Constructing an LDPC Code Containing a Given Vector

TL;DR

This paper investigates the construction of LDPC codes containing a specific vector, analyzing probabilistic conditions and uncertainty, with applications in Physical Unclonable Functions.

Contribution

It extends existing LDPC code construction methods by addressing probabilistic feasibility and uncertainty analysis, with practical implications for PUFs.

Findings

High probability of construction success under certain conditions

Quantified the uncertainty of the vector given the code

Applied findings to Physical Unclonable Functions

Abstract

The coding problem considered in this work is to construct a linear code $\mathcal{C}$ of given length $n$ and dimension $k<n$ such that a given binary vector $\mathbf{r} \in \mathbb{F}^{n}$ is contained in the code. We study a recent solution of this problem by M\"uelich and Bossert, which is based on LDPC codes. We address two open questions of this construction. First, we show that under certain assumptions, this code construction is possible with high probability if $\mathbf{r}$ is chosen uniformly at random. Second, we calculate the uncertainty of $\mathbf{r}$ given the constructed code $\mathcal{C}$. We present an application of this problem in the field of Physical Unclonable Functions (PUFs).