On an encoding system constructed on the basis of generalized Reed--Solomon codes

V. M. Sidel'nikov ; S. O. Shestakov

Diskretnaya Matematika, Tome 4 (1992) no. 3, pp. 57-63.

Résumé

In earlier papers [1, 2], based on code-theoretic constructions, methods were presented for constructing an open encoding system. They are based on the well-known $\mathfrak B$ matrix of dimension $(s+1)\times N$ with elements from a finite field $\mathrm F_q$, of the form $\mathfrak B=H\cdot\mathfrak A$, where $\mathfrak A$ is some unknown matrix that is a test matrix of a $q$-valued generalized Reed–Solomon code, in particular of a Goppa code, and $H$ is an unknown nonsingular matrix with dimension $(s+1)\times(s+1)$. In this paper we present a method for finding the unknown matrices $\mathfrak A$ and $H$ with elements from the field $\mathrm F_q$ that determine the matrix $\mathfrak B$ in $O(s^4+sN)$ operations. Thus, we establish the unreliability of the open encoding systems considered.