I have two related questions concerning cryptographically secure pseudo random number generators.
My first question is as follows: are there conditions on the definition of cryptographically secure pseudo random number generators that require distinct seeds to generate distinct (periodic) sequences? What if a generator has the property that two distinct seeds generate the same sequence, but the generator is provably secure? Can this property be ignored then or is it a requirement that distinct seeds generate distinct pseudo random sequences?
My next question concerns the paper by Blum, Blum, and Shub introducing the $x^2 \mod N$ Generator. After reading the paper and learning about the generator, it is clear that the outputs of the generator pass statistical tests for randomness, but it is unclear to me how the authors proved the output of the generator is uniformly random. Would anyone be able to walk me through a sketch of the proof showing the output of the generator is uniformly random (backwards secure)?