Reconstructing Encrypted Signals: Optimization with input from Spin Glasses and RMT

We define a (symmetric key) encryption of a signal as a random mapping known both to the sender and a recipient. In general the recipients may have access only to images corrupted by an additive noise of unknown strength. Given the encryption redundancy parameter and the signal strength parameter, we consider the problem of reconstructing the signal from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to finding the configuration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian random interaction potential. We use the Parisi replica symmetry breaking scheme for evaluating the mean overlap between the original signal and its recovered image. As a related, but separate problem, we will also briefly discuss the cost function "landscape" in the simplest random Least Square optimization on a sphere.