2018/05/17 by Yan V. Fyodorov, Yan V Fyodorov
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Chaos-based Image/Signal Encryption #Encryption #Gaussian #Gaussian noise #Mathematical Analysis and Transform Methods #Nonlinear system #Random field #Random variable #Redundancy (engineering) #Replica #Spin glass #Wireless Communication Security Techniques #cond-mat.dis-nn #cond-mat.stat-mech #cs.IT #math.IT #math.PR
paper · pdf · doi:10.1007/s10955-018-02217-9
published as Journal of Statistical Physics, vol. 175, Issue 5, 789--818 (2019) · 33 pages, 5 figures
arxiv created 2018/05/17 · openalex created_date 2018/06/01 · openalex publication_date 2019/01/12 · arxiv updated 2019/06/11 · openalex updated_date 2026/08/05
We define a (symmetric key) encryption of a signal s∈ \mathbb RN as a random mapping s↦ \mathbf y =(y1,… ,yM)T∈ \mathbb 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 si2/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 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 c-γ )3/4 and is controlled by the replica symmetry breaking mechanism.