

(University of
Maryland Baltimore County, Baltimore,
MD 1998)

[3]
Brooks M. (ed.)
Quantum Computing
& Communications
, (Springer
-
Verlag
London Ltd., L
ondon UK 1999)