In a recent paper [1], we introduced a new family of evidential distances in the framework of belief functions. Using specialization matrices as a representation of bodies of evidence, an evidential distance can be obtained by computing the norm of the difference of these matrices. Any matrix norm can be thus used to define a full metric. In particular, it has been shown that the $L^1$ norm-based specialization distance has nice properties. This distance takes into account the structure of focal elements and has a consistent behavior with respect to the conjunctive combination rule. However, if the frame of discernment on which the problem is defined has $n$ elements, then a specialization matrix size is $2^n \times 2^n$. The straightforward formula for computing a specialization distance involves a matrix product which can be consequently highly time consuming. In this article, several faster computation methods are provided for $L^p$ norm-based specialization distances. These methods are proposed for special kinds of mass functions as well as for the general case.