The task of data compression originally arises in the field of communications, where messages from a source must be delivered to a receiver in the most compact form, without losing (too much) information. Besides the countless practical applications in engineering, data compression is interesting enough to be studied on its own, its foundations being rooted in probability theory, optimization and other related disciplines which are in turn deeply linked to statistical physics. The class of problems analyzed here can be described as follows: given a random bit string of fixed length, design a compressing algorithm that casts it into a shorter string and a de-compressing algorithm that is able to reconstruct the original message up to some error.