Continuous variables quantum cryptography

Intro. Cont. Var. Information Theory CVQKD XP Next

Continuous VariableQuantum Cryptography

Towards High Speed Quantum Cryptography

Frédéric Grosshans

CNRS / ENS Cachan

Palacký University, Olomouc, 2011

1 IntroductionPrefect Secrecy and Quantum CryptographyVarious Secure Systems

2 Continuous variablesField quadraturesHomodyne Detection : Theory

3 Information TheoryXXth century CVQKDWhere are the bits ?

4 Continuous Variable Quantum Key DistributionSpyingProtocols

5 Experimental systems1st and 2nd generation demonstratorsKey-RatesIntegration with classical cryptography

6 Open problems

Conditions for Perfect Secrecy

Alice sends a secret message to Bob

through a channel observed by Eve.

She encrypts the message with a secret keyas long as the message.

Alice sends a secret message to Bobthrough a channel observed by Eve.

She encrypts the message with a secret key

as long as the message.

Quantum Key Distribution

Alice sends quantum objects to Bob

Eve’s Measurenents⇒measurable perturbations⇒ secret key generation

Eve’s Measurenents

⇒measurable perturbations⇒ secret key generation

Eve’s Measurenents⇒measurable perturbations⇒ secret key generation

Unconditionnally Secure Systems . . .

Single Photon QKD

I Long Range (∼ 100 km)I Low rate (kbit/s)

maybe a few Mbit/s in the long run

Classical One-Time-PadI Very Long Range (Paris–Olomouc)I Not so small rate :

1 CD / year = 180 bits/s1 iPod (160 GB)/ year = 40 kbit/s

I But the data has to stay here

Single Photon QKD

I Long Range (∼ 100 km)I Low rate (kbit/s) maybe a few Mbit/s in the long run

Single Photon QKD

1 CD / year = 180 bits/s

1 iPod (160 GB)/ year = 40 kbit/s

Single Photon QKD

. . . and Continuous Variable

I Medium Range :∼ 25 km

; 80 km soon ?

I Medium Rate :∼ a few kbit/s

; Mbits/s soon ?I Much less mature

⇒ Much room for improvements

I Medium Range :∼ 25 km

; 80 km soon ?

I Medium Rate :∼ a few kbit/s

; Mbits/s soon ?

I Much less mature

⇒ Much room for improvements

I Medium Range :∼ 25 km ; 80 km soon ?I Medium Rate :∼ a few kbit/s ; Mbits/s soon ?I Much less mature⇒ Much room for improvements

6 Open problems

Field quadratures

Classical fieldElectromagnetic fielddescribed by QA and PAE(t) = QA cosωt + PA sinωt

Quantum descriptionQ and P do not commute:

[Q,P] ∝ i~.Add a

“quantum noise”:Q = QA + BQ et P = PA + BP

Heisenberg =⇒ ∆BQ∆BP ≥ 1

Field quadratures

Classical fieldElectromagnetic fielddescribed by QA and PAE(t) = QA cosωt + PA sinωt

Quantum descriptionQ and P do not commute:

[Q,P] ∝ i~.Add a

“quantum noise”:Q = QA + BQ et P = PA + BP

Heisenberg =⇒ ∆BQ∆BP ≥ 1

Homodyne Detection : Theory

Photocurrents:

i± ∝ (Eosc.(t) ± Esignal(t))2

∝ Eosc.(t)2± 2Eosc.(t)Esignal(t)

after substraction:

δi ∝ Eosc.(t)Esignal(t)

∝ Eosc.(Qsignal cosϕ + Psignal sinϕ)

6 Open problems

XXth century CVQKD

At the end of XXth century it was obvious that ageneralization of QKD to continuous variables could beinteresting.Problem : discrete bits , continuous variable

XXth century CVQKD

Adapting BB84?Mark Hillery, “Quantum Cryptography withSqueezed States”,arXiv:quant-ph/9909006/PRA 61 022309

XXth century CVQKD

Natural modulation + information theory!Nicolas J. Cerf, Marc Lévy, Gilles VanAssche : “Quantum distribution of Gaussiankeys using squeezed states”,arXiv:quant-ph/0008058/PRL 63 052311

Where are the bits ?

Quite frequent discussion with discrete quantumcryptographers :

DQC : How do you encode a 0 or a 1 in CVQKD?Me : I don’t care, C. E. Shannon tells me

“∀ε > 0,∃ code of rate I − ε.”

Computation of the ideal code performance is easy !

DQC : How do you encode a 0 or a 1 in CVQKD?Me : Gilles/Jérôme/Anthony/Sébastien developed

a really efficient code, using slicedreconciliation/LDPC matrices/R8 rotations andoctonions. Only he knows how it works.

DQC : How do you encode a 0 or a 1 in CVQKD?Me : Gilles/Jérôme/Anthony/Sébastien developed

a really efficient code, using slicedreconciliation/LDPC matrices/R8 rotations andoctonions. Only he knows how it works.

They’re hidden

Availaible informationin a continuous signal

with noise ?

Differential entropy

H(X) = −∑P(x) dx logP(x) dx

∫dxP(x) logP(x)︸︷︷︸

− log dx︸︷︷︸constante

They’re hidden

Availaible informationin a continuous signal

with noise ?

They’re hidden

Availaible informationin a continuous signalwith noise ?

H(X) = log ∆X + constante

They’re hidden

Availaible informationin a continuous signalwith noise ?

Mutual information

I(X : Y) = H(Y) −H(Y|X)= H(Y) −H(Y|X)

= 12 log

∆Y2|X

6 Open problems

The spy’s power

Heisenberg :∆BEve∆BBob ≥ 1

⇒ ∆BBob gives I IEve

I IBob

The spy’s power

Heisenberg :∆BEve∆BBob ≥ 1

⇒ ∆BBob gives I IEve

I IBob

Quantum Key Distribution Protocols

Channel Evauation (noise measure)

Alice&Bob evaluate IEve

Reconciliation (error correction)

Alice&Bob share IBob identical bits.Ève knows IEve.

Privacy AmplificationAlice&Bob share IBob − IEve identical bits.

Ève knows ∼ 0.

Theoretical Progresses in the last 10 years

We went from a protocolI using squeezed states,I insecure beyond 50% losses (15 km),I proved secure against Gaussian individual attack

to a protocol

I using coherent statesI with no fundamental range limitI proved secure against collective attacksI likely secure against coherent attacksI and experimentally working

to a protocolI using coherent states

I with no fundamental range limitI proved secure against collective attacksI likely secure against coherent attacksI and experimentally working

to a protocolI using coherent statesI with no fundamental range limitI proved secure against collective attacks

I likely secure against coherent attacksI and experimentally working

to a protocolI using coherent statesI with no fundamental range limitI proved secure against collective attacksI likely secure against coherent attacks

I and experimentally working

to a protocolI using coherent statesI with no fundamental range limitI proved secure against collective attacksI likely secure against coherent attacksI and experimentally working

6 Open problems

1st generation demonstratorF. Grosshans et. al., Nature (2003) & Brevet US

Key rate I 75 kbit/s 3.1 dB (51%) lossesI 1.7 Mbit/s without losses

Integrated system

Key-Rates

Losses onlyExcess noise95% efficient code

90% efficient codeSlow code

100 kb/s

10 kb/s

1 kb/s 0 km

SECOQC Performance(2008)

Key-Rates

Losses onlyExcess noise95% efficient code

90% efficient codeSlow code

100 kb/s

10 kb/s

1 kb/s 0 km

SECOQC Performance(2008)

use GPUsIncr

, ×10

use modern codes :ocotonion based protocol+multi-edge LDPC codes

+ repetition codes

Integration with classical cryptography

6 Open problems

Open Problems

I Finite size effectsI Link with post-selection based protocols (.de, .au)I Side-channels and quantum hackingI Other cryptographic applications

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