New Techniques for Constructing Rare-Case Hard Functions

TL;DR

This paper introduces new number-theoretic constructions of rare-case hard functions that are computationally infeasible for certain classes of algorithms, with implications for complexity theory and cryptography.

Contribution

The paper presents novel techniques for constructing rare-case hard functions from NP-complete problems, extending previous methods and providing new complexity-theoretic insights.

Findings

Constructed functions are hard for polynomial-sized circuits if NP not in P/poly.

Constructed functions are hard for polynomial-time randomized algorithms if NP not in BPP.

Under RETH, these functions are hard even in subexponential time.

Abstract

We say that a function is rare-case hard against a given class of algorithms (the adversary) if all algorithms in the class can compute the function only on an $o(1)$-fraction of instances of size $n$ for large enough $n$. Starting from any NP-complete language, for each $\alpha > 0$, we construct a function that cannot be computed correctly even on a $1/n^\alpha$-fraction of instances for polynomial-sized circuit families if NP $\not \subset$ P/POLY and by polynomial-time algorithms if NP $\not \subset$ BPP - functions that are rare-case hard against polynomial-sized circuits and polynomial-time randomized algorithms. The constructed function is a number-theoretic polynomial evaluated over specific finite fields. For NP-complete languages that admit parsimonious reductions from all of NP (for example, SAT), the constructed functions are hard to compute even on a $1/n^\alpha$-fraction of instances by polynomial-time randomized algorithms and polynomial-sized circuit families simply if P# $\not \subset$ BPP and P# $\not \subset$ P/POLY, respectively. We also show that if the Randomized Exponential Time Hypothesis (RETH) is true, none of these constructed functions can be computed even on a $1/n^\alpha$-fraction of instances in subexponential time. These functions are very hard, almost always. While one may not be able to efficiently compute the values of these constructed functions themselves, in polynomial time, one can verify that the evaluation of a function, $s = f(x)$, is correct simply by asking a prover to compute $f(y)$ on targeted queries. We have extended our work to give an alternative proof of a variant of Lipton's theorem (Lipton, 1989). We also compare our techniques for constructing rare-case hard functions with two other existing methods in the literature (Sudan et al., 2001; Feige and Lund, 1996).