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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.

The Parity Problem in the Presence of Noise, Decoding Random Linear Codes, and the Subset Sum Problem