# Quantitative Assessment and Grading of Hardware Trojan Threat Based on Rough Set Theory
## 1 Introduction  
The past decade has witnessed a dramatic increase in the cognitive and organizational complexity of integrated circuit (IC) fabrication and design, pushing the IC industry towards vertical specialization. Various stages of IC fabrication and design are being outsourced to offshore foundries and relocated across national boundaries. However, this geographical dispersion of IC design activities can lead to malicious modifications in the IC supply chain [1]. There are many opportunities for perpetrators and insiders to introduce a malicious design into an IC, which can be extremely difficult to detect by conventional testing and verification methods [2]. A Hardware Trojan (HT) is a malicious design that lies inactive until it is activated by rare and unknown conditions [3]. After being activated, it can cause catastrophic damage to electronic systems or leak confidential information stored by the IC [4, 5]. Defending against HTs and detecting them faces many challenges, ranging from circuit design and testing to economic issues [6], and resource limitations prevent testing all possible HT types within specific circuits [7]. This motivates the need for a mathematical framework to understand the strategic interactions between the HT designer and the defender. Recently, a number of studies have focused on modeling their strategic interactions using game theory in order to anticipate the outcome of such interactions [8, 9]. Here, we propose a game-theoretical framework based on fuzzy theory to obtain the optimal strategy [10]. The aforementioned works have highlighted the advantages of game theory for the development of better HT detection strategies. The shortcoming of these works, however, is that the payoffs of actions must be set artificially. These works do not take into account this subjectivity, which significantly impacts the game results and affects the resulting optimal attack and defense strategies. For defenders, the works in [11; 12] quantitatively analyze the security and vulnerability of the IC. [PERSON] et al. [11] have proposed a mathematical framework that considers complex circuit-level dependencies and ranks HT insertion sites inside the IC. [PERSON] et al. [12] introduced QIF-Verilog as a new language-based framework to evaluate the trustworthiness of an IC at the register transfer level (RTL). To an extent, these can help to quantitatively assess the payoffs of the defense strategies. While interesting, most of these existing works do not consider the payoffs of HT insertion strategies. In order to fill this gap, we introduced the Rough Set Theory(RST) to quantitatively assess the HT threat for the payoff of its insertion strategies. Over the past ten years, RST has become a topic of great interest to researchers for quantitative analysis and has been applied in many domains. The degree of correlation or difference among the indexes has been applied to risk assessment [13] and safety evaluation [14]. Subsequently, other evaluation methods have been integrated when determining the weights, such as the information entropy method [15] and the fuzzy analytic hierarchy process [16]. Similar to threat assessment of HTs, these evaluate the characteristics of objects according to the objects' related attributes. This similarity motivates the application of the RST to HT threat assessment. Although interesting, the above works assume that the samples have decision attributes for quantitative assessment using RST. However, for the HT threat, it is notable that samples have no prior data. In RST, HT threats form an information system without decision attributes. Considering this situation, it is both critical and challenging to assess the threat quantitatively. In this paper, we propose a modern approach for quantitative assessment of HT threat based on RST. We introduce information content in the rough set for weight determination of all the attributes. Our method achieves quantitative assessment based on the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS).