Download - BB84
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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 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
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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
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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
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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
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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
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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
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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 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
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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”
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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.
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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
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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 classical channel
Expensive for long keys: Ω(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 managementNo danger that public key is compromised
Certificate authorities
New protocolsDigital signature
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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
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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 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
|,|
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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
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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
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Discussions……
Thank you!