Device Independent Random Number Generation

Randomness is an invaluable resource in today's life with a broad use reaching from numerical simulations through randomized algorithms to cryptography. However, on the classical level no true randomness is available and even the use of simple quantum devices in a prepare-measure setting suffers from lack of stability and controllability. This gave rise to a group of quantum protocols that provide randomness certified by classical statistical tests -- Device Independent Quantum Random Number Generators. In this paper we review the most relevant results in this field, which allow the production of almost perfect randomness with help of quantum devices, supplemented with an arbitrary weak source of additional randomness. This is in fact the best one could hope for to achieve, as with no starting randomness (corresponding to no free will in a different concept) even a quantum world would have a fully deterministic description.

[1]  Renato Renner,et al.  Security of quantum key distribution , 2005, Ausgezeichnete Informatikdissertationen.

[2]  Renato Renner,et al.  Unconditional Authenticity and Privacy from an Arbitrarily Weak Secret , 2003, CRYPTO.

[3]  Arjen K. Lenstra,et al.  Ron was wrong, Whit is right , 2012, IACR Cryptol. ePrint Arch..

[4]  Rodrigo Gallego,et al.  Full randomness from arbitrarily deterministic events , 2012, Nature Communications.

[5]  Y. Peres.  Iterating Von Neumann's Procedure for Extracting Random Bits , 1992 .

[6]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[7]  B. S. Cirel'son.  Quantum generalizations of Bell's inequality , 1980 .

[8]  Karol Horodecki,et al.  Robust Device-Independent Randomness Amplification with Few Devices , 2013, 1310.4544.

[9]  Ran Raz,et al.  Extractors with weak random seeds , 2005, STOC '05.

[10]  Umesh V. Vazirani,et al.  Certifiable quantum dice: or, true random number generation secure against quantum adversaries , 2012, STOC '12.

[11]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[12]  S. Wehner,et al.  Bell Nonlocality , 2013, 1303.2849.

[13]  Matthew Coudron,et al.  Infinite randomness expansion with a constant number of devices , 2014, STOC.

[14]  Stefano Pironio,et al.  Random numbers certified by Bell’s theorem , 2009, Nature.

[15]  Anindya De,et al.  Trevisan's Extractor in the Presence of Quantum Side Information , 2009, SIAM J. Comput..

[16]  Moni Naor,et al.  Small-Bias Probability Spaces: Efficient Constructions and Applications , 1993, SIAM J. Comput..

[17]  Jean-Daniel Bancal,et al.  More Randomness From Noisy Sources , 2014, TQC.

[18]  Ran Raz,et al.  Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography , 2008, SIAM J. Comput..

[19]  Avi Wigderson,et al.  Extractors: optimal up to constant factors , 2003, STOC '03.

[20]  Jozef Gruska,et al.  Quantum Computing , 2008, Wiley Encyclopedia of Computer Science and Engineering.

[21]  Renato Renner,et al.  True randomness from realistic quantum devices , 2013, ArXiv.

[22]  N. Bohr.  II - Can Quantum-Mechanical Description of Physical Reality be Considered Complete? , 1935 .

[23]  Joel H. Spencer,et al.  On the (non)universality of the one-time pad , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[24]  Luca Trevisan,et al.  Extracting randomness from samplable distributions , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[25]  Ronen Shaltiel,et al.  Recent Developments in Explicit Constructions of Extractors , 2002, Bull. EATCS.

[26]  Serge Fehr,et al.  Security and Composability of Randomness Expansion from Bell Inequalities , 2011, ArXiv.

[27]  Stefano Pironio,et al.  Randomness versus nonlocality and entanglement. , 2011, Physical review letters.

[28]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[29]  Le Phuc Thinh,et al.  Min-entropy sources for Bell tests , 2014 .

[30]  Amit Sahai,et al.  On the (im)possibility of cryptography with imperfect randomness , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[31]  Vijay V. Vazirani,et al.  Random polynomial time is equal to slightly-random polynomial time , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[32]  Adrian Kent,et al.  Private randomness expansion with untrusted devices , 2010, 1011.4474.

[33]  Aravind Srinivasan,et al.  Computing with very weak random sources , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[34]  Ran Raz,et al.  Improved Randomness Extraction from Two Independent Sources , 2004, APPROX-RANDOM.

[35]  Roger Colbeck,et al.  Free randomness can be amplified , 2011, Nature Physics.

[36]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[37]  Aram Wettroth Harrow,et al.  Quantum de Finetti Theorems Under Local Measurements with Applications , 2012, Communications in Mathematical Physics.

[38]  Jan Bouda,et al.  Weak randomness seriously limits the security of quantum key distribution , 2012 .

[39]  M. Plesch,et al.  Device-independent randomness amplification with a single device , 2013, 1305.0990.

[40]  Yevgeniy Dodis,et al.  Does Privacy Require True Randomness? , 2007, TCC.

[41]  Karol Horodecki,et al.  Free randomness amplification using bipartite chain correlations , 2013, 1303.5591.

[42]  Julia Kempe,et al.  Two-Source Extractors Secure Against Quantum Adversaries , 2010, Theory Comput..

[43]  A. Shimony,et al.  Bell’s theorem without inequalities , 1990 .

[44]  Caslav Brukner,et al.  Experimenter's freedom in Bell's theorem and quantum cryptography (7 pages) , 2005, quant-ph/0510167.

[45]  Jan Bouda,et al.  Improving the Hadamard extractor , 2012, Theor. Comput. Sci..

[46]  Guy Kindler,et al.  Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors , 2005, STOC '05.

[47]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[48]  Jan Bouda,et al.  Device-independent randomness extraction from an arbitrarily weak min-entropy source , 2014 .

[49]  A. Fine.  Hidden Variables, Joint Probability, and the Bell Inequalities , 1982 .

[50]  Kai-Min Chung,et al.  Physical Randomness Extractors , 2014 .

[51]  S. Pironio,et al.  Using complete measurement statistics for optimal device-independent randomness evaluation , 2013, 1309.3930.

[52]  Kiel T. Williams,et al.  Extreme quantum entanglement in a superposition of macroscopically distinct states. , 1990, Physical review letters.

[53]  David Zuckerman,et al.  Deterministic extractors for small-space sources , 2011, J. Comput. Syst. Sci..

[54]  K. Jakobsson.  Theory, Methods and Tools for Statistical Testing of Pseudo and Quantum Random Number Generators , 2014 .

[55]  Noam Nisan,et al.  Extracting Randomness: A Survey and New Constructions , 1999, J. Comput. Syst. Sci..

[56]  Umesh V. Vazirani,et al.  Strong communication complexity or generating quasi-random sequences from two communicating semi-random sources , 1987, Comb..

[57]  Anindya De,et al.  Near-optimal extractors against quantum storage , 2010, STOC '10.

[58]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[59]  Benny Pinkas,et al.  On the Impossibility of Private Key Cryptography with Weakly Random Keys , 1990, CRYPTO.

[60]  Karol Horodecki,et al.  Realistic noise-tolerant randomness amplification using finite number of devices , 2013, Nature Communications.

[61]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[62]  Stefano Pironio,et al.  Security of practical private randomness generation , 2011, 1111.6056.

[63]  A. Acín,et al.  A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations , 2008, 0803.4290.

[64]  Valerio Scarani.  The device-independent outlook on quantum physics (lecture notes on the power of Bell's theorem) , 2013 .

[65]  Matthew Coudron,et al.  Robust Randomness Amplifiers: Upper and Lower Bounds , 2013, APPROX-RANDOM.

[66]  Robert König,et al.  The Operational Meaning of Min- and Max-Entropy , 2008, IEEE Transactions on Information Theory.

[67]  V. Scarani,et al.  More randomness from the same data , 2013, 1309.3894.

[68]  Miklos Santha,et al.  Generating Quasi-random Sequences from Semi-random Sources , 1986, J. Comput. Syst. Sci..

[69]  Yevgeniy Dodis,et al.  On Extracting Private Randomness over a Public Channel , 2003, RANDOM-APPROX.

[70]  Chained Bell Inequalities , 1989 .

[71]  Gilles Brassard,et al.  Quantum cryptography: Public key distribution and coin tossing , 2014, Theor. Comput. Sci..

[72]  J. Bourgain,et al.  MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS , 2005 .

[73]  Yaoyun Shi,et al.  Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices , 2014, STOC.

[74]  Ueli Maurer,et al.  On the power of quantum memory , 2005, IEEE Transactions on Information Theory.

[75]  Umesh V. Vazirani,et al.  A classical leash for a quantum system: command of quantum systems via rigidity of CHSH games , 2012, ITCS '13.

[76]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[77]  David Zuckerman.  Simulating BPP using a general weak random source , 2005, Algorithmica.

[78]  Jaikumar Radhakrishnan,et al.  Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators , 2000, SIAM J. Discret. Math..

[79]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[80]  Thomas Lawson,et al.  Biased nonlocal quantum games , 2010, 1011.6245.

[81]  Enkatesan G Uruswami.  Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes , 2008 .

[82]  Yaoyun Shi,et al.  Optimal robust quantum self-testing by binary nonlocal XOR games , 2012, 1207.1819.

[83]  Marcus Huber,et al.  Weak randomness in device-independent quantum key distribution and the advantage of using high-dimensional entanglement , 2013, 1301.2455.

[84]  Lluis Masanes,et al.  Universally-composable privacy amplification from causality constraints , 2008, Physical review letters.

[85]  J Silman,et al.  Device-independent randomness generation in the presence of weak cross-talk. , 2012, Physical review letters.

[86]  Matthew McKague,et al.  Self-Testing Graph States , 2010, TQC.

[87]  P. Grangier,et al.  Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment : A New Violation of Bell's Inequalities , 1982 .

[88]  Andrew Chi-Chih Yao,et al.  Self testing quantum apparatus , 2004, Quantum Inf. Comput..