  We establish a connection between semi-primitive roots of the multiplicative
group of integers modulo $2^{k}$ where $k\geq 3$, and the logarithmic base in
the algorithm introduced by Fit-Florea and Matula (2004) for computing the
discrete logarithm modulo $2^{k}$. Fit-Florea and Matula used properties of the
semi-primitive root 3 modulo $2^{k}$ to obtain their results and provided a
conversion formula for other possible bases. We show that their results can be
extended to any semi-primitive root modulo $2^{k}$ and also present a
generalized version of their algorithm to find the discrete logarithm modulo
$2^{k}$. Various applications in cryptography, symbolic computation, and others
can potentially benefit from higher precision hardware integer arithmetic. The
algorithm is suitable for hardware support of applications where fast
arithmetic computation is desirable.
