I've been reading up on index calculus methods, and I think I finally understand, it's an approach for writing relations like $x^k = f_1(x)^{j_1} f_2(x)^{j_2} ...$ for irreducible $f_i(x)$ of small degree, and finding a bunch of these relations so you can first solve for $f_i(x) = x^{k_i}$, then write the relations in log form to solve $k = k_1j_1 + k_2j_2 + ... $
But my question is how to combine with Silver-Pohlig-Hellman to get discrete log for a composite order system. For example suppose you have a degree-32 primitive polynomial, which has order $M=2^{32} - 1 = 3 \cdot 5 \cdot 17 \cdot 257 \cdot 65537$; you solve $(x^k)^{(M/m)}$ for each of the factors m then combine with the Chinese Remainder Theorem.
How could I take an unknown $f(x) = x^k$, compute $g(x) = (f(x))^{65535}$, and use index calculus methods to calculate $g(x) = x^{k_g}$ in a group of order 65537 rather than in the full multiplicative group of order $2^{32} - 1$? SPH's strength is that it reduces the discrete log problem to one of its subgroups of smaller order, but I can't see how index calculus methods can take advantage of that... am I missing something?
(In this example for $2^{32}-1$ do you use polynomials that are smooth in powers of $x'=x^{65535}$ rather than in powers of $x$ itself?)