<p>The number field sieve (NFS) is the state-of-the-art algorithm for integer factorization and discrete logarithm computation over finite fields of non-small characteristic. Many variants have been proposed to enhance the effectiveness of NFS both theoretically and practically. However, the complexity analyses of these variants are all in ad hoc manners. Whenever a new variant is proposed, the overall complexity needs to be reanalyzed. In this paper, we give the complexity formulae for NFS and its variants. These formulae are not limited to any specific polynomial selection method, and have no restriction on the degree of the polynomials we sieve on or the number of fields we use. Then for a given new variant, there is no necessity to reanalyze the complexity, but only to substitute certain parameters in the complexity formulae directly, and the advantages of the new algorithm at the complexity level can be easily seen. While the previous analyses are based on the multivariate optimization methods, our analysis method is more elementary. By utilizing the formulae, we establish the lower bounds of the complexities of NFS and all its variants, which are proved to be not lower than <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L(1/3,\root 3 \of {\frac{32}{9}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>3</mn> <mo>,</mo> <mroot> <mfrac> <mn>32</mn> <mn>9</mn> </mfrac> <mn>3</mn> </mroot> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Hence, in order to achieve a theoretical breakthrough, it is necessary to break the current framework of the fields construction and introduce innovative concepts.</p>