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Published online by Cambridge University Press: 30 May 2025
Let p be a prime number and n a positive integer, and let q = pn. Let 𝔽q be the field of q elements and denote by 𝔽*q the multiplicative subgroup of 𝔽*q. Assume t and u are elements in 𝔽*q with the property that u is in the subgroup generated by t. The discrete logarithm of u with respect to the base t, written logt u, is the least non-negative integer x such that tx= u.
In this paper we describe two methods to compute discrete logarithms, both of which derive from the number field sieve (NFS) factoring algorithm described in [Stevenhagen 2008] and [Lenstra and Lenstra 1993].
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