A Tight High-Order Entropic Quantum Uncertainty Relation with Applications Serge Fehr, Christian...

A Tight High-Order Entropic Quantum Uncertainty Relationwith Applications Serge Fehr, Christian Schaffner (CWI Amsterdam, NL)

Renato Renner (ETH Zürich, CH)

Ivan Damgård, Louis Salvail (University of Århus, DK)

QIP 2008, Delhi, India

Thursday, December 20th 2007

Serge Fehr, Christian Schaffner (CWI Amsterdam, NL)

Ivan Damgård, Louis Salvail (University of Århus, DK)

Secure Identification and QKD in the Bounded-Quantum-Storage Model

QIP 2008, Delhi, India

Thursday, December 20th 2007

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~1970: Birth of Quantum Cryptography

…

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1-2 Oblivious Transfer

SC

S0;S1C 2 f0;1g

complete for 2-party computation impossible in the plain (quantum) model possible in the Bounded-Quantum-Storage Model

1-2OT

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(Randomized) 1-2 Oblivious Transfer

Rand1-2OT SC

S0;S1C 2 f0;1g

complete for 2-party computation impossible in the plain (quantum) model possible in the Bounded-Quantum-Storage Model

6 / 42

Outline

Motivation and Notation

Quantum Uncertainty Relations

Secure Identification

Man-In-The-Middle Attacks

Conclusion

7 / 42

Quantum Mechanics Notation

Measurements:

+ basis

£ basis

j0i+ j1i+

j1i£j0i£

EPR pairs:

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get X'

0

0

1

1

0

Quantum 1-2 OT ProtocoljX i£

S0;S1

F1 2R F 1F0 2R F 0

£ ;F0;F1

S1 = ?S0

C = 0£ 0= +n

Correctness

Receiver-Security against Dishonest Alice

£ 2R X 2R sendf+;£gn f0;1gn

0

1

1

1

0

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Sender-Security?jX i£

S0;S1

F1 2R F 1F0 2R F 0

£ ;F0;F1

get

½


0

1

1

1

0

Sender-Security: one of the strings looks completely random to dishonest Bob

# qubits < n=4

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get

Entanglement-Based Protocol

S0;S1

F1 2R F 1F0 2R F 0

£ ;F0;F1

½

epr n

# qubits < n=4

Sender-Security: One of the strings looks completely random to dishonest Bob


?

?

?

?

?

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get

Entanglement-Based ProtocoljX i£

S0;S1

F1 2R F 1F0 2R F 0

£ ;F0;F1

½

# qubits < n=4



0

1

1

1

0

12 / 42



?

?

?

?

?

Let Bob Act First

F1 2R F 1F0 2R F 0

½

£ ;F0;F1

getepr n

//...

# qubits < n=4

S0;S1

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Privacy Amplification:

get£ 2R X 2R send

f+;£gn f0;1gn

?

?

?

?

?

Sender-Security Uncertainty Relation

F1 2R F 1F0 2R F 0

½

£ ;F0;F1

epr n

//...

# qubits < n=4

[Renner KÄonig 05, Renner 06]

H1 (X j £ ) ¸ ?S0;S1

H1 (X j £)| {z }

¸ ?

> # qubits| {z }

<n=4

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Outline


Quantum Uncertainty Relations



Conclusion

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M aassen U±nk 88: Let ½i be a 1-qubit state.£ i 2R f+;£g, X i the outcome of measuring ½i in basis £ i . Then,

H(X i j £ i ) = 12

¡H(X i j £ i = +) + H(X i j £ i = £)| {z }

¸ 1

¢¸ 1

2:

Entropic Uncertainty Relation for One Qubit

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In general: H(¢) ¸ H1 (¢)

) H"1 (X n j £ )

n! 1¼ n ¢H(X i j £ i ) ¸ n=2

£ 2R statef+;£gn ½

...

Quantum Uncertainty Relation needed

//

H1 (X j £) ¸ ?

X i independent

H(X i j £ i ) ¸ 12


H(X i j £ i ) = 12

¡H(X i j £ i = +) + H(X i j £ i = £)| {z }

¸ 1

¢¸ 1

2:

except with prob · "

X i := X 1 : : :X i

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£ 2R statef+;£gn ½

...

Main Result

//

H1 (X j £) ¸ ?

X i dependent

H(X i j £ i ) ¸ 12

Quantum Uncertainty R elation: LetX = (X 1; : : : ;X n) be the outcome. Then,

H"1 (X j £ ) & n=2

with " negligible in n.

H(X i j £ i ;X i ¡ 1 = xi ¡ 1;£ i ¡ 1 = µi ¡ 1) ¸ 12


H(X i j £ i ) = 12

¡H(X i j £ i = +) + H(X i j £ i = £)| {z }

¸ 1

¢¸ 1

2:

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Main Technical Lemma

Z1; : : : ;Zn (dependent) random variables

Then, H"1 (Z) & n ¢h with " negligible in n

with H(Zi j Z i ¡ 1 = zi ¡ 1) ¸ h.

P roof:

² information theory

² generalized Cherno®bound (A zuma inequality)

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conjugate coding / BB84:£ 2R state

f+;£gn ½

...

classical technical lemma:

instantiate it for various quantum codings:

High-Order Entropic Uncertainty Relations

H(Zi j Z i ¡ 1 = z) ¸ h ) H"1 (Zn) & hn

//

H"1 (X j £) & n=2

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conjugate coding / BB84:

three bases / six-state:

…

classical technical lemma:

instantiate it for various quantum codings:

High-Order Entropic Uncertainty Relations

//

//

£ 2R statef+;£;ª gn ½

...

H"1 (X j £) & n=2

H"1 (X j £) & 2

3n

H(Zi j Z i ¡ 1 = z) ¸ h ) H"1 (Zn) & hn

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Outline


Quantum Uncertainty Relation



Conclusion

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Why Secure Identification?

I’m Alice my PIN is IMAB52

I want $25

Alright Alice, here you go.

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Why Secure Identification?

I’m Alice my PIN is IMAB52

I want $25

Sorry, I’m out of order

Alice: IMAB52

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Why Secure Identification?Alice: IMAB52

I’m Alice my PIN is IMAB52I want $25,000,000

Alright Alice, here you go.

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Secure Evaluation of the Equality

PIN-based identification scheme should be a secure evaluation of the equality function

A dishonest player can exclude only one possible password

=?WA WB

WA?= WBWA

?= WB

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get X'

0

0

1

1

0

Recall: Quantum 1-2 OT ProtocoljX i£

S0;S1

F1 2R F 1F0 2R F 0

£ ;F0;F1

S1 = ?S0

C = 0£ 0= +n


0

1

1

1

0

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C(0) X' C(1) X '

0 0

0 1

1 1

1 0

0 0

0 0

1 0

1 1

0 1

0 0


0

1

1

1

0

0

1

1

1

0

Quantum 1-m OT ProtocoljX i£

W 2 W

C(W)

Code C:W ! f+;£gn

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C(0) X' C(1) X '

0 0

0 1

1 1

1 0

0 0

0 0

1 0

1 1

0 1

0 0


0

1

1

1

0

0

1

1

1

0


S0

W 2 W

C(W)

Code C:W ! f+;£gn

S0

£ ;F0

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C(0) X' C(1) X '

0 0

0 1

1 1

1 0

0 0

0 0

1 0

1 1

0 1

0 0


0

1

1

1

0

0

1

1

1

0


S0

W 2 W

C(W)

S1

£ ;F0;F1; : : :

S0;S1; : : :

Code C:W ! f+;£gn

£ ;F0;F1; : : :

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Idea

correct

dishonest Bob can learn at most one SW and compare it with

(the random) SW . He can at most exclude his W.

Dishonest Alice learns nothing about W by the OT

but can e.g. set all Si = 0 , then send SW = 0

Rand1-mOTS0;S1; : : : ;Sm¡ 1

W 2 f0;1;: : : ;m¡ 1gW 2 f0;1;: : : ;m¡ 1g

W

SW

SWSW

?= SW

W 2 f0;1;: : : ;m¡ 1gW 2 f0;1;: : : ;m¡ 1gW 2 f0;1;: : : ;m¡ 1g

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TW := g(W) ©SW

Dishonest Alice learns nothing about W by the OT

with high probability: all Ti are different

Alice has to guess the right TW

Dishonest Alice

SW

g g 2R G : W ! f0;1g̀

TW?= g(W) ©SW

strongly universal

Rand1-mOT

W 2 f0;1;: : : ;m¡ 1g

W

S0;S1; : : : ;Sm¡ 1

W = ?

)

SW?= SW

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Properties of the Basic Scheme

efficient: n qubits, 3 classical messages, honest players do not require quantum memory

provably secure against: unbounded dishonest user Alice dishonest server Bob with quantum memory < n/11 both have unbounded computing power and

classical memory can be extended to tolerate noise, therefore

implementable with current technology

OTPin WPin W

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Outline


Quantum Uncertainty Relation



Conclusion

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Dishonest Alice or Bob can only exclude one possible PIN W

Eve learns nothing about W ?

Extension: Man-in-the-Middle AttackPin W Pin W

nontrivialKeyK KeyK

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Dishonest Alice or Bob can only exclude one possible PIN W, even given key K

Eve learns nothing about W and

only negligible information about key K

Extension: Man-in-the-Middle Attack Pin W Pin W

KeyK KeyK

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Application: QKD

authK (¢)

= (K 0;SK ) = (K 0;SK )

authK 0(¢)

KeyK KeyK

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Problem: Eve interferes with communication

authK (¢)

= (K 0;SK ) = (K 0;SK )

authK 0(¢) honest players run out of authentication key secure QKD no longer possible

KeyK KeyK

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Solution: QKD based on Secure Identification

authK (¢)

Pin W Pin W

=? =? K and W remain secure, can be re-used over and over again,

even if scheme is disrupted Losing key K does not compromise security

(if loss is noticed) BUT: security holds only against a

quantum-memory bounded eavesdropper

KeyK KeyK

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High-order entropic quantum uncertainty relations :

Bounded-Quantum-Storage Model:

Oblivious Transfer: Non-interactive, practical protocols for 1-2 OT and Bit Commitment secure according to strong security definitions.

Secure PIN-based Identification: ditonon-trivial extension to handle man-in-the-middle attacks

Conclusion

H"1 (X j £) ¸ 2

3nH"1 (X j £) ¸ n=2,

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High-order entropic quantum uncertainty relations:

Quantum Key Distribution: (assuming quantum-memory bounded Eve)

security proofs: for higher error rates (=longer distances)

robustness: Eve cannot make parties run out of authentication key

Conclusion

H"1 (X j £) ¸ 2

3nH"1 (X j £) ¸ n=2,

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Thanks to

you!

Follow-up work

[Wehner, Terhal, S]: a step closer to reality:Cryptography from Noisy Photonic Storage

see poster and http://arxiv.org/0711.2895

A Tight High-Order Entropic Quantum Uncertainty Relation with Applications Serge Fehr, Christian...

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