A Cryptographic Test of Quantumness and Certifiable Randomness from a Single Quantum Device

We give a protocol for producing certifiable randomness from a single untrusted quantum device that is polynomial-time bounded. The randomness is certified to be statistically close to uniform from the point of view of any computationally unbounded quantum adversary, that may share entanglement with the quantum device. The protocol relies on the existence of post-quantum secure trapdoor claw-free functions, and introduces a new primitive for constraining the power of an untrusted quantum device. We then show how to construct this primitive based on the hardness of the learning with errors (LWE) problem. The randomness protocol can also be used as the basis for an efficiently verifiable "quantum supremacy" proposal, thus answering an outstanding challenge in the field.

[1]  Roger Colbeck,et al.  Quantum And Relativistic Protocols For Secure Multi-Party Computation , 2009, 0911.3814.

[2]  Maassen,et al.  Generalized entropic uncertainty relations. , 1988, Physical review letters.

[3]  Miklós Ajtai,et al.  Generating Hard Instances of the Short Basis Problem , 1999, ICALP.

[4]  Chris Peikert,et al.  Better Key Sizes (and Attacks) for LWE-Based Encryption , 2011, CT-RSA.

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

[6]  Chris Peikert,et al.  Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller , 2012, IACR Cryptol. ePrint Arch..

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

[8]  Damien Stehlé,et al.  Classical hardness of learning with errors , 2013, STOC '13.

[9]  Lov K. Grover,et al.  Creating superpositions that correspond to efficiently integrable probability distributions , 2002, quant-ph/0208112.

[10]  Alan Mink,et al.  Experimentally generated randomness certified by the impossibility of superluminal signals , 2018, Nature.

[11]  Urmila Mahadev,et al.  Classical Verification of Quantum Computations , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[12]  Zvika Brakerski,et al.  Certifiable Randomness from a Single Quantum Device , 2018, ArXiv.

[13]  Thomas Vidick,et al.  Practical device-independent quantum cryptography via entropy accumulation , 2018, Nature Communications.

[14]  Chris Peikert,et al.  Pseudorandomness of ring-LWE for any ring and modulus , 2017, STOC.

[15]  Chris Peikert,et al.  Public-key cryptosystems from the worst-case shortest vector problem: extended abstract , 2009, STOC '09.

[16]  Mark M. Wilde,et al.  Quantum Information Theory , 2013 .

[17]  Urmila Mahadev,et al.  Classical Homomorphic Encryption for Quantum Circuits , 2017, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[18]  Mark M. Wilde,et al.  From Classical to Quantum Shannon Theory , 2011, ArXiv.

[19]  Oded Regev,et al.  On lattices, learning with errors, random linear codes, and cryptography , 2005, STOC '05.

[20]  Yaoyun Shi,et al.  N ov 2 01 4 Universal security for randomness expansion , 2014 .

[21]  Marco Tomamichel,et al.  A Fully Quantum Asymptotic Equipartition Property , 2008, IEEE Transactions on Information Theory.

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

[23]  Silvio Micali,et al.  A "Paradoxical'"Solution to the Signature Problem (Abstract) , 1984, CRYPTO.

[24]  Stephan Krenn,et al.  Learning with Rounding, Revisited: New Reduction, Properties and Applications , 2013, IACR Cryptol. ePrint Arch..

[25]  W. Banaszczyk.  New bounds in some transference theorems in the geometry of numbers , 1993 .

[26]  Aram W. Harrow,et al.  Quantum computational supremacy , 2017, Nature.

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

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

[29]  Brent Waters,et al.  Homomorphic Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based , 2013, CRYPTO.

[30]  Elham Kashefi,et al.  Delegated Pseudo-Secret Random Qubit Generator , 2018, ArXiv.

[31]  Yael Tauman Kalai,et al.  Robustness of the Learning with Errors Assumption , 2010, ICS.