Generating Hard Problems of Cellular Automata

TL;DR

This paper introduces two computationally hard problems related to cellular automata, demonstrates their connection to established cryptographic problems, and develops a proof-of-work protocol based on one of these problems.

Contribution

It defines new hard problems in cellular automata, links them to known cryptographic challenges, and proposes a novel proof-of-work protocol utilizing these problems.

Findings

DDP$^M_{n,p}$ reduces to discrete logarithm problem

SDDP$^eta_{k,n}$ reduces to short integer solution problem

Proposed proof-of-work protocol based on SDDP$^eta_{k,n}$

Abstract

We propose two hard problems in cellular automata. In particular the problems are: [DDP$^M_{n,p}$] Given two \emph{randomly} chosen configurations $t$ and $s$ of a cellular automata of length $n$, find the number of transitions $\tau$ between $s$ and $t$. [SDDP$^\delta_{k,n}$] Given two \emph{randomly} chosen configurations $s$ of a cellular automata of length $n$ and $x$ of length $k<n$, find the configuration $t$ such that $k$ number of cells of $t$ is fixed to $x$ and $t$ is reachable from $s$ within $\delta$ transitions. We show that the discrete logarithm problem over the finite field reduces to DDP$^M_{n,p}$ and the short integer solution problem over lattices reduces to SDDP$^\delta_{k,n}$. The advantage of using such problems as the hardness assumptions in cryptographic protocols is that proving the security of the protocols requires only the reduction from these problems to the designed protocols. We design one such protocol namely a proof-of-work out of SDDP$^\delta_{k,n}$.