Short answer: NO! p pidgeon holes, p pidgeons. p-2 pidgeon holes get 1 pidgeon each; pidgeon hole "0" gets no pidgeons; pidgeon hole "1" gets 2 pidgeons. So the distribution over the non-"0" pidgeon holes isn't uniform, but has a spike at "1". At 01:08 PM 10/14/2016, Henry Baker wrote:

Is this known or conjectured with a conjecture name?

If X is a uniform random distribution over [0,p-1], p prime, is g^X a uniform random distribution, for a generator g ?