Quantum Cryptography

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Quantum Cryptography

• Prafulla Basavaraja
• CS 265 Spring 2005

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Introduction

• Classical cryptography relies on time
  complexity of certain mathematical operations
  given current computing methods
• Quantum computers can solve these hard problems
  in polynomial time.
• Example Shors algorithm for finding prime
  factors of very large numbers
• Unbreakable code One-Time pad
• Practical difficulty is in key distribution
• Quantum Cryptography solves this problem by
  providing secure key distribution using quantum
  mechanics
• First suggested by C.Bennett and G.Brassard

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Quantum Mechanics

• Deals with behavior of elementary particles
  (atoms energy) in terms of probabilities
• Energy, momentum angular momentums as well as
  charges come in discrete amounts called quanta
• Photons are discrete bundles of energy that make
  up light
• Properties that describe behavior of photon
• Superposition principle
• Position or energy of the photon can
  simultaneously possess 2 or more values
• Photon Ray Gun experiment

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Photon ray gun experiment
Fig.1
Fig.3
Fig.2
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Quantum Mechanics (contd.)

• Entanglement property
• Applies to a pair of spatially separated photon
  where each is described with reference to other
• In case of entangled photons measurement of
  spin of one gives the spin of other
• Measurement problem
• Measuring the state of a photon changes it
• Polarization of Photons
• Photon has electric and magnetic fields
  represented by vectors perpendicular to each
  other and direction of travel
• Polarization
• describes the spin nature of a photon
• determined by electric vector of photon

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Quantum State based Coding Scheme

• Polaroid a filter to polarize measure the
  polarization of photons
• Using rectilinear and diagonal polarization
  schemes gives 4 quantum bits or qubits

We shall use the following notations
represents rectilinear scheme (horizontal and
vertical polaroids) - to represent 0
to represent 1 x represents diagonal
scheme (left and right inclined diagonal
polariods) / to represent 0 \
to represent 1
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Qubit transmission and binary digit selection -
example
Using the above qubit representations, a
transmission for the binary 11010011 could look
like this
Alice Scheme X X X
Alice Bits 1 1 0 1 0 0 1 1
Alice Qubit ? ? ? ? / / \ ?
Bob Scheme X X X X
Bob Qubit ? \ \ ? ? / \ ?
Bob Bits 1 1 1 1 0 0 1 1
Key Selection v v v v v
Alice sends the 1st 1 using the scheme, the 2nd
one using the X scheme, 1st 0 using the X scheme
and so on.
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BB84 protocol Quantum Key Distribution

• Step 1 Alice transmits random seq of 1s 0s
  (qubits) to Bob over quantum channel
• Alice uses random selection of rectilinear and
  diagonal schemes
• Bob also uses random schemes to detect
  polarization of received photons so interprets
  the 1s 0s correctly only sometimes
• Step 2 Over a regular channel Alice tells Bob
  the polarization scheme she used for each qubit
• Bob tells Alice when he used the same scheme
  notes down bits determined with right scheme
• Step 3 Out of bits selected Alice Bob pick a
  small subset and compare if they got the bits
  right. Eg100 out of 500 bits
• If the bits match discard bits used to compare
  use remaining as the key (for encrypting actual
  data)
• If the bits do not match it could be due to Eve
  whose detector had modified the polarization of a
  photon in transmission. So discard all bits and
  restart from step 1.

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Key Selection and Detecting Eves presence
Bit Number 1 2 3 4 5 6 7 8
Alice Bits 1 1 0 1 0 0 1 1
Alice Scheme X X X
Alice Qubit ? ? ? ? / / \ ?
Eve Scheme X X X X
Eve Qubit / ? ? ? ? / ? /
Bob Scheme X X X X
Bob Qubit ? \ \ ? ? / / ?
Bob Bits 1 1 1 1 0 0 0 0
Selection v v v v v
Here the bits 1, 4, 6, 7, 8 are selected by Alice
and Bob since both of them use the same detection
scheme. But when they randomly check bits 1, 7
and 8 they find that the values are different.
Through this they can detect the presence of the
eavesdropper.
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Practical problems with QC

• Beam Splitting attack
• Hard to produce beam of single photons
• Eve can use beam splitter when multiple photons
  are emitted
• However it is not easy for Eve to determine when
  multiple photons are emitted Splitting single
  photon will affect the state of the photon and
  give away Eves presence
• Man in the middle attack
• Bob Alice need proper authentication before
  talking to each other
• Distance limitation and media limitation
• Fibre optics
• Optical pulse travels limited distance with out
  amplification so have to be done hop by hop.
  Distance achieved - 87 kms.
• Open space communication
• higher error rate
• 20 30 kms

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Thank You!