Powers of codes and applications to cryptography

Given a linear error correcting code C, its m-th power is defined as the linear span of the set of all coordinate-wise products of m (not necessarily distinct) codewords in C. The study of powers of codes (and especially squares) is relevant in a number of recent results in several areas of cryptography where we need to bound certain parameters (such as the dimension and the minimum distance) of both a linear code and some power of it simultaneously. These areas include most notably secret sharing and multiparty computation, but also two-party cryptography and public key cryptography. In this paper, some of these applications will be discussed together with several recent results and some open challenges.