On generating independent random strings

TL;DR

This paper demonstrates methods to efficiently generate multiple independent random strings from initial strings with partial or full randomness, using Kolmogorov complexity, with polynomial time algorithms for certain cases.

Contribution

It introduces algorithms for constructing many independent random strings from partially random or fully random initial strings, advancing the understanding of randomness extraction in computational complexity.

Findings

Constructs polynomially many independent random strings from two partially random strings.

Achieves polynomial-time construction when initial strings are fully random.

Provides polynomial-time method to generate a random string from two strings with constant randomness rate.

Abstract

It is shown that from two strings that are partially random and independent (in the sense of Kolmogorov complexity) it is possible to effectively construct polynomially many strings that are random and pairwise independent. If the two initial strings are random, then the above task can be performed in polynomial time. It is also possible to construct in polynomial time a random string, from two strings that have constant randomness rate.