  Computing discrete logarithms in finite fields is a main concern in
cryptography. The best algorithms in large and medium characteristic fields
(e.g., {GF}$(p^2)$, {GF}$(p^{12})$) are the Number Field Sieve and its variants
(special, high-degree, tower). The best algorithms in small characteristic
finite fields (e.g., {GF}$(3^{6 \cdot 509})$) are the Function Field Sieve,
Joux's algorithm, and the quasipolynomial-time algorithm. The last step of this
family of algorithms is the individual logarithm computation. It computes a
smooth decomposition of a given target in two phases: an initial splitting,
then a descent tree. While new improvements have been made to reduce the
complexity of the dominating relation collection and linear algebra steps,
resulting in a smaller factor basis (database of known logarithms of small
elements), the last step remains at the same level of difficulty. Indeed, we
have to find a smooth decomposition of a typically large element in the finite
field. This work improves the initial splitting phase and applies to any
nonprime finite field. It is very efficient when the extension degree is
composite. It exploits the proper subfields, resulting in a much more smooth
decomposition of the target. This leads to a new trade-off between the initial
splitting step and the descent step in small characteristic. Moreover it
reduces the width and the height of the subsequent descent tree.
