For an error-correcting code and a distance bound, the list decoding problem is to compute all the codewords within a given distance to a received message. The bounded distance decoding problem is to find one codeword if there is at least one codeword within the given distance, or to output the empty set if there is 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 $0 0 $. We show that the discrete logarithm problem over ${\bf F}_{q^{h}}$ can be efficiently reduced by a randomized algorithm to the bounded distance decoding problem of the Reed-Solomon code $[q, g-h]_q$ with radius $q - g$. These results show that the decoding problems for the Reed-Solomon code are at least as hard as the discrete logarithm problem over certain finite fields. For the list decoding problem of Reed-Solomon codes, although the infeasible radius that we obtain is much larger than the radius, which is known to be feasible, it is the first nontrivial bound. Our result on the bounded distance decodability of Reed-Solomon codes is also the first of its kind. The main tools for obtaining these results are an interesting connection between the problem of list decoding of Reed-Solomon code, the problem of a discrete logarithm over finite fields, and a generalization of Katz’s theorem on representations of elements in an extension finite field by products of distinct linear factors.