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

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 (a ₁ , b ₁ ), (a ₂ , b ₂ )..., where each at is uniformly distributed over {0,1} ⁿ and bt 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.