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Quantum Cryptography. Brandin L Claar CSE 597E 5 December 2001. Overview. Motivations for Quantum Cryptography Background Quantum Key Distribution (QKD) Attacks on QKD. Motivations. Desire for privacy in the face of unlimited computing power

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quantum cryptography

Quantum Cryptography

Brandin L Claar

CSE 597E

5 December 2001

overview
Overview
  • Motivations for Quantum Cryptography
  • Background
  • Quantum Key Distribution (QKD)
  • Attacks on QKD

Brandin L Claar

motivations
Motivations
  • Desire for privacy in the face of unlimited computing power
  • Current cryptographic schemes based on unproven mathematical principles like the existence of a practical trapdoor function
  • Shor’s quantum factoring algorithm could break RSA in polynomial time
  • Quantum cryptography realizable with current technology

Brandin L Claar

photons
Photons
  • Photons are the discrete bundles of energy that make up light
  • They are electromagnetic waves with electric and magnetic fields represented by vectors perpendicular both to each other and the direction of travel
  • The behavior of the electric field vector determines the polarization of a photon

Brandin L Claar

polarizations
Polarizations
  • A linear polarization is always parallel to a fixed line, e.g. rectilinear and diagonal polarizations
  • A circular polarization creates a circle around the axis of travel
  • Elliptical polarizations exist in between

Brandin L Claar

the poincar sphere
The Poincaré Sphere

z

  • Any point resting on the surface of the unit sphere represents a valid polarization state for a photon
  • The x, y, and z axes represent the rectilinear, diagonal, and circular polarizations respectively

(0,0,1)

(-1,0,0)

(0,-1,0)

(0,1,0)

y

(1,0,0)

x

(0,0,-1)

Brandin L Claar

bases
Bases
  • Diametrically opposed points on the surface of the sphere form a basis
  • Here, {P,-P} and {Q,-Q} represent bases
  • Bases correspond to measurable properties
  • Conjugate bases are separated by 90

z

P

-Q

y

Q

-P

x

Brandin L Claar

quantum uncertainty
Quantum Uncertainty
  • Quantum mechanics is simply the study of very small things
  • Heisenburg’s uncertainty principle places limits on the certainty of measurements on quantum systems
  • Inherent uncertainties are expressed as probabilities

Brandin L Claar

measuring polarization
Measuring Polarization

z

  • Imagine a photon in state Q, measured by {P,-P} where  is the angle between P and Q
  • It behaves as P with probability:

P

y

Q

  • It behaves as -P with probability:

-P

x

Brandin L Claar

measuring polarization1
Measuring Polarization

z

  • This phenomenon produces some interesting behavior for cryptography
  • Prob(P) + Prob(-P) = 1
  • If  is 90 or 270, Prob(P) = Prob(-P) = .5
  • If  is 0 or 180, Prob(P) = 1

P

y

Q

-P

x

Brandin L Claar

properties for cryptography
Properties for Cryptography
  • Given 2 conjugate bases, a photon polarized with respect to one and measured in another reveals zero information
  • Dirac: this loss is permanent; the system “jumps” to a state of the measurement basis
  • Only measurement in the original basis reveals the actual state

Brandin L Claar

key to quantum cryptography
Key to Quantum Cryptography

z

  • Imagine a bit string composed from 2 distinct quantum alphabets
  • It is impossible to retrieve the entire string without knowing the correct bases
  • Random measurements by an intruder will necessarily alter polarization resulting in errors

1

(0,0,1)

(-1,0,0)

0

(0,-1,0)

(0,1,0)

y

(1,0,0)

1

x

0

(0,0,-1)

Brandin L Claar

history
History
  • Conjugate Coding, Stephen Wiesner (late 60’s)
  • CRYPTO ’82: Quantum Cryptography, or unforgeable subway tokens
  • Charles H. Bennett, Gilles Brassard: use photons to transmit instead of store

Brandin L Claar

quantum key distribution
Quantum Key Distribution
  • Experimental Quantum Cryptography, Bennett, Bessette, Brassard, Salvail, Smolin (1991)
  • Allows Alice and Bob to agree on a secure random key of arbitrary length potentially for use in a one-time pad

Brandin L Claar

the protocol
The Protocol
  • Communication over the Quantum Channel
  • Key Reconciliation
  • Privacy Amplification

Brandin L Claar

the quantum channel
The Quantum Channel

lens

free air optical

path (~32cm)

Wollaston

prism

LED

photomultiplier

tubes

pinhole

interference

filter

Pockels

cells

Brandin L Claar

basic protocol
Basic Protocol
  • Alice sends random sequence of 4 types of polarized photons over the quantum channel: horizontal, vertical, right-circular, left-circular
  • Bob measures each in a random basis
  • After full sequence, Bob tells Alice the bases he used over the public channel
  • Alice informs Bob which bases were correct
  • Alice and Bob discard the data from incorrectly measured photons
  • The polarization data is converted to a bit string (↔ = ↶ = 0 and ↕ = ↷ = 1)

Brandin L Claar

basic protocol example
Basic Protocol Example

↶ ↷ ↔ ↕ ↷ ↔ ↔ ↷ ↷

+ o + + o o + + o

↕ ↷ ↔ ↕ ↶ ↔ ↷

+ o + + o + o

Y Y Y Y

↷ ↔ ↕ ↷

1 0 1 1

Brandin L Claar

key reconciliation
Key Reconciliation
  • Data is compared and errors eliminated by performing parity checks over the public channel
  • Random string permutations are partitioned into blocks believed to contain 1 error or less
  • A bisective search is performed on blocks with incorrect parity to eliminate the errors
  • The last bit of each block whose parity was exposed is discarded
  • This process is repeated with larger and larger block sizes
  • The process ends when a number of parity checks of random subsets of the entire string agree

Brandin L Claar

privacy amplification
Privacy Amplification
  • A hash function h of the following class is randomly and publicly chosen:
  • With n bits where Eve’s expected deterministic information is l bits, and an arbitrary security parameter s, Eve’s expected information on h(x) will be less than
  • h(x) will be the final shared key between Alice and Bob

Brandin L Claar

attacking qkd
Attacking QKD
  • Intercept/Resend Attack
  • Beamsplitting Attack
  • Estimating Eve’s Information

Brandin L Claar

intercept resend attack
Intercept/Resend Attack
  • Allows Eve to determine the value of each bit with probability
  • At least 25% of intercepted pulses will generate errors when read by Bob
  • All errors are assumed to be the result of intercept/resend
  • Hence, a conservative estimate of Eve’s information on the raw quantum transmission (given t detected errors) is

Brandin L Claar

beamsplitting attack
Beamsplitting Attack
  • Ideally, each pulse sent by Alice would consist of exactly 1 photon
  • The number of expected photons per pulse is 
  • Eve is able to learn a constant fraction of the bits by splitting a pulse
  • Given N pulses, the number of bits lost to Eve through beamsplitting is estimated to be less than

Brandin L Claar

estimating eve s information
Estimating Eve’s Information
  • Given a bit error rate p and a pulse intenstity , Eve is expected to learn a fraction of the raw key:
  • Alice and Bob can estimate the number of leaked bits and use this to eliminate Eve’s information in the privacy amplification stage:

Brandin L Claar

other protocols
Other protocols
  • Quantum Oblivious Transfer
  • Einstein-Podolsky-Rosen (EPR) effect

Brandin L Claar