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We study the lazy Markov chain on $\mathbf{F}_p$ defined as $X_{n+1}=X_n$with probability $1/2$ and $X_{n+1}=f(X_n) \cdot \varepsilon_{n+1}$, where$\varepsilon_n$ are random variables distributed uniformly on $\{ \gamma^{},\gamma^{-1}\}$, $\gamma$ is a primitive root and $f(x) = \frac{x}{x-1}$ or$f(x)=\mathrm{ind} (x)$. Then we show that the mixing time of $X_n$ is$\exp(O(\log p / \log \log p))$. Also, we obtain an application to anadditive--combinatorial question concerning a certain Sidon--type family ofsets.

On multiplicative Chung--Diaconis--Graham process