Bit errors in the Kirchhoff-Law–Johnson-Noise secure key exchange

We classify and analyze bit errors in the voltage and current measurement modes of the Kirchhofflaw–Johnson-noise (KLJN) secure key distribution system. In both measurement modes, the error probability decays exponentially with increasing duration of the bit sharing period (BSP) at fixed bandwidth. We also present an error mitigation strategy based on the combination of voltagebased and current-based schemes. The combination method has superior fidelity, with drastically reduced error probability compared to the former schemes, and it also shows an exponential dependence on the duration of the BSP.


Introduction
This paper classifies and analyzes different types of errors, due to statistical inaccuracies in noise measurements, within the Kirchhoff-law-Johnson-noise (KLJN) system 1,2 and demonstrates a new and efficient way of removing these errors. 3,4 It is a summary of recent findings presented in Refs. 3

and 4.
We first present a brief description of the working principle of the KLJN system. 1-4

The Kirchhoff-law-Johnson-noise secure key distribution
The KLJN secure key exchange has been proven to give information-theoretic security. 5,6 This key distribution scheme is based on Kirchhoff's loop law of quasi-static electrodynamics and the fluctuation-dissipation theorem of statistical physics. [1][2][3][4][5][6][7][8][9] Figure 1 shows the ideal KLJN scheme. [1][2][3][4] The core channel is a wire line to which the two communicating parties, denoted "Alice" and "Bob", connect their resistors A R and B R , respectively. These resistors are randomly chosen from the set { } 0 1 , where 0 R and 1 R represent the different bit values. At the beginning of each bit sharing period (BSP), Alice and Bob-who have identical pairs of resistors-randomly select and connect one of these resistors to the wire line. Therefore there are four possible bit situations in which these two pairs of resistors can be connected to the wire line: 00, 01, 10 and 11. The cases when Alice and Bob use the same resistance values-i.e., the 00 and 11 situations-represent a non-secure bit exchange. The situations when Alice and Bob use different resistance values-i.e., the 01 and 10 bit situations-signify a secure bit exchange event because, according to the Second Law of Thermodynamics, [3][4][5][6] Eve cannot distinguish between them through measurements, and whenever Alice and Bob see the 01/10 situation they know that the other party has the complementary bit value, which means that they can infer the full bit arrangement. Thus a secure bit has been generated and shared.

KLJN bit interpretations
Assuming ideal components/conditions, we proceed as in earlier works. 3    For the sake of simplicity we assume 0 The bit interpretations of the measured mean-square channel noise voltage 3 and current 4 are 00 when u ch 2 (t) τ < u 00 2 (t) + ∆ 1 and 2 The interpretations are 11 when respectively. The secure bit situations 01/10 are interpreted as such when respectively.

Error analysis in the KLJN system
Bit errors occur when the protocol makes incorrect bit interpretations due to statistical inaccuracies in the measured mean-square noise voltage and/or current. There are different types of errors situations, as shown in Table 1 It is apparent that two types of errors need to be addressed: the 11==>01/10 errorsi.e., the errors when the actual situation 11 is interpreted as the secure noise level 01/10and the 00==>01/10 errors occurring when the actual situation 00 is interpreted as 01/10. Figure 3 shows a block diagram for the measurement process. The channel voltage and/or current first enter a squaring unit. For typical practical applications, the output signal is a voltage, because the squaring unit employs voltage-signal-based electronics. Thus when measuring the current, for the sake of simplicity and without losing generality, we assume that the numerical values of the voltage correspond to the measured current. ). We estimate the error probability with the probability that ( ) t µ is crossing a threshold during the time interval τ . To compute this probability, we follow a procedure that is analogous to the one in Refs. 3 and 4, specifically by using Rice's formula 11,12 for threshold crossings.

General approach
We suppose that ∆ is the threshold value used to determine the boundaries between the different interpretations of ( ) t µ over the time window τ . If we define , ( ) AC S f τ as the power density spectrum of ( ) t µ , the mean frequency of level crossing can be given as In the high-threshold situation, the errors follow Poisson statistics and thus the error probability during a time interval is equal to the expected number of errors within this interval provided this number is much less than one. Thus the probability ε in the case of 1

Statistical errors in the voltage and current measurement modes
In this section, the threshold values 1 ∆ , 2 ∆ and 3 ∆ , 4 ∆ have meanings that are similar to the one of the threshold value ∆ in the general approach presented above and are also defined as a fraction of the corresponding measured mean-square channel noise voltage and current, namely: respectively. Similarly, by substituting 3 ∆ and 4 ∆ in Eq. (4) we find that the error probabilities respectively. It should be noticed that-for both the voltage-based and the current-based methods-the error probabilities are exponential functions of the parameter γ , which shows that the error probability decays exponentially with increasing magnitude of τ .

Illustration of the results with practical parameters
To demonstrate the results, we assign practical values to the parameters in Eq. (4). For 100 γ = and 0.5 ϕ = , the bit error probability is 0.001 ε = . Increasing γ by a factor of two (and analogously the time average window τ ) results in 6 10 ε − ≈ , which is satisfactory for many applications.
In our case of 1 α >> the mean square noise current level at 11 is much closer to the value at 01/10 than to the value at 00 (cf. Fig. 2), and hence ,00 i ε << ,11 i ε . This situation is reversed for the case of the voltage-based method, where 00 ε >> 11 ε .

A combined current-voltage error removal strategy
We assume that Alice and Bob measure both 2 c u τ and 2 c i τ at the same time. In an ideal error-free situation, the same bit interpretations ensue from both mean-square channel noise amplitudes. However the bit interpretations can differ when there are errors, because the current and voltage amplitudes are statistically independent due to Gaussianity and the Second Law of Thermodynamics. [1][2][3][4][5][6] To eliminate errors, we keep only those cases where both the current and voltage methods interpret 01/10 bit values. Table 2 illustrates this situation. 4 Due to their independence, the error probabilities in the combined current-voltage method are the product of the error probabilities of the current-based and voltage-based schemes. 4 The resultant probabilities ,00 t ε and ,11 t ε of 00==>01/10 and 11==>01/10 types of errors are 4 2 2 ,00 00 ,00 respectively.
These error probabilities are again exponential functions of the parameter γ . i.e., γ , is increased by a factor of two, the total bit error probability ,00 t ε is decreased to Although in the experimental setup presented in Ref. 9 we used a slightly modified decision table, the joint use of voltage and current detection had already been found suitable for decreasing error ratios. In addition, the results presented in current article are in good agreement with the experimental results.

Conclusion
This paper has classified and analyzed the types of errors in the KLJN key exchange for both voltage-based and current-based schemes. These error probabilities showed an exponential dependence on the duration of the bit exchange. Furthermore, we presented an enhanced error mitigation method, based on the combination of the voltage-based and current-based schemes, which operates without any error correction algorithm.