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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!