# Elimination of a Second-Law-Attack, and All Cable-Resistance-Based Attacks, in the Kirchhoff-Law-Johnson-Noise (KLJN) Secure Key Exchange System

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## Abstract

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**PACS Codes:**PACS 72.70.+m; PACS 89.20.Ff; PACS 89.90.+n

## 1. Introduction

_{L}, R

_{H}}, R

_{L}≠ R

_{H}, where the elements represent low (L) and high (H) bit values. The Gaussian voltage noise generators—mimicking the Fluctuation-Dissipation Theorem and delivering band-limited white noise with publicly agreed bandwidth—produce enhanced thermal (Johnson) noise at a publicly agreed effective temperature T

_{eff}, typically being T

_{eff}≥10

^{9}K [3], so the temperature of the wire can be neglected. The noises are statistically independent of each other and from the noise of the former bit period.

_{c}(t) and/or current I

_{c}(t) in the cable, because both arrangements lead to the same result. In the rest of the paper we assume that one of these secure bit exchange situations (either LH or HL) apply.

_{c}is cable resistance. Clearly Δ

_{KS}scales with the square of the cable resistance, i.e., ${\mathrm{\Delta}}_{\text{KS}}\propto {R}_{\text{c}}^{2}$.

## 2. Results and Discussion

#### 2.1. The Second-Law-Attack

_{eff}much greater than the cable temperature). Consequently the Second Law of Thermodynamics cannot provide full security. The cable-heating powers by the generators at the “H” and “L” ends are different and are given by

_{Hc}and P

_{Lc}can be utilized for the Second-Law-attack, because the resistor values R

_{H}and R

_{L}are publicly known. The implementation of this attack is to measure and compare the net power flows at the two ends of the cable, as illustrated in Figure 2. The mean power flow P

_{HL}from the “H” end toward the “L” end of the cable, and the mean power flow P

_{LH}from the “L” end toward the “H” end are, respectively,

_{HL}and P

_{LH}are directly measurable by Eve, and their difference,

_{H}has the greater resistance value and R

_{L}the smaller one, i.e., R

_{L}< R

_{H}. In the ideal case, when R

_{c}= 0, one obtains ΔP

_{AB}= 0 in accordance with the Second Law of Thermodynamics, which yields 〈U

_{c}(t)I

_{c}(t)〉 = 0. However, in the practical case, with R

_{c}> 0, one finds

- (i)
- if ΔP
_{AB}> 0, then Alice has R_{H}and Bob has R_{L}, - (ii)
- if ΔP
_{AB}< 0, then Alice has R_{L}and Bob has R_{H}.

_{c}, which provides a much better situation for Eve—especially in the case of vanishing cable resistance—than the square-law scaling of the BSY attack. Moreover, it is also obvious that in a practical case [3,9,10], where R

_{c}≪ R

_{L}≪ R

_{H}, Eve’s signal-to-noise ratio is always greater in the Second-Law-attack than in the BSY attack. This is so because the BSY attack evaluates the dc fraction of $\approx {R}_{\text{c}}^{2}/\left({R}_{\text{L}}{R}_{\text{H}}\right)$ in the measured (empirical) mean-square channel noise voltage, while the Second-Law-attack evaluates the dc fraction of R

_{c}/R

_{H}in the measured mean power flow. It should be noted that the measured mean-square channel noise voltage, and the measured mean power flow, follow similar statistics because they are the time average of the products of Gaussian processes [11].

#### 2.2. Natural/“Simple” Defense

_{eff}to be the same as the cable temperature. While these steps can be taken, the KLJN scheme is no longer simple. Moreover, the mentioned defense method may be unpractical because of the requirement of a homogeneous cable temperature, small noise levels, and since it prohibits the adoption of enhanced KLJN methods wherein Alice and Bob eliminate their own contributions in order to accomplish higher speed, security [9,15] and fidelity [16].

#### 2.3. Advanced Defense, Also Eliminating All Cable Resistance Attacks

_{H}and the R

_{L}resistors so that the equation

_{eff}is the noise temperature at the R

_{H}resistors and βT

_{eff}is the noise temperature of the R

_{L}resistors. The solution of equation (10) is

_{L}< R

_{H}and β <1 for R

_{H}< R

_{L}.

## 3. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the Kirchhoff-law-Johnson-noise secure key exchange system. To defend against active and hacking attacks, the cable parameters and integrity are randomly monitored; the instantaneous voltage U

_{c}(t) and current I

_{c}(t)amplitudes in the cable are measured and compared via public authenticated data exchange; and full spectral and statistical analysis/checking is carried out by Alice and Bob R, t and T

_{eff}denote resistance, time and effective temperature, respectively. Line filters, etc., are not shown.

**Figure 2.**Scheme devised to illustrate the Bergou-Scheuer-Yariv attack and the Second-Law-attack. Alice’s and Bob’s locations are arbitrary in the figure. During the Second-Law-attack, the powers flowing out from the “H” and “L” ends of the cable are calculated and compared. The temperature of the cable resistor R

_{c}can be neglected because of the high noise temperature of the generators. The notation is consistent with that in Figure 1.

**Figure 3.**Eve’s measurements during the Second-Law-attack. The powers flowing out from the two ends of the cable are measured and compared. The notation is consistent with that in Figure 1.

**Figure 4.**Schematic for illustrating the elimination of the Second-Law-attack and the BSY-attack by introduction of a proper temperature offset. The notation is consistent with that in Figure 1.

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**MDPI and ACS Style**

Kish, L.B.; Granqvist, C.-G.
Elimination of a Second-Law-Attack, and All Cable-Resistance-Based Attacks, in the Kirchhoff-Law-Johnson-Noise (KLJN) Secure Key Exchange System. *Entropy* **2014**, *16*, 5223-5231.
https://doi.org/10.3390/e16105223

**AMA Style**

Kish LB, Granqvist C-G.
Elimination of a Second-Law-Attack, and All Cable-Resistance-Based Attacks, in the Kirchhoff-Law-Johnson-Noise (KLJN) Secure Key Exchange System. *Entropy*. 2014; 16(10):5223-5231.
https://doi.org/10.3390/e16105223

**Chicago/Turabian Style**

Kish, Laszlo B., and Claes-Göran Granqvist.
2014. "Elimination of a Second-Law-Attack, and All Cable-Resistance-Based Attacks, in the Kirchhoff-Law-Johnson-Noise (KLJN) Secure Key Exchange System" *Entropy* 16, no. 10: 5223-5231.
https://doi.org/10.3390/e16105223