The Collins diffraction transform (CDT) describes the optical wave diffraction from the generic paraxial optical system. The CDT has as special cases the diffraction domains given by the Fourier, Fresnel and fractional Fourier transforms. In this paper, we propose to describe the optical double random phase encoding (DRPE) using a nonlinear joint transform correlator (JTC) and the CDT. This new description of the nonlinear JTC-based encryption system using the CDT covers several optical processing domains, such as Fourier, Fresnel, fractional Fourier, extended fractional Fourier and Gyrator domains, among others. The maximum number of independent design parameters or new security keys of the proposed encryption system using the CDT increases three times in comparison with the same encryption system that uses the Fourier transform. The proposed encryption system using the CDT preserves the shift-invariance property of the JTC-based encryption system in the Fourier domain, with respect to the lateral displacement of both the key random mask in the decryption process and the retrieval of the primary image. The viability of this encryption system is verified and analysed by numerical simulations.

We propose the use of the Jigsaw transform (JT) and the iterative cosine transform over a finite field in order to encrypt and decrypt images. The JT is a nonlinear operation that allows one to increase the security over the encrypted images by adding new keys to the encryption and decryption systems. The finite field is a finite set of integer numbers where the basic mathematical operations are performed using modular arithmetic. The finite field used in the encryption and decryption systems has an order given by the Fermat prime number 257. The iterative finite field cosine transform (FFCT) was used in our work with the purpose of obtaining images that had an uniform random distribution. We used a security key given by an image randomly generated and uniformly distributed. The JT and iterative FFCT was utilized twice in the encryption and decryption systems. The encrypted images presented a uniformly distributed histogram and the decrypted images were the same original images used as inputs in the encryption system. The resulting decrypted images had a high level of image quality in comparison to the image quality of the decrypted images obtained by the actual optical decryption systems. The proposed encryption and decryption systems have three security keys represented by two random permutations used in the JTs and one random image. The key space of the proposed encryption and decryption systems is larger. The previous features of the security system allow a better protection of the encrypted image against brute force and statistical analysis attacks.

We propose a new encryption system based on a nonlinear joint transform correlator (JTC) using the information of two biometrics (one digital fingerprint for each user) as security keys of the encryption system. In order to perform the decryption and authentication in a proper way, it is necessary to have the two digital fingerprints from the respective users whose simultaneous authentication is pursued. The proposed security system is developed in the Fourier domain. The nonlinearity of the JTC along with the five security keys given by the three random phase masks and the two digital fingerprints of the two users allow an increase of the system security against brute force and plaintext attacks. The feasibility and validity of this proposal is demonstrated using digital fingerprints as biometrics in numerical experiments.

The gyrator transform (GT) is used for images processing in applications of light propagation. We propose new image processing operators based on the GT, these operators are: Generalized shift, convolution and correlation. The generalized shift is given by a simultaneous application of a spatial shift and a modulation by a pure linear phase term. The new operators of convolution and correlation are defined using the GT. All these image processing operators can be used in order to design and implement new optical image processing systems based on the GT. The sampling theorem for images whose resulting GT has finite support is developed and presented using the previously defined operators. Finally, we describe and show the results for an optical image encryption system using a nonlinear joint transform correlator and the proposed image processing operators based on the GT.

A well-known technique for optical image encryption is the double random phase encoding (DRPE) technique, which uses two random phase masks (RPMs), one RPM at the input plane of the encryption system and the other RPM at the Fourier plane of the optical system, in order to obtain the encrypted image. In this work, we propose to use tilted planes for the Fourier and the output planes of the optical DRPE encryption system with the purpose of adding two new security keys, which are the angles of the tilted planes. The optical diffraction on a tilted plane is computed using the angular spectrum of plane waves and the coordinate rotation in the Fourier domain. The tilted distributions at the intermediate and output planes of the optical DRPE encryption system are the second RPM and the encrypted image, respectively. The angles of the tilted planes allow improvement to the security of the encrypted image. We perform several numerical simulations with the purpose of demonstrating the validity and feasibility of the proposed image encryption system.