  The cosmic infrared background (CIB) sourced by infrared emission from dusty
star-forming galaxies is a valuable source of information on the star formation
history of the Universe. In measurements of the millimeter sky at frequencies
higher than $\sim 300$ GHz, the CIB and thermal emission from Galactic dust
dominate. A limited understanding of the CIB contribution at lower frequencies
on the other hand can hinder efforts to measure the kinetic Sunyaev-Zeldovich
spectrum on small scales as well as new physics that affects the damping tail
of the cosmic microwave background (CMB). The Planck satellite has measured
with high fidelity the CIB at 217, 353, 545 and 857 GHz. On very large scales,
this measurement is limited by our ability to separate the CIB from Galactic
dust, but on intermediate scales, the measurements are limited by sample
variance in the underlying matter field traced by the CIB. We show how
significant improvements (20-100%) can be obtained on parameters of star
formation models by cross-correlating the CIB (as measured from existing {\it
Planck} maps or upcoming CCAT-prime maps) with upcoming mass maps inferred from
gravitational lensing of the CMB. This improvement comes from improved
knowledge of the redshift distribution of star-forming galaxies as well as
through the use of the unbiased matter density inferred from CMB lensing mass
maps to cancel the sample variance in the CIB field. We also find that further
improvements can be obtained on CIB model parameters if the cross-correlation
of the CIB with CMB lensing is measured over a wider area while restricting the
more challenging CIB auto-spectrum measurement to the cleanest 5% of the sky.
