Johnson-Lindenstrauss lemma for circulant matrices

We prove a variant of a Johnson-Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimise the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space $k=O({ϵ}^{-2}{\log }^{3}n)$ instead of the classical bound $k=O({ϵ}^{-2}\log n)$.