When is $f(x) = g(x) \mod 2^k$ an odd permutation? Let $n = 2^k$, let $[n] = \{0, \ldots, n-1\}$, and let $f: [n] \to [n]$ denote a function of the form $f(x) := p(x) \mod n$, where $p(x)$ is some polynomial with coefficients in $[n]$. Does $f(x)$ have a well-known name? Moreover, I would like to know a reference that characterizes when $f(x)$ is an odd permutation (i.e. something like $f(x)$ is an odd permutation if and only if something about $p(x)$)?  What about the case when we don't require $p(x)$ to be a polynomial, but rather some arbitrary other function?
After some more research, I found the following result (due to Rivest) that characterizes when $f(x)$ is a permutation polynomial: Fix $k \geq 2$. A polynomial $p(x) = a_0 + a_1x + \ldots + a_{d}x^{d}$ with integer coefficients is a permutation polynomial modulo $2^k$ if and only if: (1) $a_1$ is odd, (2) $(a_2 + a_4 + a_6 + \ldots)$ is even, and (3) $(a_3 + a_5 + a_7 + \ldots)$ is even -- See https://people.csail.mit.edu/rivest/pubs/Riv01c.pdf. I am interested in a similar result that characterizes when this permutation is odd.
References would be appreciated!
 A: This is only a partial answer for the case of single-cycle permutations. I found the following characterization due to Wang and Qi (Linear Equations on Polynomial Single-Cycle T-functions). For convenience, let $\Delta_1 = a_2 + a_4 + \ldots$ and let $\Delta_2 = a_3 + a_5 + \ldots$.  A polynomial $p(x) = \sum_{i = 0}^d a_i x^i \mod{2^k}$ is a single-cycle permutation if and only if the following conditions hold: (1) $a_0$ and $a_1$ are odd, (2) $\Delta_1$ and $\Delta_2$ are even, (3) $\Delta_1 +\Delta_2 + 2[a_1]_1 \equiv 0 \mod{4}$, and (4) $\Delta_1 + 2[a_2]_0 + 2[a_1]_1 \equiv 0 \mod 4$. Here, the notation $[x]_j$ is used to denote the $(j+1)$-th least-significant bit of $x$. So $[a_2]_0$ is the least significant bit of $a_2$ and $[a_1]_1$ is the second least significant bit of $a_1$.
Single-cycle permutations can also be characterized in the following way, due to Anashin (Uniformly Distributed Sequences of p-adic Integers). The permutation polynomial $p(x) \mod 2^k$ has a single cycle if and only if it has a single cycle modulo 8.
