A Spin Glass Model for Reconstructing Nonlinearly Encrypted Signals Corrupted by Noise

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We define a (symmetric key) encryption of a signal s∈ RN as a random mapping s→y= (y1,...,yM)T ∈ RM 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 =i s2 i /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∞ ∈[ 0,1] between the original signal and its recovered image(known as’estimate’) as N →∞,for a given(’bare’)noise-to-signal ratio (NSR) γ ≥0. Such an overlap is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic 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 someintervalofNSR.Weshowthatencryptionswithanonvanishinglinearcomponentpermit reconstructions with p∞ > 0 for anyμ>1 and anyγ<∞, 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 γc(μ) such that p∞ = 0 forγ>γ c(μ) making the reconstruction impossible. The behaviour close to the threshold is given
Original languageEnglish
Pages (from-to)789-818
Number of pages30
JournalJournal of Statistical Physics
Issue number5
Early online date12 Jan 2019
Publication statusPublished - 15 Jun 2019


  • Inference
  • Signal reconstruction
  • Spin glass


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