<p>In this paper, we propose a generalized model of Priority Arbiter-based Physical Unclonable Function (PA-PUF) with an arbitrary number of paths inside each switch. We first develop a mathematical model for this generalized model. Experimentally, we observed that the class of Boolean functions generated from our model of PA-PUF increases proportionally with the number of paths inside each switch, and that motivated us to attempt one of the open challenges proposed by Kansal et al. (Discrete Appl Math 356:71–95, 2024). We first show that the set of Boolean functions generated from <i>i</i>-length PA-PUF with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((i+1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> number of paths is a proper super set of the set of Boolean functions generated from <i>i</i>-length PA-PUF with <i>i</i> number of paths. Based upon that, we show in our main result that we need at least <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((n+1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> numbers of paths inside each switch of an <i>n</i>-length PA-PUF to generate all the Boolean functions involving <i>n</i>-number of variables. Furthermore, we performed significant software and hardware experimentations to assess the resilience of our model against machine learning based modeling attacks.</p>