![Intro. Cont. Var. Information Theory CVQKD XP Next
Field quadratures
Classical field
Electromagnetic field
described by QA and PA
E(t) = QA cos ωt + PA sin ωt
Quantum description
Q and P do not commute:
[Q, P] ∝ i .
Add a
“quantum noise”:
Q = QA + BQ et P = PA + BP
Heisenberg =⇒ ∆BQ ∆BP ≥ 1](https://image.slidesharecdn.com/continuousvariablesquantumcryptography-121214215346-phpapp01/85/Continuous-variables-quantum-cryptography-21-320.jpg)
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Continuous variables quantum cryptography
- 1. Intro. Cont. Var. Information Theory CVQKD XP Next Continuous Variable Quantum Cryptography Towards High Speed Quantum Cryptography Frédéric Grosshans CNRS / ENS Cachan Palacký University, Olomouc, 2011
- 2. Intro. Cont. Var. Information Theory CVQKD XP Next 1 Introduction Prefect Secrecy and Quantum Cryptography Various Secure Systems 2 Continuous variables Field quadratures Homodyne Detection : Theory 3 Information Theory XXth century CVQKD Where are the bits ? 4 Continuous Variable Quantum Key Distribution Spying Protocols 5 Experimental systems 1st and 2nd generation demonstrators Key-Rates Integration with classical cryptography 6 Open problems
- 3. Intro. Cont. Var. Information Theory CVQKD XP Next Conditions for Perfect Secrecy Alice sends a secret message to Bob
- 4. Intro. Cont. Var. Information Theory CVQKD XP Next Conditions for Perfect Secrecy Alice sends a secret message to Bob through a channel observed by Eve.
- 5. Intro. Cont. Var. Information Theory CVQKD XP Next Conditions for Perfect Secrecy Alice sends a secret message to Bob through a channel observed by Eve. She encrypts the message with a secret key
- 6. Intro. Cont. Var. Information Theory CVQKD XP Next Conditions for Perfect Secrecy Alice sends a secret message to Bob through a channel observed by Eve. She encrypts the message with a secret key as long as the message.
- 7. Intro. Cont. Var. Information Theory CVQKD XP Next Quantum Key Distribution Alice sends quantum objects to Bob
- 8. Intro. Cont. Var. Information Theory CVQKD XP Next Quantum Key Distribution Alice sends quantum objects to Bob Eve’s Measurenents
- 9. Intro. Cont. Var. Information Theory CVQKD XP Next Quantum Key Distribution Alice sends quantum objects to Bob Eve’s Measurenents ⇒ measurable perturbations ⇒ secret key generation
- 10. Intro. Cont. Var. Information Theory CVQKD XP Next Unconditionnally Secure Systems . . . Single Photon QKD Long Range (∼ 100 km) Low rate (kbit/s)
- 11. Intro. Cont. Var. Information Theory CVQKD XP Next Unconditionnally Secure Systems . . . Single Photon QKD Long Range (∼ 100 km) Low rate (kbit/s) maybe a few Mbit/s in the long run
- 12. Intro. Cont. Var. Information Theory CVQKD XP Next Unconditionnally Secure Systems . . . Single Photon QKD Long Range (∼ 100 km) Low rate (kbit/s) maybe a few Mbit/s in the long run Classical One-Time-Pad Very Long Range (Paris–Olomouc) Not so small rate :
- 13. Intro. Cont. Var. Information Theory CVQKD XP Next Unconditionnally Secure Systems . . . Single Photon QKD Long Range (∼ 100 km) Low rate (kbit/s) maybe a few Mbit/s in the long run Classical One-Time-Pad Very Long Range (Paris–Olomouc) Not so small rate : 1 CD / year = 180 bits/s
- 14. Intro. Cont. Var. Information Theory CVQKD XP Next Unconditionnally Secure Systems . . . Single Photon QKD Long Range (∼ 100 km) Low rate (kbit/s) maybe a few Mbit/s in the long run Classical One-Time-Pad Very Long Range (Paris–Olomouc) Not so small rate : 1 CD / year = 180 bits/s 1 iPod (160 GB)/ year = 40 kbit/s
- 15. Intro. Cont. Var. Information Theory CVQKD XP Next Unconditionnally Secure Systems . . . Single Photon QKD Long Range (∼ 100 km) Low rate (kbit/s) maybe a few Mbit/s in the long run Classical One-Time-Pad Very Long Range (Paris–Olomouc) Not so small rate : 1 CD / year = 180 bits/s 1 iPod (160 GB)/ year = 40 kbit/s But the data has to stay here
- 16. Intro. Cont. Var. Information Theory CVQKD XP Next . . . and Continuous Variable Medium Range :∼ 25 km Medium Rate :∼ a few kbit/s
- 17. Intro. Cont. Var. Information Theory CVQKD XP Next . . . and Continuous Variable Medium Range :∼ 25 km Medium Rate :∼ a few kbit/s Much less mature
- 18. Intro. Cont. Var. Information Theory CVQKD XP Next . . . and Continuous Variable Medium Range :∼ 25 km ; 80 km soon ? Medium Rate :∼ a few kbit/s ; Mbits/s soon ? Much less mature ⇒ Much room for improvements
- 19. Intro. Cont. Var. Information Theory CVQKD XP Next 1 Introduction Prefect Secrecy and Quantum Cryptography Various Secure Systems 2 Continuous variables Field quadratures Homodyne Detection : Theory 3 Information Theory XXth century CVQKD Where are the bits ? 4 Continuous Variable Quantum Key Distribution Spying Protocols 5 Experimental systems 1st and 2nd generation demonstrators Key-Rates Integration with classical cryptography 6 Open problems
- 20. Intro. Cont. Var. Information Theory CVQKD XP Next Field quadratures Classical field Electromagnetic field described by QA and PA E(t) = QA cos ωt + PA sin ωt
- 21. Intro. Cont. Var. Information Theory CVQKD XP Next Field quadratures Classical field Electromagnetic field described by QA and PA E(t) = QA cos ωt + PA sin ωt Quantum description Q and P do not commute: [Q, P] ∝ i . Add a “quantum noise”: Q = QA + BQ et P = PA + BP Heisenberg =⇒ ∆BQ ∆BP ≥ 1
- 22. Intro. Cont. Var. Information Theory CVQKD XP Next 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 cos ϕ + Psignal sin ϕ)
- 23. Intro. Cont. Var. Information Theory CVQKD XP Next 1 Introduction Prefect Secrecy and Quantum Cryptography Various Secure Systems 2 Continuous variables Field quadratures Homodyne Detection : Theory 3 Information Theory XXth century CVQKD Where are the bits ? 4 Continuous Variable Quantum Key Distribution Spying Protocols 5 Experimental systems 1st and 2nd generation demonstrators Key-Rates Integration with classical cryptography 6 Open problems
- 24. Intro. Cont. Var. Information Theory CVQKD XP Next XXth century CVQKD At the end of XXth century it was obvious that a generalization of QKD to continuous variables could be interesting. Problem : discrete bits continuous variable
- 25. Intro. Cont. Var. Information Theory CVQKD XP Next XXth century CVQKD At the end of XXth century it was obvious that a generalization of QKD to continuous variables could be interesting. Problem : discrete bits continuous variable Adapting BB84? Mark Hillery, “Quantum Cryptography with Squeezed States”, arXiv:quant-ph/9909006/PRA 61 022309
- 26. Intro. Cont. Var. Information Theory CVQKD XP Next XXth century CVQKD At the end of XXth century it was obvious that a generalization of QKD to continuous variables could be interesting. Problem : discrete bits continuous variable Natural modulation + information theory! Nicolas J. Cerf, Marc Lévy, Gilles Van Assche : “Quantum distribution of Gaussian keys using squeezed states”, arXiv:quant-ph/0008058/PRL 63 052311
- 27. Intro. Cont. Var. Information Theory CVQKD XP Next Where are the bits ? Quite frequent discussion with discrete quantum cryptographers : 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.”
- 28. Intro. Cont. Var. Information Theory CVQKD XP Next Where are the bits ? Quite frequent discussion with discrete quantum cryptographers : DQC : How do you encode a 0 or a 1 in CVQKD? Me : Gilles/Jérôme/Anthony/Sébastien developed a really efficient code, using sliced reconciliation/LDPC matrices/R8 rotations and octonions. Only he knows how it works.
- 29. Intro. Cont. Var. Information Theory CVQKD XP Next Where are the bits ? Quite frequent discussion with discrete quantum cryptographers : DQC : How do you encode a 0 or a 1 in CVQKD? Me : Gilles/Jérôme/Anthony/Sébastien developed a really efficient code, using sliced reconciliation/LDPC matrices/R8 rotations and octonions. Only he knows how it works. Computation of the ideal code performance is easy !
- 30. Intro. Cont. Var. Information Theory CVQKD XP Next They’re hidden Availaible informationin a continuous signal
- 31. Intro. Cont. Var. Information Theory CVQKD XP Next They’re hidden Availaible informationin a continuous signal Differential entropy H(X) = − P(x) dx log P(x) dx dx P(x) log P(x) − log dx H(X) constante
- 32. Intro. Cont. Var. Information Theory CVQKD XP Next They’re hidden Availaible informationin a continuous signal with noise ? Differential entropy H(X) = − P(x) dx log P(x) dx H(X) = log ∆X + constante dx P(x) log P(x) − log dx H(X) constante
- 33. Intro. Cont. Var. Information Theory CVQKD XP Next They’re hidden Availaible informationin a continuous signal with noise ? Differential entropy Mutual information I(X : Y) = H(Y) − H(Y|X) H(X) = − P(x) dx log P(x) dx = H(Y) − H(Y|X) dx P(x) log P(x) − log dx ∆Y2 = 1 log H(X) constante 2 ∆Y2 |X
- 34. Intro. Cont. Var. Information Theory CVQKD XP Next 1 Introduction Prefect Secrecy and Quantum Cryptography Various Secure Systems 2 Continuous variables Field quadratures Homodyne Detection : Theory 3 Information Theory XXth century CVQKD Where are the bits ? 4 Continuous Variable Quantum Key Distribution Spying Protocols 5 Experimental systems 1st and 2nd generation demonstrators Key-Rates Integration with classical cryptography 6 Open problems
- 35. Intro. Cont. Var. Information Theory CVQKD XP Next The spy’s power Heisenberg : ∆BEve ∆BBob ≥ 1
- 36. Intro. Cont. Var. Information Theory CVQKD XP Next The spy’s power Heisenberg : ∆BEve ∆BBob ≥ 1 ⇒ ∆BBob gives IEve IBob
- 37. Intro. Cont. Var. Information Theory CVQKD XP Next Quantum Key Distribution Protocols Channel Evauation (noise measure) Alice&Bob evaluate IEve
- 38. Intro. Cont. Var. Information Theory CVQKD XP Next Quantum Key Distribution Protocols Channel Evauation (noise measure) Alice&Bob evaluate IEve Reconciliation (error correction) Alice&Bob share IBob identical bits. Ève knows IEve .
- 39. Intro. Cont. Var. Information Theory CVQKD XP Next 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 Amplification Alice&Bob share IBob − IEve identical bits. Ève knows ∼ 0.
- 40. Intro. Cont. Var. Information Theory CVQKD XP Next Theoretical Progresses in the last 10 years We went from a protocol using squeezed states, insecure beyond 50% losses (15 km), proved secure against Gaussian individual attack
- 41. Intro. Cont. Var. Information Theory CVQKD XP Next Theoretical Progresses in the last 10 years We went from a protocol using squeezed states, insecure beyond 50% losses (15 km), proved secure against Gaussian individual attack to a protocol using coherent states
- 42. Intro. Cont. Var. Information Theory CVQKD XP Next Theoretical Progresses in the last 10 years We went from a protocol using squeezed states, insecure beyond 50% losses (15 km), proved secure against Gaussian individual attack to a protocol using coherent states with no fundamental range limit proved secure against collective attacks
- 43. Intro. Cont. Var. Information Theory CVQKD XP Next Theoretical Progresses in the last 10 years We went from a protocol using squeezed states, insecure beyond 50% losses (15 km), proved secure against Gaussian individual attack to a protocol using coherent states with no fundamental range limit proved secure against collective attacks likely secure against coherent attacks
- 44. Intro. Cont. Var. Information Theory CVQKD XP Next Theoretical Progresses in the last 10 years We went from a protocol using squeezed states, insecure beyond 50% losses (15 km), proved secure against Gaussian individual attack to a protocol using coherent states with no fundamental range limit proved secure against collective attacks likely secure against coherent attacks and experimentally working
- 45. Intro. Cont. Var. Information Theory CVQKD XP Next 1 Introduction Prefect Secrecy and Quantum Cryptography Various Secure Systems 2 Continuous variables Field quadratures Homodyne Detection : Theory 3 Information Theory XXth century CVQKD Where are the bits ? 4 Continuous Variable Quantum Key Distribution Spying Protocols 5 Experimental systems 1st and 2nd generation demonstrators Key-Rates Integration with classical cryptography 6 Open problems
- 46. Intro. Cont. Var. Information Theory CVQKD XP Next 1st generation demonstrator F. Grosshans et. al., Nature (2003) & Brevet US m Key rate 75 kbit/s 3.1 dB (51%) losses 1.7 Mbit/s without losses
- 47. Intro. Cont. Var. Information Theory CVQKD XP Next Integrated system
- 48. Intro. Cont. Var. Information Theory CVQKD XP Next Integrated system
- 49. Intro. Cont. Var. Information Theory CVQKD XP Next Integrated system
- 50. Intro. Cont. Var. Information Theory CVQKD XP Next Key-Rates
- 51. Intro. Cont. Var. Information Theory CVQKD XP Next Key-Rates SECOQC Performance 100 kb/s (2008) L Exce osses o ss n n 95% oise ly effi 90 cie % nt c eff ode ici 10 kb/s Slo en t w co co de de 1 kb/s 0k 10 20 30 40 50 km km km km km m
- 52. Intro. Cont. Var. Information Theory CVQKD XP Next Key-Rates
- 53. Intro. Cont. Var. Information Theory CVQKD XP Next Key-Rates SECOQC Performance 100 kb/s (2008) L Exce osses o ss n n 95% oise ly Increases modulation rate : effi 90 cie ×10 easy, ×100 doable % nt c eff ode ici 10 kb/s Slo en t w co co de de use modern codes : ocotonion based protocol us +multi-edge LDPC codes eG + repetition codes PU s 1 kb/s 0k 10 20 30 40 50 km km km km km m
- 54. Intro. Cont. Var. Information Theory CVQKD XP Next Integration with classical cryptography
- 55. Intro. Cont. Var. Information Theory CVQKD XP Next Integration with classical cryptography
- 56. Intro. Cont. Var. Information Theory CVQKD XP Next 1 Introduction Prefect Secrecy and Quantum Cryptography Various Secure Systems 2 Continuous variables Field quadratures Homodyne Detection : Theory 3 Information Theory XXth century CVQKD Where are the bits ? 4 Continuous Variable Quantum Key Distribution Spying Protocols 5 Experimental systems 1st and 2nd generation demonstrators Key-Rates Integration with classical cryptography 6 Open problems
- 57. Intro. Cont. Var. Information Theory CVQKD XP Next Open Problems Finite size effects Link with post-selection based protocols (.de, .au) Side-channels and quantum hacking Other cryptographic applications