[Paper Review] Embedding Cryptographic Features in Compressive Sensing

Yushu Zhang, Kwok‐Wo Wong+3 more|arXiv (Cornell University)|Mar 25, 2014

Sparse and Compressive Sensing Techniques38 references4 citations

TL;DR

This paper proposes a two-level protection model (TLPM) for secure compressive sensing (SCS) by embedding chaos-based random permutation and a chaotic measurement matrix into parallel compressive sensing (PCS), achieving joint compression, encryption, and improved compression performance. The method ensures high security, robustness against noise and cropping, and maintains or enhances compression ratios, unlike traditional joint schemes.

ABSTRACT

Compressive sensing (CS) has been widely studied and applied in many fields. Recently, the way to perform secure compressive sensing (SCS) has become a topic of growing interest. The existing works on SCS usually take the sensing matrix as a key and the resultant security level is not evaluated in depth. They can only be considered as a preliminary exploration on SCS, but a concrete and operable encipher model is not given yet. In this paper, we are going to investigate SCS in a systematic way. The relationship between CS and symmetric-key cipher indicates some possible encryption models. To this end, we propose the two-level protection models (TLPM) for SCS which are developed from measurements taking and something else, respectively. It is believed that these models will provide a new point of view and stimulate further research in both CS and cryptography. Specifically, an efficient and secure encryption scheme for parallel compressive sensing (PCS) is designed by embedding a two-layer protection in PCS using chaos. The first layer is undertaken by random permutation on a two-dimensional signal, which is proved to be an acceptable permutation with overwhelming probability. The other layer is to sample the permuted signal column by column with the same chaotic measurement matrix, which satisfies the restricted isometry property of PCS with overwhelming probability. Both the random permutation and the measurement matrix are constructed under the control of a chaotic system. Simulation results show that unlike the general joint compression and encryption schemes in which encryption always leads to the same or a lower compression ratio, the proposed approach of embedding encryption in PCS actually improves the compression performance. Besides, the proposed approach possesses high transmission robustness against additive Gaussian white noise and cropping attack.

Motivation & Objective

• To address the lack of a concrete, operable encipher model in existing secure compressive sensing (SCS) schemes that treat the sensing matrix merely as a key.
• To systematically investigate SCS by drawing parallels between compressive sensing and symmetric-key ciphers.
• To design an efficient, secure encryption scheme for parallel compressive sensing (PCS) that integrates encryption directly into the compression process.
• To improve compression performance while ensuring high security and robustness against common attacks such as additive Gaussian white noise and cropping.
• To provide a new framework—two-level protection models (TLPM)—that stimulates further research in both compressive sensing and cryptography.

Proposed method

• The first level of protection applies a random permutation to a 2D signal using a chaotic system, ensuring that the sparsity level of each column in the permuted signal is reduced with overwhelming probability.
• The second level uses a chaotic measurement matrix with i.i.d. sub-Gaussian entries that satisfy the restricted isometry property (RIP) for PCS with overwhelming probability.
• Both the random permutation and the chaotic measurement matrix are generated under the control of a single chaotic system, ensuring key synchronization between encoder and decoder.
• The method leverages the inherent properties of chaos—sensitivity to initial conditions and ergodicity—to generate secure, unpredictable, and uniformly distributed transformations.
• The encryption is embedded directly into the PCS framework, avoiding the typical trade-off where encryption degrades compression ratio.
• Theoretical analysis proves that the permutation is acceptable (i.e., reduces maximum column sparsity) with probability approaching one, and the measurement matrix preserves signal recovery under RIP conditions.

Experimental results

Research questions

• RQ1 Can a systematic and practical model for secure compressive sensing be developed that integrates encryption directly into the compression and sampling process?
• RQ2 How can the relationship between compressive sensing and symmetric-key ciphers be exploited to design a secure and efficient encryption mechanism?
• RQ3 Does embedding encryption into the PCS framework improve or degrade the compression ratio compared to traditional joint compression-encryption schemes?
• RQ4 What is the robustness of the proposed scheme against common signal attacks such as additive Gaussian white noise and cropping?
• RQ5 Can a two-level protection model (TLPM) based on chaos provide both high security and favorable compression performance in PCS?

Key findings

• The proposed scheme improves compression performance by embedding encryption directly into the PCS process, contrary to the typical degradation seen in joint compression-encryption schemes.
• The random permutation layer reduces the maximum column sparsity with overwhelming probability, as proven by a probabilistic analysis resembling the balls-and-boxes problem.
• The chaotic measurement matrix satisfies the restricted isometry property (RIP) with overwhelming probability, ensuring stable and robust signal recovery.
• The scheme exhibits high robustness against additive Gaussian white noise and cropping attacks, maintaining signal integrity and recovery quality.
• The encryption scheme demonstrates high key sensitivity, ensuring that small changes in the secret key lead to significantly different encrypted outputs.
• Simulation results confirm that the proposed method achieves both high security and improved compression efficiency, validating the theoretical claims.