Symmetric Ciphers Based on Two-dimensional Chaotic Maps

In this paper, methods are shown how to adapt invertible two-dimensional chaotic maps on a torus or on a square to create new symmetric block encryption schemes. A chaotic map is first generalized by introducing parameters and then discretized to a finite square lattice of points which represent pixels or some other data items. Although the discretized map is a permutation and thus cannot be chaotic, it shares certain properties with its continuous counterpart as long as the number of iterations remains small. The discretized map is further extended to three dimensions and composed with a simple diffusion mechanism. As a result, a symmetric block product encryption scheme is obtained. To encrypt an N × N image, the ciphering map is iteratively applied to the image. The construction of the cipher and its security is explained with the two-dimensional Baker map. It is shown that the permutations induced by the Baker map behave as typical random permutations. Computer simulations indicate that the cipher has good diffusion properties with respect to the plain-text and the key. A nontraditional pseudo-random number generator based on the encryption scheme is described and studied. Examples of some other two-dimensional chaotic maps are given and their suitability for secure encryption is discussed. The paper closes with a brief discussion of a possible relationship between discretized chaos and cryptosystems.