Phase-Only Digital Encryption

Abstract

We study the n-dimensional deconvolution lem with an impulse response function and an (additive) noise function that are both characterised by the same phase-only stochastic spectrum. In this case, it is shown that the deconvolution problem becomes well-posed and has a general solution that is both exact and unique, subject to a re-normalisation condition relating to the scale of the solution. While the phase-only spectral model considered is of limited value in general (in particular, problems arising in the fields of digital signal processing and communications engineering, specifically with regard to the retrieval of information from noise), its application to digital cryptography has potential. One of the reasons for this (as discussed in this paper), is that it provides a method of encrypting data where the diffused plaintext can be effectively embedded in a (phase-only) cipher (subject to the floating point precision used for data processing), thereby fully dissipating the statistical signature of the plaintext in the distribution of the cipher. Further, a decrypt can be generated that is computationally efficient subject to the usual cases of sender and receiver having access to identical algorithm(s) and key(s), deconvolution being equivalent to decryption in the context of the (phase-only) encryption model that is considered. For the two-dimensional case, this approach has a potential weakness in terms of a 'correlation attack' using phase retrieval algorithms and a solution to this problem is provided by introducing a (stochastic) amplitude weighting function. Prototype MATLAB functions are provided in the Appendices that accompany this paper to give readers the opportunity to repeat the computational results presented and extend them further. The functions constitute a symmetric algorithm for encrypting and decrypting full colour images in which the key(s) have been exchanged a priori. In this context, the final part of the paper considers the application of phase-only encryption for key exchange using a Threeway Pass Protocol for which a further prototype MATLAB function is provided for validation and further development of the approach by interested readers.