The Nonlinear Curvature Wavefront Sensor (nlCWFS), first proposed by Guyon, determines wavefront shape from images of a reference beacon in a number of planes between the pupil and focal plane of a telescope. We describe a new algorithm that rapidly recovers the low-order aberrations accurately enough to allow practical use of the nlCWFS in an adaptive optics (AO) system. The algorithm was inspired by refractive strong scintillation in the interstellar medium, which behaves similarly to near-pupil linear curvature focusing, but over larger scales. The refractive component is extracted from the speckled images by binning with the lowest-order aberrations being additionally estimated through the use of first and second distribution moments. The linearity of the refractive scintillation process allows us to use a reconstructor matrix to compute an estimate of the pupil wavefront. The resulting wavefront estimate is then applied in reverse to a deformable mirror (DM), reducing the nonlinearity to the point that a single update phase retrieval algorithm such as a multi-plane version of Gerchberg-Saxton (GS) can be used to estimate the remaining wavefront error (WFE). An AO simulation of a 1.5 m telescope, a 16x16 actuator DM, and four image planes show that the scintillation algorithm works, reducing ~800 nm rms WFE to ~ 40 nm, well below the fitting error (~90 nm) in closed loop. Once corrected to this level, the image planes still show a great deal of information that can then be used with a single-update wavefront retrieval algorithm. A couple simple variants of GS are suggested, including one that can be parallelized for each camera and run in parallel with the scintillation algorithm. A Monte Carlo study will be required to determine the best approach.