  We develop the theory of cryptographic nondeterministic-secure
pseudorandomness beyond the point reached by Rudich's original work (Rudich
1997), and apply it to draw new consequences in average-case complexity and
proof complexity. Specifically, we show the following:
  *Demi-bit stretch*: Super-bits and demi-bits are variants of cryptographic
pseudorandom generators which are secure against nondeterministic statistical
tests (Rudich 1997). They were introduced to rule out certain approaches to
proving strong complexity lower bounds beyond the limitations set out by the
Natural Proofs barrier (Rudich and Razborov 1997). Whether demi-bits are
stretchable at all had been an open problem since their introduction. We answer
this question affirmatively by showing that: every demi-bit $b:\{0,1\}^n\to
\{0,1\}^{n+1}$ can be stretched into sublinear many demi-bits
$b':\{0,1\}^{n}\to \{0,1\}^{n+n^{c}}$, for every constant $0<c<1$.
  >>> see rest of abstract in paper.
