Intuition for a simple conversion

Question

I was working on a scheme in cryptography and came up with the following scenario.

To put it in proper words.

1. We have an element $\frac{1}{x+m}$.
2. The 2 elements $x$ and $x_1$ are known.
3. We want to transform $\frac{1}{x+m}$ to $\frac{1}{x_1+m}$. I.e, we need a $k$ such that $\frac{k}{x+m} = \frac{1}{x1+m}$.

You can consider $x, x_1, m$ to be elements of $\mathbb Z_p^\ast$. You can introduce any extra dummy variables if you want for the conversion. It'd be of great of help if you can give me an idea with this.

Thanks!

Answer 1

You are given $y=\frac{1}{x+m}$. Then $\frac{1}{y}= x+m$, and $x_1+m = \frac{1}{y}-x+x_1$.

So $$\frac{1}{x_1+m} = \frac{y}{1-y(x-x_1)}.$$

Therefore, simply set $$k = \frac{1}{1-y(x-x_1)},\quad \text{where}\quad y = \frac{1}{1+m},$$ which can be done since you know $x$, $x_1$, and $y$.