Quantum Cryptography

Qingqing Yuan

Outline

No-Cloning Theorem

BB84 Cryptography Protocol

Quantum Digital Signature

One Time Pad Encryption

Conventional cryptosystem:Alice and Bob share N random bits b1…bN

Alice encrypt her message m1…mN b1m1,…,bNmN

Alice send the encrypted string to BobBob decrypts the message: (mjbj)bj =

mj As long as b is unknown, this is secure

Can be passively monitored or copied

Two Qubit Bases

Define the four qubit states:

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

They are indistinguishable from each other.

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1

0

21

21

No-Cloning Theorem

|q = α|0+β|1To determine the amplitudes of an

unknown qubit, need an unlimited copiesIt is impossible to make a device

that perfectly copies an unknown qubit.

Suppose there is a quantum process that implements: |q,_|q,q

Contradicts the unitary/linearity restriction of quantum physics

Wiesner’s Quantum Money

A quantum bill contains a serial number N, and 20 random qubits from {0,1,+,-}

The Bank knows which string {0,1,+,-}20 is associated with which N

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

A counterfeiter cannot copy the bill if he does not know the 20 bases

Quantum Cryptography

In 1984 Bennett and Brassard describe how the quantum money idea with its basis {0,1} vs. {+,-} can be used in quantum key distribution protocol

Measuring a quantum system in general disturbs it and yields incomplete information about its state before the measurement

BB84 Protocol (I)

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

Alice Bob

Quantum Channel

Classical public channel

Eve

O(N) classical and quantum communication to establish N shared key bits

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 continue with 2N outcomes for which the same basis was used

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

5) Remaining N outcomes function as the secrete key

Quantum

Public & Classical

Shared Key

Security of BB84

Without knowing the proper basis, Eve not possible toCopy the qubits

Measure the qubits without disturbing

Any serious attempt by Eve will be detected when Alice and Bob perform “equality check”

Quantum Coin Tossing

Alice’s bit: 1 0 1 0 0 1 1 1 0 1 1 0

Alice’s basis: Diagonal

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

Bob’s basis: R D D R D R D R D D R R

Bob’s rect. table: 0 1 0 1 1 1

Bob’s Dia. table: 0 1 0 1 0 1

Bob guess: diagonal

Alice reply: you win

Alice sends original string to verify.

Quantum Coin Tossing (Cont.)

Alice may cheatAlice create EPR pair for each bit

She sends one member of the pair and stores the other

When Bob makes his guess, Alice measure her parts in the opposite basis

Arguments Against QKD

QKD is not public key cryptography

Eve can sabotage the quantum channel to force Alice and Bob use classical channel

Expensive for long keys: Ω(N) qubits of communication for a key of size N

Practical Feasibility of QKD

Only single qubits are involved

Simple state preparations and measurements

Commercial Availability id Quantique:

http://www.idquantique.com

Outline

No-Cloning Theorem

BB84 Cryptography Protocol

Quantum Digital Signature

Pros of Public Key Cryptography

High efficiency

Better key distribution and managementNo danger that public key is compromised

Certificate authorities

New protocolsDigital signature

Quantum One-way Function

Consider a map f: k fk. k is the private keyfk is the public key

One-way function: For some maps f, it’s impossible (theoretically) to determine k, even given many copies of fk

we can give it to many people without revealing the private key k

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)

Quantum Scheme

Gottesman & Chuang 2001Private 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).

ii kkff10

|,|

Levels of Acceptance

Suppose s keys fail the equality test If sc1M: 1-ACC: Message comes

from Alice, other recipients will agree.

If c1M < s c2M: 0-ACC: Message comes from Alice, other recipients might disagree.

If s > c2M: REJ: Message might not come from Alice

Reference

[BB84]: Bennett C. H. & Brassard G., “Quantum cryptography: Public key distribution and coin tossing”

Daniel Gottesman, Isaac Chuang, “Quantum Digital Signatures”

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

Discussions……

Thank you!