A mathematical problem for security analysis of hash functions and pseudorandom generators

TL;DR

This paper introduces Function Density Problems (FDPs), a new mathematical framework connecting the security analysis of hash functions and pseudorandom generators, aiming to enhance theoretical understanding rather than practical solutions.

Contribution

It proposes FDPs as a novel theoretical approach to evaluate hash functions and construct secure PRGs, offering new insights and mathematical discussions in cryptography.

Findings

FDPs establish connections between hash function security and pseudorandom generator constructions.

Examples of mathematical discussions on FDPs are provided, of independent interest.

The paper suggests future research directions for FDPs in cryptography and mathematics.

Abstract

In this paper, we specify a class of mathematical problems, which we refer to as "Function Density Problems" (FDPs, in short), and point out novel connections of FDPs to the following two cryptographic topics; theoretical security evaluations of keyless hash functions (such as SHA-1), and constructions of provably secure pseudorandom generators (PRGs) with some enhanced security property introduced by Dubrov and Ishai [STOC 2006]. Our argument aims at proposing new theoretical frameworks for these topics (especially for the former) based on FDPs, rather than providing some concrete and practical results on the topics. We also give some examples of mathematical discussions on FDPs, which would be of independent interest from mathematical viewpoints. Finally, we discuss possible directions of future research on other cryptographic applications of FDPs and on mathematical studies on FDPs themselves.