On the multiplicative Chung-Diaconis-Graham process

We study the lazy Markov chain on F-p defined as follows: Xn+ 1 = X-n with probability 1/2 and Xn+ 1 = f(X-n) center dot epsilon(n+1), where the epsilon(n) are random variables distributed uniformly on the set {gamma,gamma(-1)},gamma is a primitive root and f(x) = x/(x - 1) or f(x) = ind(x). Then we show that the mixing time of X-n is exp(O(log p center dot log log log p/ log log p)). Also, we obtain an application to an additive-combinatorial question concerning a certain Sidon-type family of sets.