Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness (2304.14700v1)

Abstract: 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.