Abstract
The best practical algorithm for class group computations in imaginary quadratic number fields (such as group structure, class number, discrete logarithm computations) is a variant of the quadratic sieve factoring algorithm. Paradoxical as it sounds, the principles of the number field sieve, in a strict sense, could not be applied to number field computations, yet. In this article we give an indication of the obstructions.
In particular, we first present fundamental core elements of a number field sieve for number field computations of which it is absolutely unknown how to design them in a useful way. Finally, we show that the existence of a number field sieve for number field computations with a running time asymptotics similar to that of the genuine number field sieve likely implies the existence of an algorithm for elliptic curve related computational problems with subexponential running time.
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Bauer, M.L., Hamdy, S. (2003). On Class Group Computations Using the Number Field Sieve. In: Laih, CS. (eds) Advances in Cryptology - ASIACRYPT 2003. ASIACRYPT 2003. Lecture Notes in Computer Science, vol 2894. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-40061-5_19
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DOI: https://doi.org/10.1007/978-3-540-40061-5_19
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