Convolutional block codes with cryptographic properties over the semi-direct product ${\mathbb {Z}}/N{\mathbb {Z}} \rtimes {\mathbb {Z}}/M{\mathbb {Z}}$

Abstract

Classic convolutional codes are defined as the convolution of a message and a transfer function over $\mathbb {Z}$. In this paper, we study time-varying convolutional codes over a finite group G of the form ${\mathbb {Z}}/N{\mathbb {Z}} \rtimes {\mathbb {Z}}/M{\mathbb {Z}}$. The goal of this study is to design codes with cryptographic properties. To define a message u of length k over the group G, we choose a subset E of G that changes at each encoding, and we put $u = \sum _i u_iE(i)$. These subsets E are generated chaotically by a dynamical system, walking from a starting point (x, y) on a space paved by rectangles, each rectangle representing an element of G. So each iteration of the dynamical system gives an element of the group which is saved on the current E. The encoding is done by a convolution product with a fixed transfer function. We have found a criterion to check whether an element in the group algebra can be used as a transfer function. The decoding process is realized by syndrome decoding. We have computed the minimum distance for the group $G=\mathbb {Z}/7\mathbb {Z} \rtimes \mathbb {Z}/3\mathbb {Z}$. We found that it is slightly smaller than those of the best linear block codes. Nevertheless, our codes induce a symmetric cryptosystem whose key is the starting point (x, y) of the dynamical system. Consequently, these codes are a compromise between error correction and security.