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Let $X$ be a projective K3 surface over $\mathbb C$. We prove that its Coxring $R(X)$ has a generating set whose degrees are either classes of smoothrational curves, sums of at most three elements of the Hilbert basis of the nefcone, or of the form $2(f+f')$, where $f,f'$ are classes of elliptic fibrationswith $f\cdot f'=2$. This result and techniques using Koszul's type exactsequences allow to determine a generating set for the Cox ring of all Moridream K3 surfaces of Picard number three which is minimal in most cases. Apresentation for the Cox ring is given in some special cases with fewgenerators.

Cox rings of K3 surfaces of Picard number three