Grothendieck's standard conjecture, which is a set of conjectures on algebraic cycles, is wide open. In this talk, I will prove the standard conjecture for the square of a K3 surface in positive characteristic. The new part is the Hodge standard conjecture, which predicts certain positivity of the intersection product. Our main ingredient is the Kuga-Satake period map from the moduli space of K3 surfaces to a certain period domain, called the orthogonal Shimura variety. This is joint work with Tetsushi Ito and Teruhisa Koshikawa.