# Quantitative Assessment and Grading of Hardware Trojan Threat Based on Rough Set Theory
## 2 System Model and Assessment Formulation  
### Information System and Attributes  
Rough Set Theory uses an Information System (IS) to represent knowledge, which is usually expressed as follows:  
\[IS=(U,A,V,f) \tag{1}\]  
where \(U\) represents the universal set, a finite non-empty set with \(n\) elements, \(U=\{\) x\({}_{1}\)x\({}_{2}\),...x\({}_{n}\)\(\}\), \(A\) expresses the attribute set (a non-empty finite set with \(m\) attributes), A = \(\{\) a\({}_{1}\)a\({}_{2}\),...a\({}_{m}\)\(\}\), \(V=\bigcup\limits_{a\in A}\), \(V_{a}\) is a nonempty set of attribute values, and \(f:U\times A\to V\) expresses an information function that maps an element in \(U\) to exactly one value in \(V\), which means \(\forall a\in A,x\in U,f(x,a)\in V_{a}\).  
For HT threat, we established an information system according to 94 independent HTs and their characteristics on the Trust-hub website. Over the past few years, there have been efforts to develop comprehensive HT taxonomies based on their implementation and effect [2]. [PERSON] et al. [17] have further improved the taxonomies in earlier works by including the physical characteristics of HTs. HT taxonomy can be broken down into Insertion Phase, Abstraction Level, Activation Mechanism, Effect, Location, and Physical Characteristics. The HTs on the Trust-hub website are classified based on these categories. Among them, the classification results according to physical characteristics are not independent of each other. For example, the HT coded B19-T200 belongs to both 'Distribution' and 'Layout Same'. In this case, rough sets cannot be applied to assign values to this attribute. Therefore, the information system for HTs was established using the other five attributes.  
It is worth noting that the value assigned to each HT class may be different depending on the host IC and the relevant use case. The value should be determined according to the specific IC and its application scenario. Here, we take a Field Programmable Gate Array (FPGA) used as a communication system as an example to demonstrate our weightened assessment and grading scheme. The values of each class of attributes are shown in Table 1. Table 1 is ordered based on its first and second column according to the 'Chip-level Trojan Taxonomy' catalogue from Trust-hub. This sorting can help to quickly find the value corresponding to the HT Taxonomy.  
\begin{table}
\begin{tabular}{c c c} \hline
**Attribute** & **Class** & **Value** \\ \hline Insertion Phase & Design & 1 \\ Fabrication & 2 \\ \hline \multirow{4}{*}{Abstraction Level} & Register Transfer & 1 \\  & Gate & 2 \\  & Layout & 3 \\  & Physical & 4 \\ \hline \multirow{4}{*}{Activation Mechanism} & Always On & 4 \\  & Time-Based & 2 \\ \cline{1-1}  & Physical-Condition-Based & 1 \\ \cline{1-1}  & User Input & 3 \\ \hline \multirow{4}{*}{Effect} & Change Functionality & 2 \\ \cline{1-1}  & Degrade Performance & 1 \\ \cline{1-1}  & Information Leakage & 3 \\ \cline{1-1}  & Denial of Service & 4 \\ \hline \multirow{4}{*}{Location} & Processor & 6 \\ \cline{1-1}  & Memory & 5 \\ \cline{1-1}  & I/O & 4 \\ \cline{1-1}  & Power Supply & 2 \\ \cline{1-1}  & Clock Grid & 3 \\ \cline{1-1}  & Others & 1 \\ \hline \end{tabular}
\end{table}
Table 1: Selected Attributes Analysis for HT Threats.