### Abstract

We prove that any algorithm for learning parities requires either a memory of quadratic size or an exponential number of samples. This proves a recent conjecture of Steinhardt, Valiant and Wager [15] and shows that for some learning problems a large storage space is crucial. More formally, in the problem of parity learning, an unknown string x ϵ {0,1}n was chosen uniformly at random. A learner tries to learn x from a stream of samples (a1, b1), (a2, b2)..., where each at is uniformly distributed over {0,1}n and bt is the inner product of at and x, modulo 2. We show that any algorithm for parity learning, that uses less than n2/25 bits of memory, requires an exponential number of samples. Previously, there was no non-trivial lower bound on the number of samples needed, for any learning problem, even if the allowed memory size is O(n) (where n is the space needed to store one sample). We also give an application of our result in the field of bounded-storage cryptography. We show an encryption scheme that requires a private key of length n, as well as time complexity of n per encryption/decryption of each bit, and is provenly and unconditionally secure as long as the attacker uses less than n2/25 memory bits and the scheme is used at most an exponential number of times. Previous works on bounded-storage cryptography assumed that the memory size used by the attacker is at most linear in the time needed for encryption/decryption.

Original language | English (US) |
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Title of host publication | Proceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 |

Publisher | IEEE Computer Society |

Pages | 266-275 |

Number of pages | 10 |

ISBN (Electronic) | 9781509039333 |

DOIs | |

State | Published - Dec 14 2016 |

Externally published | Yes |

Event | 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 - New Brunswick, United States Duration: Oct 9 2016 → Oct 11 2016 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2016-December |

ISSN (Print) | 0272-5428 |

### Other

Other | 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 |
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Country | United States |

City | New Brunswick |

Period | 10/9/16 → 10/11/16 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)

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## Cite this

*Proceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016*(pp. 266-275). [7782939] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 2016-December). IEEE Computer Society. https://doi.org/10.1109/FOCS.2016.36