2. Cryptography.
• Transmitting information with access
restricted to the intended recipient even if
the message is intercepted by others.
• Cryptography is of increasing importance
in our technological age using broadcast,
network communications, Internet ,e-mail,
cell phones which may transmit sensitive
information related to finances, politics,
business and private confidential matters.
3. The process
• Sender Plaintext
Cryptotext
Decryption
Plaintext
Recipient
Message encryption
Key
Key ready for use
Secure key distribution
Encryption
Secure
transmission
Hard Problem for conventional
encryption
4. The classic cryptography
• Encryption algorithm and related key are kept
secret.
• Breaking the system is hard due to large
numbers of possible keys.
• For example: for a key 128 bits long
• there are
38
128
10
2
keys to check using brute force.
The fundamental difficulty is key distribution to parties
who want to exchange messages.
5. PKC :the modern cryptography
• In 1970s the Public Key Cryptography
emerged.
• Each user has two mutually inverse
keys,
• The encryption key is published;
• The decryption key is kept secret.
• Anybody can send a message to Bob
but only Bob can read it.
6. RSA
• The most widely used PKC is the RSA
algorithm based on the difficulty of
• factoring a product ot two large primes.
• Easy Problem Hard Problem
Given two large
primes p and q
compute
q
p
n
Given n
compute p and q.
7. Factoring a product of two large
primes
• The best known conventional algorithm
requires the solution time proportional to:
]
)
ln
(ln
)
(ln
exp[
)
( 3
/
2
3
/
1
n
n
c
n
T
For p & q 65 digits long T(n) is approximately
one month using cluster of workstations.
For p&q 200 digits long T(n) is astronomical.
8. Quantum Computing algorithm for
factoring.
• In 1994 Peter Shor from the AT&T Bell
Laboratory showed that in principle a
quantum computer could factor a very long
product of primes in seconds.
• Shor’s algorithm time computational
complexity is
]
)
[(ln
)
( 3
n
O
n
T
Once a quantum computer is built
the RSA method
would not be safe.
9. Elements of the Quantum Theory
• Light waves are propagated as discrete
quanta called photons.
• They are massless and have energy,
momentum and angular momentum called
spin.
• Spin carries the polarization.
• If on its way we put a polarization filter
a photon may pass through it or may not.
• We can use a detector to check of a photon
has passed through a filter.
11. Heisenberg Uncertainty Principle
• Certain pairs of physical properties are related
in such a way that measuring one property
prevents the observer from knowing the value
of the other.
When measuring the polarization of a photon,
the choice of what direction to measure affects
all subsequent measurements.
• If a photon passes through a vertical filter
it will have the vertical orientation regardless of
its initial direction of polarization.
12. Photon Polarization
Vertical
filter
Tilted filter at
the angle
The probability of a photon appearing after the second
filter depends on the angle and becomes 0 at
= 90 degrees.
The first filter randomizes the measurements of the
second filter.
13. Polarization by a filter
• A pair of orthogonal filters such as
vertical/horizontal is called a basis.
• A pair of bases is conjugate if the
measurement in the first basis
completely randomizes the
measurements in the second basis.
• As in the previous slide example for
=45deg.
14. Sender-receiver of photons
• Suppose Alice uses 0-deg/90-deg polarizer
sending photons to Bob. But she does not
reveal which.
• Bob can determine photons by using
filter aligned to the same basis.
• But if he uses 45deg/135 deg polarizer to
measure the photon he will not be able to
determine any information about the initial
polarization of the photon.
• The result of his measurement will be completely
random
15. Eavesdropper Eve
• If Eve uses the filter aligned with
Alice’s she can recover the original
polarization of the photon.
• If she uses the misaligned filter she
will receive no information about the
photon .
• Also she will influence the original
photon and be unable to retransmit it
with the original polarization.
• Bob will be able to deduce Ave’s
presence.
16. Binary information
• Each photon carries one qubit of information
• Polarization can be used to represent a 0 or 1.
• In quantum computation this is called
qubit.
To determine photon’s polarization the
recipient must measure the polarization by
,for example, passing it through a filter.
17. Binary information
• A user can suggest a key by sending a
stream of randomly polarized photons.
• This sequence can be converted to a
binary key.
• If the key was intercepted it could be
discarded and a new stream of
randomly polarized photons sent.
18. The Main contribution of Quantum
Cryptography.
• It solved the key distribution problem.
• Unconditionally secure key distribution
method proposed by:
• Charles Bennett and Gilles Brassard in
1984.
• The method is called BB84.
• Once key is securely received it can be
used to encrypt messages transmitted
by conventional channels.
19. Quantum key distribution
• (a)Alice communicates with Bob via a
quantum channel sending him photons.
• (b) Then they discuss results using a
public channel.
• (c) After getting an encryption key Bob can
encrypt his messages and send them by
any public channel.
20. Quantum key distribution
• Both Alice and Bob have two polarizers
each.
• One with the 0-90 degree basis (+) and one
with 45-135 degree basis ( )
• (a) Alice uses her polarizers to send
randomly photons to Bob in one of the four
possible polarizations 0,45,90,135 degree.
• (b)
b) Bob uses his polarizers to measure each
polarization of photons he receives.
He can use the( + )basis or the ( ) but not both
simultaneously.
22. Security of quantum key
distribution
• Quantum cryptography obtains its
fundamental security from the fact that
each qubit is carried by a single
photon, and each photon will be altered
as soon as it is read.
• This makes impossible to intercept
message without being detected.
23. Noise
• The presence of noise can impact
detecting attacks.
• Eavesdropper and noise on the
quantum channel are
indistinguishable.
• (1) Malicious eavesdropper can
prevent communication.
• (2) Detecting eavesdropper in the
presence of noise is hard.
24. State of the Quantum
Cryptography technology.
• Experimental implementations have
existed since 1990.
• Current (2004) QC is performed over
distances of 30-40 kilometers using
optical fiber.
In general we need two capabilities.
(1) Single photon gun.
(2) Being able to measure single photons.
25. State of the QC technology.
• Efforts are being made to use Pulsed
Laser Beam with low intensity for firing
single photons.
• Detecting and measuring photons is hard.
• The most common method is exploiting
Avalanche Photodiodes in the Geiger
mode where single photon triggers a
detectable electron avalanche.
26. State of the QC technology.
• Key transmissions can be achieved for
about 80 km distance ( Univ of Geneva
2001).
• (2)For longer distances we can use
repeaters. But practical repeaters are a
long way in the future.
• Another option is using satellites.
• Richard Hughes at LOS ALAMOS NAT
LAB (USA) works in this direction.
• The satellites distance from earth is in
hundreds of kilometers.
27. NIST System
• Uses an infrared laser to generate
photons
• and telescopes with 8-inch mirrors to
send and receive photons over the air.
• Using the quantum transmitted key
messages were encrypted at the rate
1 million bits per second.
The speed was impressive but the distance
between two NIST buildings was only 730
meters.
28. Commercial QC providers
• id Quantique, Geneva Switzerland
• Optical fiber based system
• Tens of kilometers distances
• MagiQ Technologies, NY City
• Optical fiber-glass
• Up to 100 kilometers distances
• NEC Tokyo 150 kilometers
• QinetiQ Farnborough, England
• Through the air 10 kilometers.
• Supplied system to BBN in Cambridge Mass.
![Factoring a product of two large
primes
• The best known conventional algorithm
requires the solution time proportional to:
]
)
ln
(ln
)
(ln
exp[
)
( 3
/
2
3
/
1
n
n
c
n
T
For p & q 65 digits long T(n) is approximately
one month using cluster of workstations.
For p&q 200 digits long T(n) is astronomical.](https://image.slidesharecdn.com/introductiontoquantumcryptography-250720161544-ee29fcce/85/Introduction-to-quantum-cryptography-ppt-7-320.jpg)
![Quantum Computing algorithm for
factoring.
• In 1994 Peter Shor from the AT&T Bell
Laboratory showed that in principle a
quantum computer could factor a very long
product of primes in seconds.
• Shor’s algorithm time computational
complexity is
]
)
[(ln
)
( 3
n
O
n
T
Once a quantum computer is built
the RSA method
would not be safe.](https://image.slidesharecdn.com/introductiontoquantumcryptography-250720161544-ee29fcce/85/Introduction-to-quantum-cryptography-ppt-8-320.jpg)
