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As we examine the behaviour of the number field sieve (NFS) in the medium prime case,we notice various patts that can be exploited to improve the running time of the sieving stage.The contributions of these observations to the computational mathematics community are twofold.Firstly,we clarify the understanding of the true practical effectiveness of the algorithm.Secondly,we propose a test for a better choice of the polynomials used in the NFS.These results are of particular interest to cryptographers as the run-time of the NFS directly determines the security level of some discrete logarithm problem based protocols.