Finite abelian hidden-shift promise

The input group is a finite abelian group $A$ supplied in explicit invariant-factor coordinates $A=\mathbb{Z}\mathrm{/}{n}_1\mathbb{Z}\times \cdots \times \mathbb{Z}\mathrm{/}{n}_d\mathbb{Z}$, with componentwise addition and the empty product allowed. The finite set $S$ is supplied with an injective fixed-length bit encoding. The input functions $f,g:A\to S$ are injective and have coherent XOR oracles with explicitly declared access and costs.

The promise is that there is a unique $s\in A$ such that

$$f(a)=g(a+s)\text{for every }a\in A\mathrm{.}$$

The output is the coordinate encoding of $s$. The definition makes no claim that a group presented only by black-box operations comes with invariant-factor coordinates, efficient uniform preparations, efficient group operations, or an efficient implementation of either oracle.