Developed by L. S. Hill in 1929, the Hill cipher is a polygraphic substitution cipher based on matrix multiplication. This cipher has been proved vulnerable to many attacks, especially the known-plaintext attack, while only few ciphertext-only attacks have been developed. The aim of our work is to study a new kind of ciphertext-only attack for the Hill cipher which is based on a restricted search over an explicit set of texts, called orbits, and not on a search over the key-space; it is called Orbit-Based Attack (OBA). To explain in a convenient setting this approach, we make use of basic notions from group action theory ; we present then in details an algorithm for this attack and finally results from experiments. We demonstrate experimentally that this new method can be efficient in terms of time-execution and can even be faster on average than the classical Brute-Force Attack in the considered settings.