A Spin Glass Model for Reconstructing Nonlinearly Encrypted Signals Corrupted by Noise
Abstract
We define a (symmetric key) encryption of a signal s\in {R^N} as a random mapping s\mapsto y =(y_1,\ldots ,y_M)^T\in R^M known both to the sender and a recipient. In general the recipients may have access only to images y corrupted by an additive noise of unknown strength. Given the encryption redundancy parameter (ERP) μ =M/N≥1 and the signal strength parameter R=√{\sum _i {s_i^2/N}}, we consider the problem of reconstructing s 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 to evaluate the mean overlap p_{∞}\in [0,1] between the original signal and its recovered image (known as 'estimate') as N→ ∞, for a given ('bare') noisetosignal ratio (NSR) γ ≥ 0. Such an overlap is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linearquadratic family of random mappings and discuss the full p_{∞ } (γ ) curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in some interval of NSR. We show that encryptions with a nonvanishing linear component permit reconstructions with p_{∞}>0 for any μ >1 and any γ <∞, with p_{∞}∼ γ ^{1/2} as γ → ∞. In contrast, for the case of purely quadratic nonlinearity, for any ERP μ >1 there exists a threshold NSR value γ _c(μ ) such that p_{∞}=0 for γ >γ _c(μ ) making the reconstruction impossible. The behaviour close to the threshold is given by p_{∞}∼ (γ _cγ )^{3/4} and is controlled by the replica symmetry breaking mechanism.
 Publication:

Journal of Statistical Physics
 Pub Date:
 June 2019
 DOI:
 10.1007/s10955018022179
 arXiv:
 arXiv:1805.06982
 Bibcode:
 2019JSP...175..789F
 Keywords:

 Spin glass;
 Signal reconstruction;
 Inference;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 Computer Science  Information Theory;
 Mathematics  Probability
 EPrint:
 33 pages, 5 figures