SIMULTANEOUS COMPRESSIVE SENSING AND OPTICAL ENCRYPTION OF SIGNALS AND IMAGES

Transcription

1 SIMULTANEOUS COMPRESSIVE SENSING AND OPTICAL ENCRYPTION OF SIGNALS AND IMAGES Dr. Ertan Atar Türk Telekom İstanbul-I Area Offices İstanbul, Turkey Prof. Dr. Okan K. Ersoy Scool of Electrical &Computer Engineering, Purdue University Indiana, USA and Bogazici University, Electrical and Electronic Engineering Department, İstanbul, Turkey Assist. Prof. Dr. Lale Özyılmaz Electronics&Com. Eng. Dept. Yıldız Technical University İstanbul, Turkey

2 CONTENTS Abstract I. Introduction II. Compressive Sensing Orthogonal Matching Pursuit III. Cryptography IV. Details of Implementation and Methods V. Experimental Results VI. Conclusions References

3 ABSTRACT Compressive Sensing (CS) makes it possible to reduce the amount of data and thereby to greatly simplify the sensor system. In this method, data is compressed before measurements whereas data is first measured and then compressed in the current technology. This approach leads to reducing the number of sensors. In this study, simultaneous compressive sensing and optical cryptography is developed as a new approach for encryption and decryption of signals and images.

4 ABSTRACT CS is carried out with the Orthogonal Matching Pursuit (OMP) algorithm. The CS-OMP algorithm is next combined with the Double Random Phase/Amplitude Encyrption (DRPAE) method to achieve both compression and encyrption of data. In order to achieve high security, the keys used in CS and DRPAE are transmitted to the receiver by the asymmetric RSA cryptography method.

5 ABSTRACT Thus, the overall cryptographic system is a hybrid optical system (both symmetric and asymmetric) since DRPAE is a symmetric optical encyrption method. In this approach, data size is decreased and use of detection matrix in CS also protects DRPAE optical keys against well-known attacks. Experimental results both with 1-D signals and images are presented.

6 The signal is transformed by a measurement matrix to another space and measured in compressed form, the measurement matrix must be such that the original signal can be recovered without loss. I. INTRODUCTION According to classical signal processing methods, when the signal is sampled, Nyquist criteria must be satisfied to avoid loss of information. After sampling, if the signal is compressible, the signal is further compressed typically by a transform method. In the CS method, the signal is supposed to be sparse in the sense that most signal values are assumed to be negligible (zero).

7 I. INTRODUCTION

8 I. INTRODUCTION Several advantages of compression detection are given below: Reducing the number of sensors, thereby transferring more information with fewer examples. Reducing the data loading process which is especially costly in image processing. Reduced power consumption and hardware complexity.

9 I. INTRODUCTION Cryptography involves encryption and decryption of information. The mathematical methods for security of information, data integrity and identity validation are significant topics of cryptography. In this study, we propose a method for simultaneous compressive sensing, encyrption and decryption of signals and images using orthogonal matching pursuit and optical cryptography.

10 II. COMPRESSIVE SENSING ORTHOGONAL MATCHING PURSUIT Orthogonal Matching Pursuit (OMP) is an iterative method to recover the nonzero elements of a sparse signal from the measured signal. If the length m of the measurement vector is larger than k, the number of nonzero elements of the sparse signal, the sparse signal can be recovered by OMP with a very large probability.

11 II. COMPRESSIVE SENSING ORTHOGONAL MATCHING PURSUIT

12 II. COMPRESSIVE SENSING ORTHOGONAL MATCHING PURSUIT In the OMP program, the detection matrix was generated by using one of a set of fast transforms, namely, Fast Fourier transform (FFT), Discrete Cosine Transform (DCT), Real Sinesoidal Transform (RST), Haar wavelet transform, Hadamard transform, Discrete Sine Transform (DST) and Discrete Cosine-III Transform (DC3T). In the experiments discussed in Section 5, the ranking of transforms in terms of accuracy of results were DC3T, DCT, FFT, RST, DST, Hadamard and Haar. In addition, DC3T has fewer number of multiplications than DCT.

13 III. CRYPTOGRAPHY Cryptography refers to the methods of encryption and decryption of information. Cryptoanalysis involves methods to investigate the strengths and weaknesses of cryptographic systems. A cryptosystem involves five elements: P=Plaintext K=Key E=Encryption C=Ciphertext D=Decryption

14 III. CRYPTOGRAPHY

15 III. CRYPTOGRAPHY Symmetric encryption algorithms use the same key for encryption and decryption operations. Symmetric encryption is very fast and easy to implement in electronic devices. In asymmetric encryption, one general key is used for encryption, and another special key is used for decryption. Asymmetric encryption algorithms are usually developed with very large prime numbers. In the proposed method, the RSA algorithm was used for asymmetric encryption of the keys.

16 III. CRYPTOGRAPHY

17 RSA; III. CRYPTOGRAPHY

18 III. CRYPTOGRAPHY Using asymmetric and symmetric encryptions together is named hybrid cryptography. Hybrid cryptography increases the advantages and decreases the disadvantages of the asymmetric and symmetric encryptions.

19 IV. DETAILS OF IMPLEMENTION AND METHODS Simultaneous Compressive Sensing and Hybrid Optical Cryptography (SCOSC) is proposed for effective and efficient compression and secure transmission of signals and images. In this work, CS is achieved with the orthogonal matching pursuit (OMP) algorithm. Then, the CS-OMP algorithm is combined with the double random phase plus amplitude encyrption (DRPAE) method to achieve both compression and encyrption of data.

20 IV. DETAILS OF IMPLEMENTION AND METHODS In order to achieve high security, the keys used in CS and DRPAE are transmitted to the receiver by an asymmetric cryptography method (currently the RSA method). Thus, the overall cryptographic system is a hybrid optical system (both symmetric and asymmetric) since DRPAE is a symmetric optical encryption method. In this approach, data size is decreased, and use of detection matrix in CS also improves DRPAE optical keys against attacks.

21 IV. DETAILS OF IMPLEMENTION AND METHODS

22 IV. DETAILS OF IMPLEMENTION AND METHODS

23 IV. DETAILS OF IMPLEMENTION AND METHODS Next the sparse signal per block is generated with length M satisfying M K* log(n) by keeping M largest elements in magnitude and zeroing the others. The detection matrix is generated by pre and post multiplying a fast transform matrix with random permutation matrices. The SCOSC method is applied per block of information.

24 IV. DETAILS OF IMPLEMENTION AND METHODS The optical phase/amplitude keys p1 and p2 are generated with random complex numbers. The phase covers 360 degrees. This is repeated once more for a 4-f optical encryption system for additional security. The resulting encrypted signal with the overall system is of smaller size than the original signal.

25 V. EXPERIMENTAL RESULTS

26 V. EXPERIMENTAL RESULTS

27 V. EXPERIMENTAL RESULTS

28 V. EXPERIMENTAL RESULTS

29 V. EXPERIMENTAL RESULTS Phase recovery and dirac delta function methods were studied for cryptoanalysis of the proposed method. Phase recovery methods are not effective when the keys of the 4f system are phase/amplitude keys. The dirac delta function method is effective in recovering the complete encryption matrix, but this is not usable for decryption since OMP requires the measurement matrix which is not available.

30 VI. CONCLUSIONS In this study, a method for simultaneous compressive sensing using OMP and optical cryptography using double random phase/amplitude keys was proposed. The RSA algorithm involving asymmetric cryptography was used to encrypt/decrypt the keys.

31 VI. CONSLUSIONS The phase/amplitude keys were used in a 4f optical cryptography system. The measurement matrix for the OMP algorithm was also constructed using Fourier related fast transforms such as Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT), Real Sinusoidal Transform (RST), Haar wavelet transform, Hadamard transform, Discrete Sine Transform (DST) and Discrete Cosine-III Transform (DC3T) together with random permutation matrices.

32 VI. CONCLUSIONS The experimental results with encryption/decryption of 1-D audio signals and 2-D images showed that the method is capable of high accuracy encryption and decryption with correct keys. It was not possible to achieve correct decryption with 1-D and 2-D input signals when the keys were wrong. Cryptoanalysis studies with phase recovery and dirac delta function methods showed that the proposed method is extremely difficult to attack.

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37 Dr. Ertan Atar Türk Telekom İstanbul-I Area Offices İstanbul, Turkey Prof. Dr. Okan K. Ersoy Electric&Computer Eng. Dept. Purdue University Indiana, USA Assist. Prof. Dr. Lale Özyılmaz Electronics&Com. Eng. Dept. Yıldız Technical University İstanbul, Turkey