The interest in subspace codes has increased recently due to their application in error correction for random network coding. In order to study their properties and find good constructions, the notion of cyclic subspace codes was introduced by using the extension field structure of the ambient space. However, to this date there exists no general construction with a known relation between k, the dimension of the codewords, and n, the dimension of the entire space.

In this talk we present a construction of a full orbit cyclic subspace code in which n is quadratic in k. This construction is based on Sidon sets, which are sets of integers in which all pairwise sums are distinct. We prove the existence of a full orbit code where n is linear in k, and show that any such code induces a Sidon set. Finally, we show a surprising application of our techniques for devising a private-key cryptosystem, and lay the grounds for a public-key one.