  We describe a general algorithm for the computation of the remainder function
for n-gluon scattering in multi-Regge kinematics for strongly coupled planar
N=4 super Yang-Mills theory. This regime is accessible through the infrared
physics of an auxiliary quantum integrable system describing strings in
AdS5xS5. Explicit formulas are presented for n=6 and n=7 external gluons. Our
results are consistent with expectations from perturbative gauge theory. This
paper comprises the technical details for the results announced in
arXiv:1405.3658 .
