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Quantum Cryptography

Qingqing Yuan

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Outline

No-Cloning Theorem

BB84 Cryptography Protocol

Quantum Digital Signature

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One Time Pad Encryption

Conventional cryptosystem:

Alice and Bob share N random bits b1bN

Alice encrypt her message m1mNb1m1,,bNmN Alice send the encrypted string to Bob

Bob decrypts the message: (mjbj)bj = mj As long as b is unknown, this is secure

Can be passively monitored or copied

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Two Qubit Bases

Define the four qubit states:

{0,1}(rectilinear) and {+,-}(diagonal) form anorthogonal qubit state.

They are indistinguishable from each other.

!

!

)10(

)10(

1

0

2

1

2

1

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No-Cloning Theorem

|q = |0+|1

To determine the amplitudes of an unknownqubit, need an unlimited copies

It is impossible to make a device thatperfectly copies an unknown qubit.

Suppose there is a quantum process that

implements: |q,_p|q,q Contradicts the unitary/linearity restriction of

quantum physics

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Wiesners Quantum Money

A quantum bill contains a serial number N, and20 random qubits from {0,1,+,-}

The Bank knows which string {0,1,+,-}20

isassociated with which N

The Bank can check validity of a bill N bymeasuring the qubits in the proper 0/1 or +/-

bases A counterfeiter cannot copy the bill if he

does not know the 20 bases

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Quantum Cryptography

In 1984 Bennett and Brassarddescribe how the quantum money idea

with its basis {0,1} vs. {+,-} can be usedin quantum key distribution protocol

Measuring a quantum system in general

disturbs it and yields incompleteinformation about its state before themeasurement

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BB84 Protocol (I)

Central Idea: Quantum Key Distribution(QKD) via the {0,1,+,-} states betweenAlice and Bob

Alice Bob

Quantum Channel

Classical public channel

Eve

O(N) classical and quantum communicationto establish N shared key bits

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BB84 Protocol (II)

1) Alice sends 4N random qubits {0,1,+,-} to Bob

2) Bob measures each qubit randomly in 0/1 or +/-basis

3) Alice and Bob compare their 4N basis, and continuewith b2N outcomes for which the same basis wasused

4) Alice and Bob verify the measurement outcomes onrandom (size N) subset of the 2N bits

5) Remaining N outcomes function as the secrete key

Quantum

Public & Classical

Shared Key

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Security of BB84

Without knowing the proper basis, Evenot possible to

Copy the qubits Measure the qubits without disturbing

Any serious attempt by Eve will be

detected when Alice and Bob performequality check

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Quantum Coin Tossing

Alices bit: 1 0 1 0 0 1 1 1 0 1 1 0

Alices basis: Diagonal

Alice sends: - + - + + - - - + - - +

Bobs basis: R D D R D R D R D D R R

Bobs rect. table: 0 1 0 1 1 1

Bobs Dia. table: 0 1 0 1 0 1

Bob guess: diagonalAlice reply: you win

Alice sends original string to verify.

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Quantum Coin Tossing (Cont.)

Alice may cheat

Alice create EPR pair for each bit

She sends one member of the pair andstores the other

When Bob makes his guess, Alice measure

her parts in the opposite basis

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Arguments Against QKD

QKD is not public key cryptography

Eve can sabotage the quantum channel

to force Alice and Bob use classicalchannel

Expensive for long keys: (N) qubits

of communication for a key of size N

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Practical Feasibility of QKD

Only single qubits are involved

Simple state preparations and

measurements Commercial Availability

id Quantique: http://www.idquantique.com

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Outline

No-Cloning Theorem

BB84 Cryptography Protocol

Quantum Digital Signature

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Pros of Public Key Cryptography

High efficiency

Better key distribution and management

No danger that public key is compromised

Certificate authorities

New protocols

Digital signature

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Quantum One-way Function

Consider a map f: k pfk.

k is the private key

fk is the public key One-way function: For some maps f, its

impossible (theoretically) to determinek, even given many copies of fk

we can give it to many people withoutrevealing the private key k

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Digital Signature (Classical scheme)

Lamport 1979

One-way function f(x)

Private key (k0, k1)

Public key (0,f(k0)), (1,f(k1))

Sign a bit b: (b, kb)

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Quantum Scheme

Gottesman & Chuang 2001

Private key (k0(i), k1(i)) (i=1, ..., M)

Public key To sign b, send (b, kb(1), kb(2), ..., kb(M)).

To verify, measure fk to check k = kb(i).

_ aii kk ff 10 |,|

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Levels of Acceptance

Suppose s keys fail the equality test

If sec1M: 1-ACC: Message comes from

Alice, other recipients will agree. If c1M < s e c2M: 0-ACC: Message

comes from Alice, other recipients mightdisagree.

If s > c2M: REJ: Message might notcome from Alice

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Reference

[BB84]: Bennett C. H. & Brassard G.,Quantum cryptography: Public keydistribution and coin tossing

Daniel Gottesman, Isaac Chuang,Quantum Digital Signatures

http://www.perimeterinstitute.ca/personal/dgottesman/Public-key.ppt

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Discussions

Thank you!