Recent progress on the elliptic curve discrete logarithm problem
Cheng and Wan have related the decoding of Reed-Solomon codes to the computation of discrete logarithms over finite fields, with the aim of proving the hardness of their decoding. In this work, we experiment with solving the discrete logarithm over F qh using Reed-Solomon decoding. For fixed h and q going to infinity, we introduce an algorithm (RSDL) needing Õ(h! · q) operations over Fq, operating on a q × q matrix with (h+ 2)q nonzero coefficients. We give faster variants including an incremental version and another one that uses auxiliary finite fields that need not be subfields of F qh ; this variant is very practical for moderate values of q and h. We include some numerical results of our first implementations.