  The Number Field Sieve and its numerous variants is the best algorithm to
compute discrete logarithms in medium and large characteristic finite fields.
When the extension degree n is composite and the characteristic p is of medium
size, the Tower variant (TNFS) is asymptotically the most efficient one. Our
work deals with the last main step, namely the individual logarithm step, that
computes a smooth decomposition of a given target T in the finite field thanks
to two distinct phases: an initial splitting step, and a descent tree. In this
article, we improve on the current state-of-the-art Guillevic's algorithm
dedicated to the initial splitting step for composite n. While still exploiting
the proper subfields of the target finite field, we modify the lattice
reduction subroutine that creates a lift in a number field of the target T. Our
algorithm returns lifted elements with lower degrees and coefficients,
resulting in lower norms in the number field. The lifted elements are not only
much likely to be smooth because they have smaller norms, but it permits to set
a smaller smoothness bound for the descent tree. Asymptotically, our algorithm
is faster and works for a larger area of finite fields than Guillevic's
algorithm, being now relevant even when the medium characteristic p is such
that Lpn (1/3) $\le$ p < Lpn (1/2). In practice, we conduct experiments on
500-bit and 700-bit composite finite fields: Our method becomes more efficient
as the largest non trivial divisor of n grows, being thus particularly adapted
to even extension degrees.
