Download QUANTUM KEY DISTRIBUTION 1. Cryptography In the course of

QUANTUM KEY DISTRIBUTION
NICO VERHART EN WILLEM ONDERWAATER
1. Cryptography
In the course of history it has always been necessary to transmit encoded messages. Encoding ensures that the information is only available to whom the message
is intended. To others the message is unintelligible noise. Encoding has been done
in a large variety of methods but often it involves a key. A certain protocol on how
to code and decode understandable information.
One example of modern methods of encoding information is the Rivest-ShamirAdleman (RSA) Code [1]. The foundation of the RSA algorithm is the current
inability to factorize large numbers. The product of two large primes represents
a public known key which is used to encode messages. To decode messages the
prime factors need to be known. If factorization could be done in limited time the
RSA algorithm would no longer function. Theoretically quantum computers are
able to factorize large numbers. This would mean that once quantum computers
are operational the RSA code would cease to be useful.
Luckily the quantum world supplies a solution. Quantum phenomena can be
used to construct a key needed for the encoding and decoding of a message. A
key can be safely constructed while the sender and recipient of the message are far
apart.
2. BB84
In quantum key distribution (QKD) protocols we make use of the rules of quantum physics to ensure that the key used to encrypt and decipher a message is
known only to the sender (Alice), and the receiver (Bob) and not to any third
party (Eve). For quantum key distribution we need both a quantum and a classical
communication channel.
We first consider the BB84 protocol (published by Bennet & Brassard 1984,
hence the name)[2]. Over
√ qubits in two orthonormal
the
quantum channel
we send
0̃ , 1̃ such that ij̃ = 1/ 2, where we interpret |0i and
bases
{|0i
,
|1i}
and
0̃ as 0 and |1i and 1̃ as 1. Such bases are called conjugate. The bases could
o
n
for example be {|0i , |1i} and
√1
2
(|0i + |1i) , √12 (|0i − |1i) . Physically this can
for example be spin up/down along the z- and x-direction, or photon polarization
at 0/90 degrees and 45/-45 degrees. The BB84 protocol is now as follows:
(1) Alice creates a random sequence of 0’s and 1’s.
(2) For each bit in this sequence Allice randomly chooses one of the two bases
and then sends the bit to Bob in the form of qubit in that basis.
(3) Similarly Bob chooses one of the two bases for each qubit, and then measures the qubit in that basis. If he measures the qubit in the same basis
as Alice has send it he will obtain the correct bit. If on the other hand he
1
2
NICO VERHART EN WILLEM ONDERWAATER
1
2
3
4
5
random bits
0
1
1
1
0
0
0
0
1
basis chosen by A
x
x
z
x
z
z
x
z
z
qubit send to B
|→i |←i |↓i |←i |↑i |↑i |→i |↑i |↓i
basis chosen by B
z
x
x
x
z
x
x
z
z
bits measured by B
1
1
1
1
0
1
0
0
1
compare bases
OK OK OK OK OK OK
bits kept
1
1
0
0
0
1
bits used for comparison
1
0
1
key
1
0
0
Table 1. Illustration of the BB84 protocol using spin up/down
along the z-direction {|↑i , |↓i} and along the x-direction
{|→i , |←i}, in the absence of eavesdropping and noice.
measures in a different basis, he will get either 0 or 1, both with probability
1/2. Hence with probability 1/2 he will get a different result.
(4) Next Alice and Bob publicly compare the bases they used. In doing so
Alice and Bob now which of their bits are certain to be the same and they
discard the other bits. Note that an outsider cannot use this information
to determine which bits they obtained.
(5) In order to verify that no one has been measuring their qubits Alice and
Bob randomly pick some of the bits for public comparison. If a third party
Eve has measured the qubits about 1/4 of the bits will no longer match
and Alice and Bob know that their key has been compromised, which will
stop them from using this key to encrypt the message.
An example of BB84 key distribution using spin-1/2 objects is given in table 1. In
theory the BB84 key distribution work perfectly. Heisenbergs uncertainty principle
guarantees that Eve cannot measure the two chosen bases at the same time, and
the no-cloning theorem states that Eve cannot make identical copies of the qubit to
measure each basis on another identical qubit. The fact that the two chosen bases
are conjugate ensures that when the transmitted qubit is measured in the wrong
basis, it completely randomizes the qubit in the other basis. This is important
because it allows Alice an Bob to determine if Eve tried to eavesdrop with maximal
probability. In practice there are technical issues to be considered[3]. Quantum
channels are always noisy. This means that Bob will measure different data than
Alice has sent. In addition it is very difficult to send light pulses containing only
one photon. It is much easier to produce coherent pulses of of number of photons.
But this leaves Eve the ability to split the beam and learn a fraction of the bits,
without disturbing the qubit sent to Bob.
References
1. J. Preskill, Lecture Notes for Physics Quantum Information and Computation, Caltech. Chapter 6.10.2
2. C.Bennett, G Brassard, ‘Quantum cryptography: Public key distribution and coin tossing’,in
Proceedings of IEEE International Conference on Computers, Systems and Signal processing
(Bangalore, India, 1984), pp 175-179
3. Bennett, C.H., F. Bessette, G. Brassard, L. Salvail, and J. Smolin, ‘Experimental quantum
cryptography’, Journal of Cryptology 5(1), 328 (1992)