On the List and Bounded Distance Decodibility of the Reed-Solomon Codes (Extended Abstract)

For an error-correcting code and a distance bound, the list decoding problem is to compute all the codewords within the given distance to a received message. The bounded distance decoding problem, on the other hand, is to find one codeword if there exists one or more codewords within the given distance, or to output the empty set if there does not. Obviously the bounded distance decoding problem is not as hard as the list decoding problem. For a Reed-Solomon code [n, k]q, a simple counting argument shows that for any integer g < n, there exists at least one Hamming ball of radius n − g, which contains at least ( n g) qg−k many codewords. Let ĝ(n, k, q) be the smallest integer g such that (ng) qg−k < 1. For the distance bound between n− √ nk and n− ĝ(n, k, q), we do not know whether the Reed-Solomon code is list, or bounded distance decodable, nor do we know whether there are polynomially many codewords in all balls of the radius. It is generally believed that the answers to both questions are no. There are public key cryptosystems proposed recently, whose security is based on the assumptions. In this paper, we prove: (1) List decoding can not be done for radius n− ĝ(n, k, q) or larger, otherwise the discrete logarithm over Fqĝ(n,k,q)−k is easy. (2) Let h be a positive integer satisfying h < q − 2. We show that the discrete logarithm problem over Fqh can be efficiently reduced to the bounded distance decoding problem of the Reed-Solomon code [q, 3h+4]q with radius q−4h−4. These results show that the decoding problems for the Reed-Solomon code are at least as hard as the discrete logarithm problem over finite fields. The main tools to obtain these results are an interesting connection between the problems of list-decoding of Reed-Solomon code and the problems of discrete logarithms over finite fields, and a generalization of the Katz’s theorem, which concerns representations of elements in an extension finite field by products of linear factors.