14618442 quantum cryptography

8/3/2019 14618442 Quantum Cryptography

1/31

QUANTUM CRYPTOGRAPHYQUANTUM CRYPTOGRAPHY

PRESENTED BYPRESENTED BY

V.LACHIRAMV.LACHIRAM --06141A053806141A0538


2/31

Quantum cryptographyQuantum cryptography--the final battle?the final battle?

CS4236 Principles of Computer SecurityCS4236 Principles of Computer Security

National University of SingaporeNational University of Singapore

Jonas Rundberg, NT030157AJonas Rundberg, NT030157A


3/31

This presentationThis presentation Quantum mechanicsQuantum mechanics

IntroductionIntroduction

NotationNotation Polarized photonsPolarized photons

ExperimentExperiment

Quantum cryptologyQuantum cryptology

Key distributionKey distribution

EavesdroppingEavesdropping Detecting eavesdroppingDetecting eavesdropping

NoiseNoise

Error correctionError correction

Privacy AmplificationPrivacy Amplification

EncryptionEncryption


4/31

IntroductionIntroduction Spawned during the last centurySpawned during the last century

DescribesDescribes properties and interactionproperties and interactionbetween matter at small distance scalesbetween matter at small distance scales

Quantum state determined byQuantum state determined by(among others)(among others) PositionsPositions

VelocitiesVelocities PolarizationsPolarizations

SpinsSpins

qubitsqubits


5/31

NotationNotation BraBra/Ket notation (pronounced bracket)/Ket notation (pronounced bracket)

From Dirac 1958From Dirac 1958 Each state represented by a vectorEach state represented by a vector

denoted by a arrow pointing in thedenoted by a arrow pointing in thedirection of the polarizationdirection of the polarization


6/31

NotationNotation Simplified Bra/KetSimplified Bra/Ket--notation in thisnotation in this

presentationpresentation

Representation of polarized photons:Representation of polarized photons:

horizontally:horizontally: vertically:vertically: diagonally:diagonally: andand


7/31

Polarized photonsPolarized photons

Polarization can bePolarization can bemodeled as a linearmodeled as a linear

combination of basiscombination of basisvectorsvectors andand

Only interested inOnly interested indirectiondirection

aa+ b+ bwill result inwill result ina unit vectora unit vector suchsuchthat |a|that |a|22 + |b|+ |b|22 = 1= 1

b

a


8/31

Polarized photonsPolarized photons

Measurement of a stateMeasurement of a statenot onlynot only measuresmeasures butbutactually transforms thatactually transforms thatstate to one of the basisstate to one of the basisvectorsvectors andand

If we chose the basisIf we chose the basisvectorsvectors andand whenwhenmeasuring the state ofmeasuring the state ofthe photon, the result willthe photon, the result willtell us that the photon'stell us that the photon'spolarization is eitherpolarization is either oror, nothing in between., nothing in between.

b

a


9/31


Classical experimentClassical experiment

Equipment:Equipment: laser pointerlaser pointer

three polarization filtersthree polarization filters

The beam of light i pointed toward aThe beam of light i pointed toward a

screen.screen. The three filters are polarized atThe three filters are polarized at

,, andand respectivelyrespectively


10/31


TheThe filter is put in front of the screenfilter is put in front of the screen L

ight on outgoing side of filter is now 50%L

ight on outgoing side of filter is now 50%of original intensityof original intensity


11/31


Next we insert aNext we insert a filter whereas no lightfilter whereas no lightcontinue on the output sidecontinue on the output side


12/31


Here is the puzzling partHere is the puzzling part

We insert aWe insert a filter in betweenfilter in between ThisThis increasesincreases the number of photonsthe number of photons

passing throughpassing through


13/31

Experiment explainedExperiment explained

FilterFilter is hit by photons in randomis hit by photons in randomstates. It will measure half of the photonsstates. It will measure half of the photonspolarized aspolarized as


14/31


FilterFilter is perpendicular to that and willis perpendicular to that and willmeasure the photons with respect tomeasure the photons with respect to ,,which none of the incoming photonswhich none of the incoming photonsmatchmatch


15/31


FilterFilter measures the state with respect tomeasures the state with respect tothe basis {the basis {,, }}


16/31


Photons reaching filterPhotons reaching filter will be measuredwill be measuredasas with 50% chance. These photonswith 50% chance. These photonswill be measured by filterwill be measured by filter asas with 50%with 50%probability and thereby 12,5% of theprobability and thereby 12,5% of theoriginal light pass through all three filters.original light pass through all three filters.


17/31

Quantum cryptologyQuantum cryptology


18/31

Key distributionKey distribution

Alice and Bob first agree on twoAlice and Bob first agree on tworepresentations for ones and zeroesrepresentations for ones and zeroes

One for each basis used, {One for each basis used, {,,}}and {and {,, }.}. This agreement can be done in publicThis agreement can be done in public

DefineDefine1 =1 = 0 =0 = 1 =1 = 0 =0 =


19/31

Key distributionKey distribution -- BB84BB84

1.1. Alice sends a sequence of photons to Bob.Alice sends a sequence of photons to Bob.Each photon in a state with polarization correspondingEach photon in a state with polarization correspondingto 1 or 0, but with randomly chosen basis.to 1 or 0, but with randomly chosen basis.

2.2. Bob measures the state of the photons he receives,Bob measures the state of the photons he receives,with each state measured with respect to randomlywith each state measured with respect to randomlychosen basis.chosen basis.

3.3. Alice and Bob communicates via an open channel. ForAlice and Bob communicates via an open channel. For

each photon, they reveal which basis was used foreach photon, they reveal which basis was used forencoding and decoding respectively. All photons whichencoding and decoding respectively. All photons whichhas been encoded and decoded with the same basis arehas been encoded and decoded with the same basis arekept, while all those where the basis don't agree arekept, while all those where the basis don't agree arediscarded.discarded.


20/31

EavesdroppingEavesdropping

Eve has to randomly select basis for her measurementEve has to randomly select basis for her measurement

Her basis will be wrong in 50% of the time.Her basis will be wrong in 50% of the time.

Whatever basis Eve chose she will measure 1 or 0Whatever basis Eve chose she will measure 1 or 0 When Eve picks the wrong basis, there is 50% chanceWhen Eve picks the wrong basis, there is 50% chance

that she'll measure the right value of the bitthat she'll measure the right value of the bit

E.g. Alice sends a photon with state corresponding to 1E.g. Alice sends a photon with state corresponding to 1in the {in the {,,}} basis. Eve picks the {basis. Eve picks the {,, } basis for her} basis for hermeasurement which this time happens to give a 1 asmeasurement which this time happens to give a 1 asresult, which is correct.result, which is correct.


21/31

EavesdroppingEavesdroppingAlicesAlicesbasisbasis

AlicesAlicesbitbit

AlicesAlicesphotonphoton

EvesEvesbasisbasis

CorrectCorrect EvesEvesphotonphoton

EvesEvesbitbit

CorrectCorrect

{{,,}}

11 {{,,}} YesYes 11 YesYes{{,, }} NoNo 11 YesYes

00 NoNo

00 {{,,}} YesYes 00 YesYes{{,, }} NoNo 11 NoNo

00 YesYes

{{,, }}

11

{{,,}} NoNo 11 YesYes

00 NoNo{{,, }} YesYes 11 YesYes

00 {{,,}} NoNo 11 NoNo

00 YesYes{{,, }} yesyes 00 YesYes


22/31

Eves problemEves problem

Eve has to reEve has to re--send all the photons to Bobsend all the photons to Bob

Will introduce an error, since Eve don'tWill introduce an error, since Eve don't

know the correct basis used by Aliceknow the correct basis used by Alice Bob will detect an increased error rateBob will detect an increased error rate

Still possible for Eve to eavesdrop just aStill possible for Eve to eavesdrop just a

few photons, and hope that this will notfew photons, and hope that this will notincrease the error to an alarming rate. Ifincrease the error to an alarming rate. Ifso, Eve would have at least partialso, Eve would have at least partialknowledge of the key.knowledge of the key.


23/31

Detecting eavesdroppingDetecting eavesdropping

When Alice and Bob need to test for eavesdroppingWhen Alice and Bob need to test for eavesdropping

By randomly selecting a number of bits from theBy randomly selecting a number of bits from thekey and compute its error ratekey and compute its error rate

Error rate < EError rate < Emaxmax assume no eavesdroppingassume no eavesdropping

Error rate > EError rate > Emaxmax assume eavesdroppingassume eavesdropping(or the channel is unexpectedly noisy)(or the channel is unexpectedly noisy)

Alice and Bob should then discard the whole keyAlice and Bob should then discard the whole keyand start overand start over


24/31

NoiseNoise

Noise might introduce errorsNoise might introduce errors

A detector might detect a photon even thoughA detector might detect a photon even though

there are no photonsthere are no photons Solution:Solution:

send the photons according to a time schedule.send the photons according to a time schedule.

then Bob knows when to expect a photon, and canthen Bob knows when to expect a photon, and candiscard those that doesn't fit into the scheme's timediscard those that doesn't fit into the scheme's timewindow.window.

There also has to be some kind of errorThere also has to be some kind of errorcorrection in the over all process.correction in the over all process.


25/31

Error correctionError correction

Suggested by HoiSuggested by Hoi--Kwong Lo. (Shortened version)Kwong Lo. (Shortened version)

1.1. Alice and Bob agree on a random permutation of the bitsAlice and Bob agree on a random permutation of the bitsin the keyin the key

2.2. They split the key into blocks of length kThey split the key into blocks of length k

3.3. Compare the parity of each block. If they compute theCompare the parity of each block. If they compute thesame parity, the block is considered correct. If theirsame parity, the block is considered correct. If theirparity is different, they look for the erroneous bit, usingparity is different, they look for the erroneous bit, using

a binary search in the block. Alice and Bob discard thea binary search in the block. Alice and Bob discard thelast bit of each block whose parity has been announcedlast bit of each block whose parity has been announced

4.4. This is repeated with different permutations and blockThis is repeated with different permutations and blocksize, until Alice and Bob fail to find any disagreement insize, until Alice and Bob fail to find any disagreement inmany subsequent comparisonsmany subsequent comparisons


26/31

Privacy amplificationPrivacy amplification

Eve might have partial knowledge of the key.Eve might have partial knowledge of the key.

Transform the key into a shorter but secure keyTransform the key into a shorter but secure key

Suppose there are n bits in the key and Eve hasSuppose there are n bits in the key and Eve hasknowledge of m bits.knowledge of m bits.

Randomly chose a hash function whereRandomly chose a hash function where

h(x): {0,1h(x): {0,1\\}}nnll

{0,1{0,1\\}}nn--mm--ss

Reduces Eve's knowledge of the key to 2Reduces Eve's knowledge of the key to 2 s / ln2s / ln2

bitsbits


27/31


28/31

What to comeWhat to come

Theory for quantum cryptography alreadyTheory for quantum cryptography alreadywell developedwell developed

Problems:Problems:

quantum cryptography machine vulnerable toquantum cryptography machine vulnerable tonoisenoise

photons cannot travel long distances withoutphotons cannot travel long distances withoutbeing absorbedbeing absorbed


29/31

SummarySummary

The ability to detect eavesdroppingThe ability to detect eavesdroppingensures secure exchange of the keyensures secure exchange of the key

The use of oneThe use of one--timetime--pads ensures securitypads ensures security

Equipment can only be used over shortEquipment can only be used over shortdistancesdistances

Equipment is complex and expensiveEquipment is complex and expensive


30/31

ReferencesReferences

[RP00] Eleanor Rie_el, Wolfgang Polak,[RP00] Eleanor Rie_el, Wolfgang Polak,ACM Computing surveys,Vol. 32, No.3.September 2000ACM Computing surveys,Vol. 32, No.3.September 2000

[WWW1] Math Pages, Spin & Polarization[WWW1] Math Pages, Spin & Polarizationhttp://www.mathpages.com/rr/s9http://www.mathpages.com/rr/s9--04/904/9--04.htm04.htm

[WWW2] Luisiana Tech University,[WWW2] Luisiana Tech University,Quantum ComputationQuantum Computationhttp://www2.latech.edu/~dgao/CNSM/quantumcomput.htmlhttp://www2.latech.edu/~dgao/CNSM/quantumcomput.html

[WWW3] Edmonton Community Network,[WWW3] Edmonton Community Network,Quantum CryptographyQuantum Cryptographyhttp://home.ecn.ab.ca/~jsavard/crypto/mi060802.htmhttp://home.ecn.ab.ca/~jsavard/crypto/mi060802.htm

[WIK1] Wikipedia[WIK1] Wikipedia --The free encyclopediaThe free encyclopediahttp://www.wikipedia.org/wiki/Brahttp://www.wikipedia.org/wiki/Bra--ket_notationket_notation


31/31

ReferencesReferences

[WIK2] Wikipedia[WIK2] Wikipedia --The free encyclopediaThe free encyclopediahttp://www.wikipedia.org/wiki/Interpretation_of_quantum_mechanihttp://www.wikipedia.org/wiki/Interpretation_of_quantum_mechanicscs

[WIK3] Wikipedia[WIK3] Wikipedia --The free encyclopediaThe free encyclopedia

http://www.wikipedia.org/wiki/Copenhagen_interpretationhttp://www.wikipedia.org/wiki/Copenhagen_interpretation [GIT] Georgia Institute of Technology,[GIT] Georgia Institute of Technology,

The fundamental postulates of quantum mechanicsThe fundamental postulates of quantum mechanicshttp://www.physics.gatech.edu/academics/Classes/spring2002/6107http://www.physics.gatech.edu/academics/Classes/spring2002/6107

/Resources/The fundamental postulates of quantum mechanics.pdf/Resources/The fundamental postulates of quantum mechanics.pdf [HP] Hoi[HP] Hoi--Kwong Lo, Networked Systems Department,Kwong Lo, Networked Systems Department,

Hewlett Packard, Bristol, December 1997, Quantum CryptologyHewlett Packard, Bristol, December 1997, Quantum Cryptology [SS99] Simon Singh, Code Book, p349[SS99] Simon Singh, Code Book, p349--382,382,

Anchor Books, 1999Anchor Books, 1999 [FoF] Forskning och Framsteg,[FoF] Forskning och Framsteg,

No. 3, April 2003No. 3, April 2003

14618442 quantum cryptography

Documents