Abstract

We investigate the possibility of using multiple-scattering optical media, as resources of randomness in cryptographic tasks pertaining to commitments and auctions. The proposed commitment protocol exploits standard wavefront-shaping and heterodyne-detection techniques, and can be implemented with current technology. Its security is discussed in the framework of a tamper-resistant trusted setup.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  25. S. R. Huisman, T. J. Huisman, S. A. Goorden, A. P. Mosk, and P. W. H. Pinkse, “Programming balanced optical beam splitters in white paint,” Opt. Express 22(7), 8320–8332 (2014).
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  29. N. H. Y. Ng, S. K. Joshi, C. Chen Ming, C. Kurtsiefer, and S. Wehner, “Experimental implementation of bit commitment in the noisy-storage model,” Nat. Commun. 3(1), 1326 (2012).
    [Crossref]
  30. A. Kent, “Unconditionally secure bit commitment by transmitting measurement outcomes,” Phys. Rev. Lett. 109(13), 130501 (2012).
    [Crossref]
  31. Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
    [Crossref]
  32. T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
    [Crossref]
  33. S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programming linear quantum networks with a multimode fiber,” (2018). ArXiv:1802.07573.

2018 (2)

G. M. Nikolopoulos, “Continuous-variable quantum authentication of physical unclonable keys: Security against an emulation attack,” Phys. Rev. A 97(1), 012324 (2018).
[Crossref]

C. Mesaritakis, M. Akriotou, A. Kapsalis, E. Grivas, C. Chaintoutis, T. Nikas, and D. Syvridis, “Physical unclonable function based on a multi-mode optical waveguide,” Sci. Rep. 8(1), 9653 (2018).
[Crossref]

2017 (2)

G. M. Nikolopoulos and E. Diamanti, “Continuous-variable quantum authentication of physical unclonable keys,” Sci. Rep. 7(1), 46047 (2017).
[Crossref]

M. Herrero-Collantes and J. C. Garcia-Escartin, “Quantum random number generators,” Rev. Mod. Phys. 89(1), 015004 (2017).
[Crossref]

2016 (3)

H. Zhang and S. Tzortzakis, “Robust authentication through stochastic femtosecond laser filament induced scattering surfaces,” Appl. Phys. Lett. 108(21), 211107 (2016).
[Crossref]

A. Broadbent and C. Schaffner, “Quantum cryptography beyond quantum key distribution,” Des. Codes Cryptogr. 78(1), 351–382 (2016).
[Crossref]

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2(1), e1501054 (2016).
[Crossref]

2015 (1)

2014 (5)

H. Defienne, M. Barbieri, B. Chalopin, B. Chatel, I. A. Walmsley, B. J. Smith, and S. Gigan, “Nonclassical light manipulation in a multiple-scattering medium,” Opt. Lett. 39(21), 6090–6093 (2014).
[Crossref]

S. A. Goorden, M. Horstmann, A. P. Mosk, B. Škorić, and P. W. H. Pinkse, “Quantum-secure authentication of a physical unclonable key,” Optica 1(6), 421–424 (2014).
[Crossref]

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

B. R. Anderson, R. Gunawidjaja, and H. Eilers, “Effect of experimental parameters on optimal transmission of light through opaque media,” Phys. Rev. A 90(5), 053826 (2014).
[Crossref]

S. R. Huisman, T. J. Huisman, S. A. Goorden, A. P. Mosk, and P. W. H. Pinkse, “Programming balanced optical beam splitters in white paint,” Opt. Express 22(7), 8320–8332 (2014).
[Crossref]

2013 (3)

H. Yilmaz, W. L. Vos, and A. P. Mosk, “Optimal control of light propagation through multiple-scattering media in the presence of noise,” Biomed. Opt. Express 4(9), 1759–1768 (2013).
[Crossref]

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

R. Horstmayer, B. Judkewitz, I. M. Vellekoop, S. Assawaworrarit, and C. Yan, “Physical key-protected one-time pad,” Sci. Rep. 3(1), 3543 (2013).
[Crossref]

2012 (4)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

C. H. Yeh, P. Y. Sung, C. H. Kuo, and R. N. Yeh, “Robust laser speckle recognition system for authenticity identification,” Opt. Express 20(22), 24382–24393 (2012).
[Crossref]

N. H. Y. Ng, S. K. Joshi, C. Chen Ming, C. Kurtsiefer, and S. Wehner, “Experimental implementation of bit commitment in the noisy-storage model,” Nat. Commun. 3(1), 1326 (2012).
[Crossref]

A. Kent, “Unconditionally secure bit commitment by transmitting measurement outcomes,” Phys. Rev. Lett. 109(13), 130501 (2012).
[Crossref]

2011 (1)

S. K. Poppoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13(12), 123021 (2011).
[Crossref]

2008 (1)

G. M. Nikolopoulos, “Applications of single-qubit rotations in quantum public-key cryptography,” Phys. Rev. A 77(3), 032348 (2008).
[Crossref]

2006 (1)

2005 (2)

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

P. Lodahl, A. P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95(17), 173901 (2005).
[Crossref]

2002 (1)

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297(5589), 2026–2030 (2002).
[Crossref]

Akriotou, M.

C. Mesaritakis, M. Akriotou, A. Kapsalis, E. Grivas, C. Chaintoutis, T. Nikas, and D. Syvridis, “Physical unclonable function based on a multi-mode optical waveguide,” Sci. Rep. 8(1), 9653 (2018).
[Crossref]

Allwood, D. A.

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

Amitonova, L. V.

L. V. Amitonova, T. B. H. Tentrup, I. M. Vellekoop, and P. W. H. Pinkse, “Quantum key establishment with a multimode fiber,” (2018). ArXiv:1801.07180.

Anderson, B. R.

B. R. Anderson, R. Gunawidjaja, and H. Eilers, “Effect of experimental parameters on optimal transmission of light through opaque media,” Phys. Rev. A 90(5), 053826 (2014).
[Crossref]

Assawaworrarit, S.

R. Horstmayer, B. Judkewitz, I. M. Vellekoop, S. Assawaworrarit, and C. Yan, “Physical key-protected one-time pad,” Sci. Rep. 3(1), 3543 (2013).
[Crossref]

Atkinson, D.

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

Barbieri, M.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2(1), e1501054 (2016).
[Crossref]

H. Defienne, M. Barbieri, B. Chalopin, B. Chatel, I. A. Walmsley, B. J. Smith, and S. Gigan, “Nonclassical light manipulation in a multiple-scattering medium,” Opt. Lett. 39(21), 6090–6093 (2014).
[Crossref]

Boccara, A. C.

S. K. Poppoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13(12), 123021 (2011).
[Crossref]

Broadbent, A.

A. Broadbent and C. Schaffner, “Quantum cryptography beyond quantum key distribution,” Des. Codes Cryptogr. 78(1), 351–382 (2016).
[Crossref]

Bryan, M. T.

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

Buchanan, J. D. R.

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

Bussières, F.

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

Cabello, A.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Cao, Y.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Chaintoutis, C.

C. Mesaritakis, M. Akriotou, A. Kapsalis, E. Grivas, C. Chaintoutis, T. Nikas, and D. Syvridis, “Physical unclonable function based on a multi-mode optical waveguide,” Sci. Rep. 8(1), 9653 (2018).
[Crossref]

Chalopin, B.

Chatel, B.

Chen, T.-Y.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Chen Ming, C.

N. H. Y. Ng, S. K. Joshi, C. Chen Ming, C. Kurtsiefer, and S. Wehner, “Experimental implementation of bit commitment in the noisy-storage model,” Nat. Commun. 3(1), 1326 (2012).
[Crossref]

Cowburn, R. P.

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

Cui, K.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Curty, M.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Defienne, H.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2(1), e1501054 (2016).
[Crossref]

H. Defienne, M. Barbieri, B. Chalopin, B. Chatel, I. A. Walmsley, B. J. Smith, and S. Gigan, “Nonclassical light manipulation in a multiple-scattering medium,” Opt. Lett. 39(21), 6090–6093 (2014).
[Crossref]

S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programming linear quantum networks with a multimode fiber,” (2018). ArXiv:1802.07573.

Diamanti, E.

G. M. Nikolopoulos and E. Diamanti, “Continuous-variable quantum authentication of physical unclonable keys,” Sci. Rep. 7(1), 46047 (2017).
[Crossref]

Eilers, H.

B. R. Anderson, R. Gunawidjaja, and H. Eilers, “Effect of experimental parameters on optimal transmission of light through opaque media,” Phys. Rev. A 90(5), 053826 (2014).
[Crossref]

Fenton, K.

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

Ferraro, A.

S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programming linear quantum networks with a multimode fiber,” (2018). ArXiv:1802.07573.

Fink, M.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

S. K. Poppoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13(12), 123021 (2011).
[Crossref]

Garcia-Escartin, J. C.

M. Herrero-Collantes and J. C. Garcia-Escartin, “Quantum random number generators,” Rev. Mod. Phys. 89(1), 015004 (2017).
[Crossref]

Gershenfeld, N.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297(5589), 2026–2030 (2002).
[Crossref]

Gigan, S.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2(1), e1501054 (2016).
[Crossref]

H. Defienne, M. Barbieri, B. Chalopin, B. Chatel, I. A. Walmsley, B. J. Smith, and S. Gigan, “Nonclassical light manipulation in a multiple-scattering medium,” Opt. Lett. 39(21), 6090–6093 (2014).
[Crossref]

S. K. Poppoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13(12), 123021 (2011).
[Crossref]

S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programming linear quantum networks with a multimode fiber,” (2018). ArXiv:1802.07573.

Gisin, N.

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

Goldreich, O.

O. Goldreich, Foundations of cryptography: Basic Techniques (Cambridge University, 2004).

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

Goorden, S. A.

Grivas, E.

C. Mesaritakis, M. Akriotou, A. Kapsalis, E. Grivas, C. Chaintoutis, T. Nikas, and D. Syvridis, “Physical unclonable function based on a multi-mode optical waveguide,” Sci. Rep. 8(1), 9653 (2018).
[Crossref]

Gunawidjaja, R.

B. R. Anderson, R. Gunawidjaja, and H. Eilers, “Effect of experimental parameters on optimal transmission of light through opaque media,” Phys. Rev. A 90(5), 053826 (2014).
[Crossref]

Herrero-Collantes, M.

M. Herrero-Collantes and J. C. Garcia-Escartin, “Quantum random number generators,” Rev. Mod. Phys. 89(1), 015004 (2017).
[Crossref]

Horstmann, M.

Horstmayer, R.

R. Horstmayer, B. Judkewitz, I. M. Vellekoop, S. Assawaworrarit, and C. Yan, “Physical key-protected one-time pad,” Sci. Rep. 3(1), 3543 (2013).
[Crossref]

Houlmann, R.

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

Huisman, S. R.

Huisman, T. J.

Innocenti, L.

S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programming linear quantum networks with a multimode fiber,” (2018). ArXiv:1802.07573.

Jausovec, A.

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

Joshi, S. K.

N. H. Y. Ng, S. K. Joshi, C. Chen Ming, C. Kurtsiefer, and S. Wehner, “Experimental implementation of bit commitment in the noisy-storage model,” Nat. Commun. 3(1), 1326 (2012).
[Crossref]

Judkewitz, B.

R. Horstmayer, B. Judkewitz, I. M. Vellekoop, S. Assawaworrarit, and C. Yan, “Physical key-protected one-time pad,” Sci. Rep. 3(1), 3543 (2013).
[Crossref]

Juffmann, T.

S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programming linear quantum networks with a multimode fiber,” (2018). ArXiv:1802.07573.

Kaniewski, J.

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

Kapsalis, A.

C. Mesaritakis, M. Akriotou, A. Kapsalis, E. Grivas, C. Chaintoutis, T. Nikas, and D. Syvridis, “Physical unclonable function based on a multi-mode optical waveguide,” Sci. Rep. 8(1), 9653 (2018).
[Crossref]

Kent, A.

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

A. Kent, “Unconditionally secure bit commitment by transmitting measurement outcomes,” Phys. Rev. Lett. 109(13), 130501 (2012).
[Crossref]

Kuo, C. H.

Kurtsiefer, C.

N. H. Y. Ng, S. K. Joshi, C. Chen Ming, C. Kurtsiefer, and S. Wehner, “Experimental implementation of bit commitment in the noisy-storage model,” Nat. Commun. 3(1), 1326 (2012).
[Crossref]

Lagendijk, A.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

P. Lodahl, A. P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95(17), 173901 (2005).
[Crossref]

Leedumrongwatthanakun, S.

S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programming linear quantum networks with a multimode fiber,” (2018). ArXiv:1802.07573.

Leonhardt, U.

U. Leonhardt, Essential Quantum Optics: From Quantum Measurements to Black Holes (Cambridge University, 2010).

Lerosey, G.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

S. K. Poppoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13(12), 123021 (2011).
[Crossref]

Li, D.-D.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Li, Y.-H.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Liao, S.-K.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Lin, Z.-H.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Liu, Y.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Lodahl, P.

P. Lodahl, “Quantum correlations induced by multiple scattering of quadrature squeezed light,” Opt. Express 14(15), 6919–6929 (2006).
[Crossref]

P. Lodahl, A. P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95(17), 173901 (2005).
[Crossref]

Lunghi, T.

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

Menezes, A.

A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography (CRC, 1996).

Mesaritakis, C.

C. Mesaritakis, M. Akriotou, A. Kapsalis, E. Grivas, C. Chaintoutis, T. Nikas, and D. Syvridis, “Physical unclonable function based on a multi-mode optical waveguide,” Sci. Rep. 8(1), 9653 (2018).
[Crossref]

Mosk, A. P.

Ng, N. H. Y.

N. H. Y. Ng, S. K. Joshi, C. Chen Ming, C. Kurtsiefer, and S. Wehner, “Experimental implementation of bit commitment in the noisy-storage model,” Nat. Commun. 3(1), 1326 (2012).
[Crossref]

Nikas, T.

C. Mesaritakis, M. Akriotou, A. Kapsalis, E. Grivas, C. Chaintoutis, T. Nikas, and D. Syvridis, “Physical unclonable function based on a multi-mode optical waveguide,” Sci. Rep. 8(1), 9653 (2018).
[Crossref]

Nikolopoulos, G. M.

G. M. Nikolopoulos, “Continuous-variable quantum authentication of physical unclonable keys: Security against an emulation attack,” Phys. Rev. A 97(1), 012324 (2018).
[Crossref]

G. M. Nikolopoulos and E. Diamanti, “Continuous-variable quantum authentication of physical unclonable keys,” Sci. Rep. 7(1), 46047 (2017).
[Crossref]

G. M. Nikolopoulos, “Applications of single-qubit rotations in quantum public-key cryptography,” Phys. Rev. A 77(3), 032348 (2008).
[Crossref]

Pan, J.-W.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Pappu, R.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297(5589), 2026–2030 (2002).
[Crossref]

Paternostro, M.

S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programming linear quantum networks with a multimode fiber,” (2018). ArXiv:1802.07573.

Peng, C.-Z.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Petit, D.

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

Pinkse, P. W. H.

Poppoff, S. K.

S. K. Poppoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13(12), 123021 (2011).
[Crossref]

Recht, B.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297(5589), 2026–2030 (2002).
[Crossref]

Schaffner, C.

A. Broadbent and C. Schaffner, “Quantum cryptography beyond quantum key distribution,” Des. Codes Cryptogr. 78(1), 351–382 (2016).
[Crossref]

Seem, P.

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

Škoric, B.

Smart, N. P.

N. P. Smart, Cryptography Made Simple (Springer International Publishing, 2014).

Smith, B. J.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2(1), e1501054 (2016).
[Crossref]

H. Defienne, M. Barbieri, B. Chalopin, B. Chatel, I. A. Walmsley, B. J. Smith, and S. Gigan, “Nonclassical light manipulation in a multiple-scattering medium,” Opt. Lett. 39(21), 6090–6093 (2014).
[Crossref]

Sun, Q.-C.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Sung, P. Y.

Syvridis, D.

C. Mesaritakis, M. Akriotou, A. Kapsalis, E. Grivas, C. Chaintoutis, T. Nikas, and D. Syvridis, “Physical unclonable function based on a multi-mode optical waveguide,” Sci. Rep. 8(1), 9653 (2018).
[Crossref]

Taylor, J.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297(5589), 2026–2030 (2002).
[Crossref]

Tentrup, T. B. H.

L. V. Amitonova, T. B. H. Tentrup, I. M. Vellekoop, and P. W. H. Pinkse, “Quantum key establishment with a multimode fiber,” (2018). ArXiv:1801.07180.

Tomamichel, M.

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

Tzortzakis, S.

H. Zhang and S. Tzortzakis, “Robust authentication through stochastic femtosecond laser filament induced scattering surfaces,” Appl. Phys. Lett. 108(21), 211107 (2016).
[Crossref]

van Oorschot, P.

A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography (CRC, 1996).

Vanstone, S.

A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography (CRC, 1996).

Vellekoop, I. M.

I. M. Vellekoop, “Feedback-based wavefront shaping,” Opt. Express 23(9), 12189–12206 (2015).
[Crossref]

R. Horstmayer, B. Judkewitz, I. M. Vellekoop, S. Assawaworrarit, and C. Yan, “Physical key-protected one-time pad,” Sci. Rep. 3(1), 3543 (2013).
[Crossref]

L. V. Amitonova, T. B. H. Tentrup, I. M. Vellekoop, and P. W. H. Pinkse, “Quantum key establishment with a multimode fiber,” (2018). ArXiv:1801.07180.

Vos, W. L.

Walmsley, I. A.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2(1), e1501054 (2016).
[Crossref]

H. Defienne, M. Barbieri, B. Chalopin, B. Chatel, I. A. Walmsley, B. J. Smith, and S. Gigan, “Nonclassical light manipulation in a multiple-scattering medium,” Opt. Lett. 39(21), 6090–6093 (2014).
[Crossref]

Wang, J.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Wehner, S.

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

N. H. Y. Ng, S. K. Joshi, C. Chen Ming, C. Kurtsiefer, and S. Wehner, “Experimental implementation of bit commitment in the noisy-storage model,” Nat. Commun. 3(1), 1326 (2012).
[Crossref]

Xiong, G.

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

Yan, C.

R. Horstmayer, B. Judkewitz, I. M. Vellekoop, S. Assawaworrarit, and C. Yan, “Physical key-protected one-time pad,” Sci. Rep. 3(1), 3543 (2013).
[Crossref]

Yeh, C. H.

Yeh, R. N.

Yilmaz, H.

Zbinden, H.

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

Zhang, H.

H. Zhang and S. Tzortzakis, “Robust authentication through stochastic femtosecond laser filament induced scattering surfaces,” Appl. Phys. Lett. 108(21), 211107 (2016).
[Crossref]

Zhang, H.-F.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Zhang, Q.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Zhao, Y.

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

Appl. Phys. Lett. (1)

H. Zhang and S. Tzortzakis, “Robust authentication through stochastic femtosecond laser filament induced scattering surfaces,” Appl. Phys. Lett. 108(21), 211107 (2016).
[Crossref]

Biomed. Opt. Express (1)

Des. Codes Cryptogr. (1)

A. Broadbent and C. Schaffner, “Quantum cryptography beyond quantum key distribution,” Des. Codes Cryptogr. 78(1), 351–382 (2016).
[Crossref]

Nat. Commun. (1)

N. H. Y. Ng, S. K. Joshi, C. Chen Ming, C. Kurtsiefer, and S. Wehner, “Experimental implementation of bit commitment in the noisy-storage model,” Nat. Commun. 3(1), 1326 (2012).
[Crossref]

Nat. Photonics (1)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

Nature (1)

J. D. R. Buchanan, R. P. Cowburn, A. Jausovec, D. Petit, P. Seem, G. Xiong, D. Atkinson, K. Fenton, D. A. Allwood, and M. T. Bryan, “Fingerprinting documents and packaging,” Nature 436(7050), 475 (2005).
[Crossref]

New J. Phys. (1)

S. K. Poppoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13(12), 123021 (2011).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (3)

B. R. Anderson, R. Gunawidjaja, and H. Eilers, “Effect of experimental parameters on optimal transmission of light through opaque media,” Phys. Rev. A 90(5), 053826 (2014).
[Crossref]

G. M. Nikolopoulos, “Continuous-variable quantum authentication of physical unclonable keys: Security against an emulation attack,” Phys. Rev. A 97(1), 012324 (2018).
[Crossref]

G. M. Nikolopoulos, “Applications of single-qubit rotations in quantum public-key cryptography,” Phys. Rev. A 77(3), 032348 (2008).
[Crossref]

Phys. Rev. Lett. (4)

P. Lodahl, A. P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95(17), 173901 (2005).
[Crossref]

A. Kent, “Unconditionally secure bit commitment by transmitting measurement outcomes,” Phys. Rev. Lett. 109(13), 130501 (2012).
[Crossref]

Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental unconditionally secure bit commitment,” Phys. Rev. Lett. 112(1), 010504 (2014).
[Crossref]

T. Lunghi, J. Kaniewski, F. Bussières, R. Houlmann, M. Tomamichel, A. Kent, N. Gisin, S. Wehner, and H. Zbinden, “Experimental bit commitment based on quantum communication and special relativity,” Phys. Rev. Lett. 111(18), 180504 (2013).
[Crossref]

Rev. Mod. Phys. (1)

M. Herrero-Collantes and J. C. Garcia-Escartin, “Quantum random number generators,” Rev. Mod. Phys. 89(1), 015004 (2017).
[Crossref]

Sci. Adv. (1)

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2(1), e1501054 (2016).
[Crossref]

Sci. Rep. (3)

C. Mesaritakis, M. Akriotou, A. Kapsalis, E. Grivas, C. Chaintoutis, T. Nikas, and D. Syvridis, “Physical unclonable function based on a multi-mode optical waveguide,” Sci. Rep. 8(1), 9653 (2018).
[Crossref]

R. Horstmayer, B. Judkewitz, I. M. Vellekoop, S. Assawaworrarit, and C. Yan, “Physical key-protected one-time pad,” Sci. Rep. 3(1), 3543 (2013).
[Crossref]

G. M. Nikolopoulos and E. Diamanti, “Continuous-variable quantum authentication of physical unclonable keys,” Sci. Rep. 7(1), 46047 (2017).
[Crossref]

Science (1)

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297(5589), 2026–2030 (2002).
[Crossref]

Other (7)

L. V. Amitonova, T. B. H. Tentrup, I. M. Vellekoop, and P. W. H. Pinkse, “Quantum key establishment with a multimode fiber,” (2018). ArXiv:1801.07180.

O. Goldreich, Foundations of cryptography: Basic Techniques (Cambridge University, 2004).

N. P. Smart, Cryptography Made Simple (Springer International Publishing, 2014).

U. Leonhardt, Essential Quantum Optics: From Quantum Measurements to Black Holes (Cambridge University, 2010).

A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography (CRC, 1996).

S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programming linear quantum networks with a multimode fiber,” (2018). ArXiv:1802.07573.

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

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Figures (6)

Fig. 1.
Fig. 1. Schematic representation of the PUK analyzer considered in the proposed commitment scheme. The transverse spatial wavefront of the probe is shaped by means of phase-only spatial light modulator (SLM). The shaped light is focused on the random PUK, and the scattered (reflected) light is collected by means of a polarizing beam splitter (PBS). The output field is imaged onto a plane, where a single-mode fiber (SMF) can be translated in a controlled manner, and the phase-mask of the SLM is optimized so that the speckle exhibits a single enhanced speckle grain, at the position of the SMF (target mode) [18]. The overall imaging system is optimized so that the SMF collects light from a single speckle grain. A joint measurement of both quadratures of the collected light is performed by means of a balanced dual-homodyne detection system.
Fig. 2.
Fig. 2. Contour plot of the typical response of a PUK for optimized (blue) and non-optimized (red) SLM, for two different values of the mean number of photons: (a) $\mu =1500$, and (b) $\mu =2650$. The dashed vertical and horizontal lines show the overlap of the corresponding marginal distributions for the quadratures, with the acceptance region ${\mathcal {A}}$ (gray rectangle). Parameters: $N=625$, $w = 8/\sqrt {\eta }$, $l/L=0.2$, $\tau = 0.05$, $\eta =0.6$.
Fig. 3.
Fig. 3. Schematic representation of the commit and the reveal phases in the commitment scheme under consideration.
Fig. 4.
Fig. 4. Phase-space representation of the response of a PUK for optimized and non-optimized SLM. The blue star shows the response $\langle {\hat {\boldsymbol Z}_s} \rangle _\textrm {o}$ for the target mode $s$, according to which the phase mask of the SLM has been optimized. The red disks show the response of the PUK for all of the other output modes $s^{\prime }\neq s$, while keeping the phase mask to its optimal configuration for $s$. (a) $N=256$, $\mu =1500$; (b) $N=625$, $\mu =1500$; (c) $N=256$, $\mu =2650$, (d) $N=625$, $\mu =2650$. The dashed curves show circular areas of radius $\rho _\textrm {o}$ and $\rho$, with $\rho _\textrm {o}>\rho$. The gray area shows the acceptance region ${\mathcal {A}}(\langle {\hat {\boldsymbol Z}_{s}} \rangle _\textrm {o}, 8/\sqrt {\eta })$. Other parameters: $l/L=0.2$, $\tau = 0.05$, $\eta =0.6$.
Fig. 5.
Fig. 5. Probability of successful cheating as a function of the mean number of photons in the pulse for $N=256$ (a) and $N=625$ (b). The curves show the theoretical upper bound on the probability of successful cheating for different values of $\nu$. The symbols show the maximum probabilities that have been extracted from the numerical data of Fig. 4 (empty symbols) and Fig. 6 (filled symbols) in the case of $\nu =1$, and they are always below the theoretically expected bound (solid black curve). Squares (empty and filled) refer to $\mu =1500$ and the triangles to $\mu =2650$. Other parameters as in Fig. 4.
Fig. 6.
Fig. 6. Phase-space representation of the response of the PUK for optimized and non-optimized SLM. The blue star shows the response for the reference PUK-target pair $({\cal K},s)$, according to which the SLM has been optimized. The red disks show the responses of PUK-target pairs $({\cal K}^{\prime },s^{\prime })$ that maximize the probability of acceptance, when the phase-mask of SLM is set to its optimal configuration for $({\cal K},s)$. The data have been obtained from simulations on 500 random keys ${\cal K}^{\prime }$, and all of the possible output modes $s^{\prime } \neq s$. Other parameters as in Fig. 4.

Equations (12)

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b ^ s = j = 1 N r s , j e i ϕ j a ^ j ,
Z s ¯ = ( 0 , 0 ) , Var ( Z ^ s ) = 2 μ τ N ( 1 l L ) := 2 μ V ,
ρ := 4 μ V
( X ^ s o ) 2 + ( Y ^ s o ) 2 = 2 E μ V := 2 ρ o 2 ,
A ( Z ~ s ( o ) , w ) := { X ~ s ( o ) w / 2 x X ~ s ( o ) + w / 2 Y ~ s ( o ) w / 2 y Y ~ s ( o ) + w / 2 } ,
P in = 1 4 [ Erf ( w ~ 2 2 + ( X ~ s (o) X ^ s ) η 2 ) + Erf ( w ~ 2 2 ( X ~ s (o) X ^ s ) η 2 ) ] × [ Erf ( w ~ 2 2 + ( Y ~ s (o) Y ^ s ) η 2 ) + Erf ( w ~ 2 2 ( Y ~ s (o) Y ^ s ) η 2 ) ] .
Δ := ( ρ o ρ ) η Ω ,
μ Ω 2 η V ( E 4 ) 2 .
P accept ( ν ) := j = 0 ν / 2 ( ν j ) [ 1 p in ( o ) ] j [ p in ( o ) ] ν j ,
p in ( o ) [ Erf ( w ~ 2 2 ) ] 2 ,
P cheat ( ν ) := j = 0 ν / 2 ( ν j ) ( 1 p in ) j ( p in ) ν j
p in ( max ) = 1 4 [ Erf ( w ~ 2 2 + Δ 2 ) + Erf ( w ~ 2 2 Δ 2 ) ] 2 ,

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