Constructions for Quantum Indistinguishability Obfuscation

An indistinguishability obfuscator is a probabilistic polynomial-time algorithm that takes a circuit as input and outputs a new circuit that has the same functionality as the input circuit, such that for any two circuits of the same size that compute the same function, the outputs of the indistinguishability obfuscator are indistinguishable. Here, we study schemes for indistinguishability obfuscation for quantum circuits. We present two definitions for indistinguishability obfuscation: in our first definition (qiO) the outputs of the obfuscator are required to be indistinguishable if the input circuits are perfectly equivalent, while in our second definition (qiOD), the outputs are required to be indistinguishable as long as the input circuits are approximately equivalent with respect to a pseudo-distance D. Our main results provide (1) a computationally-secure scheme for qiO where the size of the output of the obfuscator is exponential in the number of non-Clifford (T gates), which means that the construction is efficient as long as the number of T gates is logarithmic in the circuit size and (2) a statistically-secure qiOD, for circuits that are close to the kth level of the Gottesman-Chuang hierarchy (with respect to D); this construction is efficient as long as k is small and fixed.

[1]  Isaac L. Chuang,et al.  Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations , 1999, Nature.

[2]  Ron Steinfeld,et al.  GGHLite: More Efficient Multilinear Maps from Ideal Lattices , 2014, IACR Cryptol. ePrint Arch..

[3]  Amit Sahai,et al.  On the (im)possibility of obfuscating programs , 2001, JACM.

[4]  Florian Speelman,et al.  Instantaneous Non-Local Computation of Low T-Depth Quantum Circuits , 2015, TQC.

[5]  Scott Aaronson,et al.  Improved Simulation of Stabilizer Circuits , 2004, ArXiv.

[6]  Ran Canetti,et al.  Obfuscation of Probabilistic Circuits and Applications , 2015, TCC.

[7]  Craig Gentry,et al.  Graph-Induced Multilinear Maps from Lattices , 2015, TCC.

[8]  Tal Malkin,et al.  The Power of Negations in Cryptography , 2015, TCC.

[9]  Scott Aaronson,et al.  Quantum Copy-Protection and Quantum Money , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[10]  Brent Waters,et al.  How to use indistinguishability obfuscation: deniable encryption, and more , 2014, IACR Cryptol. ePrint Arch..

[11]  Peter Selinger,et al.  Generators and relations for n-qubit Clifford operators , 2013, Log. Methods Comput. Sci..

[12]  Christian Schaffner,et al.  Quantum Homomorphic Encryption for Polynomial-Sized Circuits , 2016, CRYPTO.

[13]  D. Gottesman.  The Heisenberg Representation of Quantum Computers , 1998, quant-ph/9807006.

[14]  Stacey Jeffery,et al.  Quantum Homomorphic Encryption for Circuits of Low T-gate Complexity , 2014, CRYPTO.

[15]  Eitan M. Gurari,et al.  Introduction to the theory of computation , 1989 .

[16]  Bill Fefferman,et al.  On Quantum Obfuscation , 2016, ArXiv.

[17]  Craig Gentry,et al.  Cryptanalyses of Candidate Branching Program Obfuscators , 2017, EUROCRYPT.

[18]  Jean-Sébastien Coron,et al.  Practical Multilinear Maps over the Integers , 2013, CRYPTO.

[19]  Nir Bitansky,et al.  ZAPs and Non-Interactive Witness Indistinguishability from Indistinguishability Obfuscation , 2015, TCC.

[20]  Dmitri Maslov,et al.  Polynomial-Time T-Depth Optimization of Clifford+T Circuits Via Matroid Partitioning , 2013, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[21]  Rolando L. La Placa,et al.  Secure Software Leasing , 2020, EUROCRYPT.

[22]  Raymond Laflamme,et al.  An Introduction to Quantum Computing , 2007, Quantum Inf. Comput..

[23]  I. Chuang,et al.  Quantum Teleportation is a Universal Computational Primitive , 1999, quant-ph/9908010.

[24]  R. A. Low,et al.  Learning and testing algorithms for the Clifford group , 2009, 0907.2833.

[25]  Michele Mosca,et al.  Parallelizing quantum circuit synthesis , 2016, 1606.07413.

[26]  Guy N. Rothblum,et al.  On Best-Possible Obfuscation , 2007, Journal of Cryptology.

[27]  Amit Sahai,et al.  On the (im)possibility of obfuscating programs , 2012, JACM.

[28]  Martin R. Albrecht,et al.  A Subfield Lattice Attack on Overstretched NTRU Assumptions - Cryptanalysis of Some FHE and Graded Encoding Schemes , 2016, CRYPTO.

[29]  Amit Sahai,et al.  Indistinguishability Obfuscation Without Multilinear Maps: New Paradigms via Low Degree Weak Pseudorandomness and Security Amplification , 2019, IACR Cryptol. ePrint Arch..

[30]  Mark Zhandry,et al.  Multiparty Key Exchange, Efficient Traitor Tracing, and More from Indistinguishability Obfuscation , 2014, CRYPTO.

[31]  Robert Wille,et al.  Efficient synthesis of quantum circuits implementing clifford group operations , 2014, 2014 19th Asia and South Pacific Design Automation Conference (ASP-DAC).

[32]  M. Mosca,et al.  A Meet-in-the-Middle Algorithm for Fast Synthesis of Depth-Optimal Quantum Circuits , 2012, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[33]  A. Broadbent,et al.  Uncloneable Quantum Encryption via Oracles , 2019, TQC.

[34]  Kazuyuki Amano,et al.  Representation of Quantum Circuits with Clifford and $\pi/8$ Gates , 2008, 0806.3834.

[35]  Serge Fehr,et al.  Theory of Quantum Computation, Communication, and Cryptography , 2012, Lecture Notes in Computer Science.

[36]  Zvika Brakerski,et al.  Impossibility of Quantum Virtual Black-Box Obfuscation of Classical Circuits , 2020, ArXiv.

[37]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[38]  Zvika Brakerski.  Quantum FHE (Almost) As Secure as Classical , 2018, IACR Cryptol. ePrint Arch..

[39]  Ronald Cramer,et al.  Recovering Short Generators of Principal Ideals in Cyclotomic Rings , 2016, EUROCRYPT.

[40]  Andris Ambainis,et al.  Private quantum channels , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[41]  Peter Selinger,et al.  Remarks on Matsumoto and Amano's normal form for single-qubit Clifford+T operators , 2013, ArXiv.

[42]  Amit Sahai,et al.  Indistinguishability Obfuscation from Well-Founded Assumptions , 2020, IACR Cryptol. ePrint Arch..