This question follows-up from this question/comment.
Suppose, you are given $X \odot Y$, where $X~(\neq 0) \in \operatorname{GF}(2^{128})$ is random, $Y~(\neq 0) \in \operatorname{GF}(2^{128})$ is random, and $\odot$ is a multiplication over $\operatorname{GF}(2^{128}$) with some modulus. As explained in this answer, the product $X \odot Y$ preserves privacy about its factors.
Therefore, assume you are given some extra information (see below). Will it be possible to recover (at least some non-trivial information) $X$, $Y$ under each of the scenarios:
- $X \oplus Y$ is given. Since $\oplus$ can be stated as the addition operation over any $\operatorname{GF}(\cdot)$, we can also say the sum of $X$ and $Y$ over $\operatorname{GF}(2^{128})$ is given.
- Arithmetic sum $X + Y$ is given.
- Modular addition, $X+Y \pmod{2^{128}}$ is given. Or, may be with any other divider (except $2$ as this coincides with $\oplus$), say $2^{8}$ or $2^{16}$, instead of $2^{128}$.
I considered each of the above scenarios to be exclusive. Please feel free to combine the scenarios or propose a new scenario if that is useful.
