Anindya De
ABSTRACT
We show that Trevisan's extractor and its variants [22,19] are secure against bounded quantum storage adversaries. One instantiation gives the first such extractor to achieve an output length Θ(K-b), where K is the source's entropy and b the adversary's storage, together with a poly-logarithmic seed length. Another instantiation achieves a logarithmic key length, with a slightly smaller output length Θ((K-b)/Kγ) for any γ>0. In contrast, the previous best construction [21] could only extract (K/b)1/15 bits.
Some of our constructions have the additional advantage that every bit of the output is a function of only a polylogarithmic number of bits from the source, which is crucial for some cryptographic applications.
Our argument is based on bounds for a generalization of quantum random access codes, which we call quantum functional access codes. This is crucial as it lets us avoid the local list-decoding algorithm central to the approach in [21], which was the source of the multiplicative overhead.
- A. Ambainis, A. Nayak, A. Tashma, and U. V. Vazirani. Dense quantum coding and quantum finite automata. Journal of the ACM, 49(4):496--511, 2002. Preliminary version in phProc. of STOC 1999. Google Scholar
Digital Library
- A. Ben-Aroya, O. Regev, and R. de Wolf. A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs . In Proceedings of the 49th IEEE Symposium on Foundations of Computer Science, pages 477--486, 2008. Full version at arXiv:0705.3806. Google ScholarDigital Library
- R. Cleve, W. v. Dam, M. Nielsen, and A. Tapp. Quantum entanglement and the communication complexity of the inner product function. In QCQC '98: Selected papers from the First NASA International Conference on Quantum Computing and Quantum Communications, pages 61--74, London, UK, 1998. Springer-Verlag. Google ScholarDigital Library
- A. De, C. Portmann, R. Renner, and T. Vidick. Trevisan's extractor in the presence of quantum side information. Technical report arXiv:0912.5514, 2009.Google Scholar
- A. De and L. Trevisan. Extractors using hardness amplification. In Proc. of APPROX-RANDOM, pages 462--475, 2009. Google ScholarDigital Library
- S. Dziembowski and U. Maurer. Optimal randomizer efficiency in the bounded-storage model. Journal of Cryptology, 17(1):5--26, 2004. Google ScholarDigital Library
- S. Fehr and C. Schaffner. Randomness extraction via delta-biased masking in the presence of a quantum attacker. In R. Canetti, editor, TCC, volume 4948 of Lecture Notes in Computer Science, pages 465--481. Springer, 2008. Google ScholarDigital Library
- D. Gavinsky, J. Kempe, I. Kerenidis, R. Raz, and R. de Wolf. Exponential separation for one-way quantum communication complexity, with applications to cryptography. SIAM Journal of Computing, 38(5):1695--1708, 2008. Preliminary version in Proc. of STOC 2007. Google ScholarDigital Library
- V. Guruswami, J. Håstad, M. Sudan, and D. Zuckerman. Combinatorial bounds for list decoding. IEEE Transactions on Information Theory, 48(5):1021--1034, 2002. Google ScholarDigital Library
- T. Hartman and R. Raz. On the distribution of the number of roots of polynomials and explicit weak designs. Random Structures and Algorithms, 23(3):235--263, 2003. Google ScholarDigital Library
- A. Holevo. Information-theoretic aspects of quantum measurement. Problems of Information Transmission, 9(2):31--42, 1973.Google Scholar
- R. Impagliazzo, R. Jaiswal, and V. Kabanets. Approximately List-Decoding Direct Product Codes and Uniform Hardness Amplification. In Proceedings of the 47th IEEE Symposium on Foundations of Computer Science, pages 187--196, 2006. Full version at http://www1.cs.columbia.edu/$\sim$rjaiswal/. Google ScholarDigital Library
- R. König, U. Maurer, and R. Renner. On the power of quantum memory. IEEE Transactions on Information Theory, 51(7):2391--2401, 2005. Google ScholarDigital Library
- R. König and B. Terhal. The bounded storage model in presence of a quantum adversary. IEEE Transactions on Information Theory, 54(2):749--762, 2008. Google ScholarDigital Library
- C.-J. Lu. Encryption against storage-bounded adversaries from on-line strong extractors. Journal of Cryptology, 17(1):27--42, 2004. Google ScholarDigital Library
- U. M. Maurer. Conditionally-perfect secrecy and a provably-secure randomized cipher. Journal of Cryptology, 5(1):53--66, 1992. Google ScholarCross Ref
- A. Nayak and J. Salzman. Limits on the ability of quantum states to convey classical messages. Journal of the ACM, 53(1):184--206, 2006. Google ScholarDigital Library
- N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49:149--167, 1994. Preliminary version in Proc. of FOCS'88. Google ScholarDigital Library
- R. Raz, O. Reingold, and S. P. Vadhan. Extracting all the randomness and reducing the error in trevisan's extractors. J. Comput. Syst. Sci., 65(1):97--128, 2002. Preliminary version in Proc. of STOC 1999. Google ScholarDigital Library
- M. Sudan, L. Trevisan, and S. Vadhan. Pseudorandom generators without the XOR lemma. Journal of Computer and System Sciences, 62(2):236--266, 2001. Preliminary version in STOC and CCC 1999. Google ScholarDigital Library
- A. Tashma. Short seed extractors against quantum storage. In Proceedings of the 41st ACM Symposium on Theory of Computing, pages 401--409, 2009. Google ScholarDigital Library
- L. Trevisan. Extractors and pseudorandom generators. Journal of the ACM, 48(4):860--879, 2001. Google ScholarDigital Library
- S. P. Vadhan. Constructing locally computable extractors and cryptosystems in the bounded-storage model. Journal of Cryptology, 17(1):43--77, 2004. Google ScholarDigital Library
- E. Viola. The complexity of constructing pseudorandom generators from hard functions. Computational Complexity, 13(3-4):147--188, 2004. Google ScholarDigital Library
Index Terms
- Near-optimal extractors against quantum storage
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