A lower bound on the average entropy of a function determined up to a diagonal linear map on F_q^n

In this note, it is shown that if $f\text{\hspace{0.17em}⁣}:{\text{efq}}^n\to {\text{efq}}^n$ is any function and $\text{bA}=(A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_n)$ is uniformly distributed over ${\text{efq}}^n$, then the average over $(k_1,\mathrm{.}\mathrm{.}\mathrm{.},k_n)\in {\text{efq}}^n$ of the Renyi (and hence, of the Shannon) entropy of $f(\text{bA})+(k_1{A}_1,\mathrm{.}\mathrm{.}\mathrm{.},k_n{A}_n)$ is at least about ${\log }_2(q^n)-n$. In fact, it is shown that the average collision probability of $f(\text{bA})+(k_1{A}_1,\mathrm{.}\mathrm{.}\mathrm{.},k_n{A}_n)$ is at most about $2^n\mathrm{/}{q}^n$.