Fixed base exponentiation with precomputations

Question

I'm trying to compute $g^m$ mod $n$ where $m,n$ are 1024-bit numbers. The method I want to use is fixed base exponentiation with precomputations, also known as fixed-base windowing.The paper I'm following is Brickell et. al. "Fast Exponentiation with Precomputation: Algorithms and Lower Bounds" which can be found here.

Basically the idea is to precompute $g^{b^i}$ where $b$ is a radix. Now if we write $m$ in the radix base: $m=\sum_{i=0}^k e_i b^i$ then $g^m=\prod (g^{b^i})^{e_i}$. For example: say we want to compute $g^{142}$ and we have chosen as radix $b=3$ (I'm aware of the more natural example of radix 2, but I'm experimenting with different radices to see if there is any speed-up). I write $m=(12021)_3$ and compute $g^{142}=g^{3^4}\times(g^{3^3})^2\times (g^3)^2 \times g^3$.

The pseudo code given in the paper is:

b ← 1
a ← 1
for d = h to 1 by −1
    for each i such that a_i = d
        b ← b ∗ g^{x_i}
    a ← a ∗ b.
return a.

where $h$ is the range of $e_i$, and $a_i=e_i$ and $x_i=g^{b^i}$.

My question is: instead of going through $e_i$ by tier as the pseudo code suggested, why don't we go through the base $b$ representation of $m$ one digit at a time and multiply as we go?

When I looked up fixed-base exponentiation in Henk and Tilborg "Encyclopedia of Cryptography and Security," they say that "there is time saved by multiplying together powers with identical coefficients, and then raising the intermediate products to powers step by step." I believe this is hinting at the answer to my question, but I don't understand the reasoning.