The class of the two-loop massless crossed boxes, with light-like external
legs, is the final unresolved issue in the program of computing the
scattering amplitudes of 2 –> 2 massless particles at
next-to-next-to-leading order. In this paper, we describe an algorithm for
the tensor reduction of such diagrams. After connecting tensor integrals
to scalar ones with arbitrary powers of propagators in higher dimensions,
we derive recurrence relations from integration-by-parts and
Lorentz-invariance identities, that allow us to write the scalar integrals
as a combination of two master crossed boxes plus simpler-topology
diagrams. We derive the system of differential equations that the two
master integrals satisfy using two different methods, and we use one of
these equations to express the second master integral as a function of the
first one, already known in the literature. We then give the analytic
expansion of the second master integral as a function of epsilon=(4-D)/2,
where D is the space-time dimension, up to order O(epsilon^0).