The generalized Fibonacci transformations and application to image scrambling

This paper introduces a subfamily of the generalized Fibonacci sequence family, which we call the distinguished generalized Fibonacci sequence. Two members of this subfamily, the Fibonacci sequence and the Lucas sequence, are considered and two transformations, based on these sequences, are introduced. The applications of these transformations to image scrambling are studied in detail. It is found that these transformations have the desirable property of uniformity, that is, pixels that are equidistant in the original image remain equidistant after scrambling, albeit with different distance values. These transforms also spread adjacent pixels as far as possible. Besides totally decorrelating the image, these transformations also have the advantage of ease of implementation. This renders them useful for real-time and low cost implementations.