## 2 The New Attack Scheme Utilizing Deterministic Loop Currents  
### The Situation that [PERSON] Utilizes for the Attack  
In practical KLJN systems, in order to save a wire, the common end of the resistors (see Figure 1) is often connected to the ground. In practical situations there is often an imbalance, a voltage difference between various locations of the ground that is due, for example, to ground loop currents or electromagnetic interference (EMI) [53]. This potential information leak was pointed out in [53] as a potential source of information leaks in the case of significant cable resistance. However, it was not realized in [53] that information leaks can exist even at zero cable resistance. The present study is directly relevant for DC current-based ground loops (such as during secure communication between different units in instruments [33; 34]). For EMI-induced ground loops, our DC-limited study is only a first step in addressing a more general situation (which should be investigated in future works). In this paper, we explore this new information leak in the DC parasitic voltage limit. Hence, consideration was given to situations where during the bit exchange period, the relative change in the parasitic voltage is small. For the sake of simplicity but without the limitation of generality, we assume that the imperfection is represented by a positive DC voltage generator located at [PERSON]'s end (see Figure 2). Due to [PERSON]'s principle of security, that is, the assumption that the enemy knows everything except the momentary key, we must assume that [PERSON] knows the polarity and value of this DC voltage (if she does not know it at first, she will be able to extract it via long-time averaging). The direction of the current \(I(t)\) is assumed to point from [PERSON] to [PERSON]. The voltage \(U(t)\) and current \(I(t)\) in the wire contain the sum of a DC component and an AC (stochastic, that is, noise) component. Let us analyze the resulting voltages and currents. The current in the wire is  
\[I(t)=I_{\text{DC}}+I_{\text{n}}(t) \tag{3}\]  
where \(I_{\text{DC}}\) is its DC component  
\[I_{\text{DC}}=\frac{U_{\text{DC}}}{R_{\text{A}}+R_{\text{B}}} \tag{4}\]  
and \(I_{\text{n}}(t)\) is its AC (noise) component  
Figure 2: The KLJN system with ground loop voltage. Here \(U_{\text{An}}\in\{U_{\text{LA}};U_{\text{HAn}}\}\) and \(U_{\text{Bn}}\in\{U_{\text{LB}};U_{\text{HBn}}\}\) are the voltage noises belonging to the randomly chosen resistors, \(R_{\text{A}}\&R_{\text{B}}\in\{R_{\text{L}};R_{\text{H}}\}\), belonging to [PERSON] and [PERSON], respectively. \(U_{\text{DC}}\) is the ground loop DC voltage source and \(U(t)\) and \(I(t)\) are the voltage and current on the wire, respectively. \[I_{\rm{n}}(t)=\frac{U_{\rm{An}}(t)-U_{\rm{Bn}}(t)}{R_{\rm{A}}+R_{\rm{B}}} \tag{5}\]  
in which \(U_{\rm{An}}\) and \(U_{\rm{Bn}}\), with \(U_{\rm{An}}\in\{U_{\rm{LAn}};U_{\rm{HAn}}\}\) and \(U_{\rm{Bn}}\in\{U_{\rm{LBn}};U_{\rm{HBn}}\}\), are the voltage noise sources of the chosen resistors, \(R_{\rm{A}}\) and \(R_{\rm{B}}\), respectively. The voltage on the wire is  
\[U(t)=I(t)R_{\rm{B}}+U_{\rm{Bn}}(t). \tag{6}\]  
From Equations (3) and (6) we obtain  
\[U(t)=U_{\rm{DCw}}+U_{\rm{ACw}}(t)=I_{\rm{DC}}R_{\rm{B}}+I_{\rm{n}}(t)R_{\rm{B} }+U_{\rm{Bn}}(t) \tag{7}\]  
where \(U_{\rm{DCw}}\) and \(U_{\rm{ACw}}(t)\) represent the DC and AC voltage components in the wire, respectively. The DC component can be written as  
\[U_{\rm{DCw}}=I_{\rm{DC}}R_{\rm{B}}=\frac{U_{\rm{DC}}}{R_{\rm{A}}+R_{\rm{B}}}R _{\rm{B}}. \tag{8}\]  
The DC component is different during [PERSON]'s and [PERSON]'s LH and HL bit situations of secure bit exchange, which yields information leaks.