**Fast Learning Requires Good Memory: A Time-Space Lower Bound for Parity Learning**

*Ran Raz*

**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 and
shows that for some learning problems a large storage space is crucial.

More formally, in the problem of parity learning, an unknown string $x \in \{0,1\}^n$ was chosen uniformly at random. A learner tries to learn $x$ from a stream of samples $(a_1, b_1), (a_2, b_2)... $, where each $a_t$ is uniformly distributed over $\{0,1\}^n$ and $b_t$ is the inner product of $a_t$ and $x$, modulo 2. We show that any algorithm for parity learning, that uses less than $n^2/25$ 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 $n^2/25$ 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.

**Category / Keywords: **bounded storage model

**Date: **received 19 Feb 2016

**Contact author: **ran raz mail at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20160219:201958 (All versions of this report)

**Short URL: **ia.cr/2016/170

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