Cryptography with Streaming Algorithms.

We put forth the question of whether cryptography is feasible using streaming devices. We give constructions and prove lower bounds. In streaming cryptography (not to be confused with streamciphers) everything-the keys, the messages, and the seeds-are huge compared to the internal memory of the device. These streaming algorithms have small internal memory size and make a constant number of passes over big data maintained in a constant number of read/write external tapes. Typically, the internal memory size is O(log n) and we use 2 external tapes; whereas 1 tape is provably insufficient. In this setting we cannot compute instances of popular intractability assumptions. Nevertheless, we base cryptography on these assumptions by employing non-black-box techniques, and study its limitations. We introduce new techniques to obtain unconditional lower bounds showing that no super-linear stretch pseudorandom generator exists, and no Public Key Encryption (PKE) exists with private-keys of size sub-linear in the plaintext length. For possibility results, assuming the existence of one-way functions computable in NC1-e.g. factoring, lattice assumptions-we obtain streaming algorithms computing one-way functions and pseudorandom generators. Given the Learning With Errors (LWE) assumption we construct PKE where both the encryption and decryption are streaming algorithms. The starting point of our work is the groundbreaking work of Applebaum-Ishai-Kushilevitz on Cryptography in NC0. In the end, our developments are technically orthogonal to their work; e.g. there is a PKE where the decryption is a streaming algorithm, whereas no PKE decryption can be in NC0.