In Crypto 1997, Goldreich, Goldwasser and Halevi (GGH) proposed a lattice analogue of McEliece public key cryptosystem, which security is related to the hardness of approximating the closest vector problem (CVP) in a lattice. Furthermore, they also described how to use the same principle of their encryption scheme to provide a signature scheme. Practically, this cryptosystem uses the euclidean norm,
-norm, which has been used in many algorithms based on lattice theory. Nonetheless, many drawbacks have been studied and these could lead to cryptanalysis of the scheme. In this paper, we present a novel method of reducing a vector under the
-norm and propose a digital signature scheme based on it. Our scheme takes advantage of the
-norm to increase the resistance of the GGH scheme and to decrease the signature length. Furthermore, after some other improvements, we obtain a very efficient signature scheme, that trades the security level, speed and space.