Abstract
In [2], Blum et al. demonstrated the first sub-exponential algorithm for learning the parity function in the presence of noise. They solved the length-n parity problem in time 2O(n/logn) but it required the availability of 2O(n/logn) labeled examples. As an open problem, they asked whether there exists a 2o(n) algorithm for the length-n parity problem that uses only poly(n) labeled examples. In this work, we provide a positive answer to this question. We show that there is an algorithm that solves the length-n parity problem in time 2O(n/loglogn) using n 1 + ε labeled examples. This result immediately gives us a sub-exponential algorithm for decoding n × n 1 + ε random binary linear codes (i.e. codes where the messages are n bits and the codewords are n 1 + ε bits) in the presence of random noise. We are also able to extend the same techniques to provide a sub-exponential algorithm for dense instances of the random subset sum problem.
Research supported in part by NSF grant CCR-0093029.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ajtai, M.: Generating hard instances of lattice problems. In: Proc. 28th Annual Symposium on Theory of Computing, pp. 99–108 (1996)
Blum, A., Kalai, A., Wasserman, H.: Noise-tolerant learning, the parity problem, and the statistical query model. JACM 50(4), 506–519 (2003)
Flaxman, A., Przydatek, B.: Solving medium-density subset sum problems in expected polynomial time. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 305–314. Springer, Heidelberg (2005)
Frieze, A.: On the Lagarias-Odlyzko algorithm for the subset sum problem. SIAM J. Comput. 15, 536–539 (1986)
Goldreich, O., Levin, L.A.: Hard-core predicates for any one-way function. In: Proc. 21st Annual Symposium on Theory of Computing, pp. 25–32 (1989)
Impagliazzo, R., Naor, M.: Efficient cryptographic schemes provably as secure as subset sum. Journal of Cryptology 9(4), 199–216 (1996)
Impagliazzo, R., Zuckerman, D.: How to recycle random bits. In: Proc. 30th Annual Symposium on Foundations of Computer Science, pp. 248–253 (1989)
Lagarias, J.C., Odlyzko, A.M.: Solving low density subset sum problems. J. of the ACM 32, 229–246 (1985)
Lyubashevsky, V.: On random high density subset sums, Technical Report TR05-007, Electronic Coloquium on Computational Complexity, ECCC (2005)
Micciancio, D., Regev, O.: Worst-case to average-case reductions based on gaussian measure. In: Proc. 45th Annual Symposium on Foundations of Computer Science, pp. 372–381 (2004)
Schroeppel, R., Shamir, A.: A T = O(2(n/2)), S = O(2(n/4)) algorithm for certain NP-complete problems. SIAM J. Comput. 10, 456–464 (1981)
Shoup, V.: A Computational Introduction to Number Theory and Algebra. Cambridge University Press, Cambridge (2005)
Wagner, D.: A generalized birthday problem. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 288–303. Springer, Heidelberg (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lyubashevsky, V. (2005). The Parity Problem in the Presence of Noise, Decoding Random Linear Codes, and the Subset Sum Problem. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_32
Download citation
DOI: https://doi.org/10.1007/11538462_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28239-6
Online ISBN: 978-3-540-31874-3
eBook Packages: Computer ScienceComputer Science (R0)