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 $0<c<1$.
?*Average-case hardness*: Using work by Santhanam (2020), we apply our results to obtain new average-case Kolmogorov complexity results: we show that K$^{\rm poly}[n-O(1)]$ is zero-error average-case hard against NP/poly machines iff K$^{\rm poly}[n-o(n)]$ is, where for a function $s(n):\mathbb{N}\to\mathbb{N}$, K$^{\rm poly}[s(n)]$ denotes the languages of all strings $x\in\{0,1\}^n$ for which there are (fixed) poly-time Turing machines of description-length at most $s(n)$ that output $x$.
?*Characterising super-bits by nondeterministic unpredictability:* In the deterministic setting, Yao proved that super-polynomial hardness of pseudorandom generators is equivalent to ("next-bit") unpredictability. Unpredictability roughly means that given any strict prefix of a random string, it is infeasible to predict the next bit. We initiate the study of unpredictability beyond the deterministic setting (in the cryptographic regime), and characterise the nondeterministic hardness of generators from an unpredictability perspective. Specifically, we propose four stronger notions of unpredictability:
NP/poly-unpredictability, coNP/poly-unpredictability, $\cap$-unpredictability and $\cup$-unpredictability, and show that super-polynomial nondeterministic hardness of generators lies between $\cap$-unpredictability and $\cup$-unpredictability.
?*Characterising super-bits by nondeterministic hard-core predicates:* We introduce a nondeterministic variant of hard-core predicates, called *super-core* predicates. We show that the existence of a super-bit is equivalent to the existence of a super-core of some non-shrinking function. This serves as an analogue of the equivalence between the existence of a strong pseudorandom generator and the existence of a hard-core of some one-way function (Goldreich and Levin 1989, Hastad, Impagliazzo, Levin and Luby 1999), and provides a first alternative characterisation of super-bits. We also prove that a certain class of functions, which may have hard-cores, cannot possess any super-core.
