  The square $C^{*2}$ of a linear error correcting code $C$ is the linear code
spanned by the component-wise products of every pair of (non-necessarily
distinct) words in $C$. Squares of codes have gained attention for several
applications mainly in the area of cryptography, and typically in those
applications one is concerned about some of the parameters (dimension, minimum
distance) of both $C^{*2}$ and $C$. In this paper, motivated mostly by the
study of this problem in the case of linear codes defined over the binary
field, squares of cyclic codes are considered. General results on the minimum
distance of the squares of cyclic codes are obtained and constructions of
cyclic codes $C$ with relatively large dimension of $C$ and minimum distance of
the square $C^{*2}$ are discussed. In some cases, the constructions lead to
codes $C$ such that both $C$ and $C^{*2}$ simultaneously have the largest
possible minimum distances for their length and dimensions.
