  The power spectrum of cosmic microwave background (CMB) lensing will be
measured to sub-percent precision with upcoming surveys, enabling tight
constraints on the sum of neutrino masses and other cosmological parameters.
Measuring the lensing power spectrum involves the estimation of the connected
trispectrum of the four-point function of the CMB map, which requires the
subtraction of a large Gaussian disconnected noise bias. This reconstruction
noise bias receives contributions both from CMB and foreground fluctuations as
well as instrument noise (both detector and atmospheric noise for ground-based
surveys). The debiasing procedure therefore relies on the quality of
simulations of the instrument noise which may be expensive or inaccurate. We
propose a new estimator that makes use of at least four splits of the CMB maps
with independent instrument noise. This estimator makes the CMB lensing power
spectrum completely insensitive to any assumptions made in modeling or
simulating the instrument noise. We show that this estimator, in many practical
situations, leads to no substantial loss in signal-to-noise. We provide an
efficient algorithm for its computation that scales with the number of splits
$m$ as $\mathcal{O}(m^2)$ as opposed to a naive $\mathcal{O}(m^4)$ expectation.
