Randomness, pseudorandomness and models of arithmetic

TL;DR

This paper explores how models of arithmetic can be used to interpret pseudorandomness as genuine randomness, providing a framework that links number theory, complexity, and cryptography.

Contribution

It constructs a set of models and a probability distribution to demonstrate that pseudorandom sequences appear truly random within these models.

Findings

Pseudorandom sequences have a 50% chance of being 1 in a random model.

Models of arithmetic can explain pseudorandomness as actual randomness.

The approach bridges number theory, complexity, and cryptography through model construction.

Abstract

Pseudorandmness plays an important role in number theory, complexity theory and cryptography. Our aim is to use models of arithmetic to explain pseudorandomness by randomness. To this end we construct a set of models $\cal M$, a common element $\iota$ of these models and a probability distribution on $\cal M$, such that for every pseudorandom sequence $s$, the probability that $s(\iota)=1$ holds true in a random model from $\cal M$ is equal to 1/2.