The parity problem in the presence of noise, decoding random linear codes, and the subset sum problem

Authors:
Vadim Lyubashevsky
Affiliations:
University of California at San Diego, La Jolla, CA
Venue:
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
Year:
2005

Citing 10
Cited 19

Solving low-density subset sum problems

Journal of the ACM (JACM)
On the Lagarias-Odlyzko algorithm for the subset sum problem

SIAM Journal on Computing
A hard-core predicate for all one-way functions

STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Generating hard instances of lattice problems (extended abstract)

STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
A Generalized Birthday Problem

CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
Noise-tolerant learning, the parity problem, and the statistical query model

Journal of the ACM (JACM)
Worst-Case to Average-Case Reductions Based on Gaussian Measures

FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
A computational introduction to number theory and algebra

A computational introduction to number theory and algebra
How to recycle random bits

SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Solving medium-density subset sum problems in expected polynomial time

STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science

Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions

Computational Complexity
Cryptography with constant computational overhead

STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
On agnostic boosting and parity learning

STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
SWIFFT: A Modest Proposal for FFT Hashing

Fast Software Encryption
Unclonable Lightweight Authentication Scheme

ICICS '08 Proceedings of the 10th International Conference on Information and Communications Security
Two Attacks against the Ff RFID Protocol

INDOCRYPT '09 Proceedings of the 10th International Conference on Cryptology in India: Progress in Cryptology
PUF-HB: a tamper-resilient HB based authentication protocol

ACNS'08 Proceedings of the 6th international conference on Applied cryptography and network security
An improved multi-set algorithm for the dense subset sum problem

ANTS-VIII'08 Proceedings of the 8th international conference on Algorithmic number theory
Local list-decoding and testing of random linear codes from high error

Proceedings of the forty-second ACM symposium on Theory of computing
Some recent results on local testing of sparse linear codes

Property testing
Some recent results on local testing of sparse linear codes

Property testing
Practical attacks on HB and HB+ protocols

WISTP'11 Proceedings of the 5th IFIP WG 11.2 international conference on Information security theory and practice: security and privacy of mobile devices in wireless communication
On noise-tolerant learning of sparse parities and related problems

ALT'11 Proceedings of the 22nd international conference on Algorithmic learning theory
Public-key cryptographic primitives provably as secure as subset sum

TCC'10 Proceedings of the 7th international conference on Theory of Cryptography
Security problems of systems of extremely weak devices

Annales UMCS, Informatica - Security Systems
Cryptography from learning parity with noise

SOFSEM'12 Proceedings of the 38th international conference on Current Trends in Theory and Practice of Computer Science
Lapin: an efficient authentication protocol based on Ring-LPN

FSE'12 Proceedings of the 19th international conference on Fast Software Encryption
Hidden bits approach for authentication in RFID systems

RFIDSec'12 Proceedings of the 8th international conference on Radio Frequency Identification: security and privacy issues
Linear-time encodable codes meeting the gilbert-varshamov bound and their cryptographic applications

Proceedings of the 5th conference on Innovations in theoretical computer science

Quantified Score

Hi-index	0.00

Visualization

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 n1+ε labeled examples. This result immediately gives us a sub-exponential algorithm for decoding n × n1+ε random binary linear codes (i.e. codes where the messages are n bits and the codewords are n1+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.