A new nonlinear encryption-decryption scheme for two images using: double random phase encoding (DRPE), random phase masks (RPMs), amplitude and phase encoding, correlation operation in Fresnel domain (FrD) and truncation operations (amplitude truncation, AT and phase truncation, PT), is proposed. AT and PT are operations that work on complex-valued functions (images) in a nonlinear way, with the purpose of selecting the information contained in a complex-valued image, respectively. The first step of the encryption scheme is encoding two original images in amplitude and phase. The encryption or decryption scheme utilizes the correlation operation in FrD, AT, PT and two RPMs in a specific way to obtain the encrypted or decrypted images. The use of the AT and PT operations allow a better security of the encryption and decryption schemes because the nonlinearities introduced by these operations and the generation of two new encryption keys. The encrypted image of this proposal is real valued and the correlation operation in FrD introduces two new keys improving the security of the encryption and decryption schemes. The keys of the proposed security schemes are six and the right values of these all keys have to be used in the decryption scheme to recover the two initial images used in the input of the encryption scheme.

The Fresnel transform (FrT) is commonly used to describe the free-space propagation of optical waves. In this work, we present new definitions for the convolution, correlation and generalized shift operations based on the FrT. The generalized shift operation is defined by using simultaneous space and phase shifts. The generalized shift operation is useful for centred optical systems in the Fresnel domain (FrD) when the data distributions at the input plane of the optical system are shifted. The new convolution and correlation operations defined in terms of the FrT, the wavelength and the propagation distance, can be considered as a generalization of the usual convolution and correlation operations. The sampling theorem for distributions, whose resulting FrT has finite support, is formulated by using the new convolution operation introduced in this work and a new definition of the Dirac comb function. These new definitions and results could be applied to describe, design and implement optical processing systems related to the FrT. Finally, we present a centred optical systems used in holography and optical security systems that can be described or modelled by the new definitions of the operations proposed in this paper.