2.8-2.11

1. Why is the does the LSB of the output of the trapdoor function on predictable (consecutive, no less) input give a pseudorandom number sequence? Wouldn't it depend on the function or is this characteristic of trapdoor functions?

2. All of this should be prefaced, "if I'm not mistaken..." According to Shannon's theory of entropy and information, a sequence with the highest entropy (and so information content) is a totally random one since it cannot be predicted. So why can a sequence generated from a relatively small number of bits be cryptographically secure? Isn't it impossible for such a sequence to have more information than its inputs? (I'll do you a favor and stay away from the sophomoric pontification: "Is anything really random?")