The asymptotically fastest algorithm to solve the Discrete Logarithm Problem in finite fields is the Number Field Sieve (NFS). This work presents a summary of the Number Field Sieve and its practical experimental implementation to solve the discrete logarithm problem in finite fields of degree six. This particular problem arises e.g. when one tries to solve DLP in XTR cryptosystem. As shown, the degree six instance of the DLP is practically more difficult to solve with NFS as a classical DLP. Also contained in this book are some specific remarks to the related topic of the polynomial selection for the NFS. A three dimensional adaptation of the line sieving algorithm is described as well as the parametrization choices for the sieve region, contribution of small primes and exclusion of higher degree ideals. Although the results of this work are related to the specific instance of NFS, they can influence also the mainstream NFS applications (the factoring of integers or the classical DLP).

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