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A Static-loop-current Attack Against the Kirchhoff-Law-Johnson-Noise (KLJN) Secure Key Exchange System

A Static-loop-current Attack Against the Kirchhoff-Law-Johnson-Noise (KLJN) Secure Key Exchange... applied sciences Article A Static-loop-current Attack Against the Kirchhoff-Law-Johnson-Noise (KLJN) Secure Key Exchange System Mutaz Y. Melhem * and Laszlo B. Kish Department of Electrical and Computer Engineering, Texas A & M University, 3128 TAMU, College Station, TX 77843, USA; laszlokish@tamu.edu * Correspondence: yar111@tamu.edu; Tel.: +1-979-847-9071 Received: 10 October 2018; Accepted: 12 February 2019; Published: 15 February 2019 Abstract: In this study, a new attack against the Kirchhoff-Law-Johnson-Noise (KLJN) key distribution system is explored. The attack is based on utilizing a parasitic voltage-source in the loop. Relevant situations often exist in the low-frequency limit in practical systems, especially when the communication is over a distance, or between different units within an instrument, due to a ground loop and/or electromagnetic interference (EMI). Our present study investigates the DC ground loop situation when no AC or EMI effects are present. Surprisingly, the usual current/voltage comparison-based defense method that exposes active attacks or parasitic features (such as wire resistance allowing information leaks) does not function here. The attack is successfully demonstrated and proposed defense methods against the attack are shown. Keywords: KLJN key exchange; unconditional security; ground loop vulnerability; passive attack 1. Introduction 1.1. On Secure Communications Communications systems, standards, and technologies have been developed since ancient times. Today we have the internet, Internet-of-Things (IoT), operating fourth generation wireless networks (LTE), and the expected fifth generation wireless networks. An important requirement of any communication paradigm between these devices is to accomplish secure communication, i.e., to protect the privacy and integrity of users’ data that is transferred over the network. To achieve the security of transferred data which can contain sensitive information (e.g., bank account credentials, social security number, etc.) it is of utmost importance to defend against attacks. These attacks might be launched by an eavesdropper (Eve) who has access to the information channel between the communicating parties A (Alice) and B (Bob). The attack is passive if it eavesdrops without disturbing the channel. The attack is active (invasive) if Eve disturbs or changes the channel, such as with a man-in-the-middle attack. In the present paper, we introduce a new passive attack against the Kirchhoff-Law-Johnson-Noise (KLJN) secure key exchange scheme. 1.1.1. Secure Key Exchange Secure communication systems employ ciphers to encrypt messages (plaintext) and to decrypt encrypted messages (cyphertext). While the creation of a secure and efficient cipher is a complex problem, this problem may be solved simply. Ciphers operate with secure keys that form a momentary shared secret between Alice and Bob. Sharing (exchanging) the key securely is the difficult task. A communicator system cannot be more secure than its key. The security of the key exchange can be conditional or information-theoretic (unconditional). Appl. Sci. 2019, 9, 666; doi:10.3390/app9040666 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 666 2 of 10 1.1.2. Conditional Security Conditionally secure key exchange systems are the ones used generally nowadays. They are software protocols installed at Alice and Bob. Such algorithms utilize computational complexity and achieve only (computationally) conditional security (see e.g., [1,2]). The system is temporarily secure provided the adversary has limited computational resources. A major goal of quantum computer developments is to crack these types of key exchange systems (e.g., the Shor algorithm). From an information-theoretic point of view, security is non-existent because Eve has all the information to crack the encryption, but she needs a long time to do that unless she has a quantum computer or a yet-to-be-discovered classical algorithm that can do the job in a short time. The security is not future-proof. 1.1.3. Unconditional (Information-Theoretic) Security In order to achieve unconditional (information-theoretic) security at the key exchange, proper laws of physics with a special hardware are utilized. Two major classes of physics-based schemes have emerged for unconditional security: (i) Quantum key distribution (QKD) [3,4] concepts assume single photons and utilize quantum physics. The underlying laws of physics are Heisenberg’s uncertainty principle and the related quantum no-cloning theorem [5]. Even though there are serious debates about the actual level of unconditional security a practical QKD can offer (see e.g., [6–29]), most scientists agree that QKD is unique in its offering information-theoretic security via (a dark) optical fiber and also through air at night, provided the visibility is good. (ii) The Kirchhoff-Law-Johnson-Noise key distribution method that is based on the statistical physical features of the thermal noise of resistors [30–56]. The related law of physics is the fluctuation-dissipation-theorem (FDT). Some of its advantages are: It works via wire connections including power, phone, and internet lines, which can be used as information channels [31,32] to connect all homes and other establishments. It can be integrated on a chip, which implies excellent robustness, low price, and applicability in bankcards, computers, instruments, and physical unclonable function (PUF) hardware keys [33,34]. Its low price allows general applications such as unconditional security for the control of autonomous vehicular networks [35,36]. 1.2. On the KLJN Secure Key Distribution The KLJN scheme [30–52] utilizes the thermal noise of resistors (or the emulation of that by a specific hardware). In the core scheme Alice and Bob have two identical pairs of resistors, R and R L H (R < R ), respectively (see Figure 1). L H The key exchange protocol of a single secure bit is as follows: Alice and Bob randomly pick one of their resistors (R or R ), connect it to the wire channel, and keep them there during the bit exchange L H period while they execute voltage and/or current measurements to learn the resistor value at the other end (see below). The noise voltage generators shown in Figure 1 with each resistor can be the resistors’ own thermal noise, or an external noise generator emulating a much higher, common noise-temperature that is publicly agreed. The power density spectra of the voltage and current in the channel are given by the Johnson-Nyquist formulas [11] 4kTR R A B S ( f ) = , (1) R + R A B 4kT S ( f ) = (2) R + R A B where k is the Boltzmann’s constant, T is the common temperature, and R and R are the actually A B connected resistances at Alice’s and Bob’s ends, respectively, with R , R 2 fR , R g. After the A B L H measurement and spectral analysis, Equations (1) and (2) have two unknown variables, namely, Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 10 4kT (2) S ( f ) = R + R A B where k is the Boltzmann’s constant, T is the common temperature, and R and R are the actually connected resistances at Alice’s and Bob’s ends, respectively, with R , R ∈ R , R . After {} A B L H the measurement and spectral analysis, Equations (1) and (2) have two unknown variables, namely, the values of R and R , and thus Eve can find the values of the connected resistors, but not Appl. Sci. 2019, 9, 666 3 of 10 necessarily their locations, by solving these equations. We can represent the four different situations of the connected resistors ( R and/or R ) at the values of R and R , and thus Eve can find the values of the connected resistors, but L not necessarily H A B Alice's and Bob's ends by the indices of the connected resistors, LL, LH, HL, and HH, respectively. their locations, by solving these equations. As all the resistors have the same (noise) temperature, the ideal system is in thermal equilibrium, We can represent the four different situations of the connected resistors (R and/or R ) at L H where the second law of thermodynamics guarantees zero net power-flow. Hence, Eve cannot use Alice’s and Bob’s ends by the indices of the connected resistors, LL, LH, HL, and HH, respectively. the evaluation of power flow to determine the locations of the momentarily connected resistors As all the resistors have the same (noise) temperature, the ideal system is in thermal equilibrium, unless they have the same resistance values. On the other hand, Alice and Bob can determine the where the second law of thermodynamics guarantees zero net power-flow. Hence, Eve cannot use the connected resistor values by using Equations (1) or (2) since they know the value of their own evaluation of power flow to determine the locations of the momentarily connected resistors unless they connected resistors. have the same resist When ance R values. = R , whi On the ch ha other ppens a hand, t 5Alice 0% of and the bi Bob t exchange a can determine ttempts, the resul the connected ts A B resistor values by using Equations (1) or (2) since they know the value of their own connected resistors. are discarded. When R = R , which happens at 50% of the bit exchange attempts, the results are discarded. A B Figure 1. The core of the Kirchhoff-Law-Johnson-Noise (KLJN) system. U (t), U (t), U (t), HAn LAn HBn and Figure 1. U (tThe ) ar core e the of (thermal) the Kirchhoff-Law noise voltage -Johnson-Noise generators for(KLJ theN) related system. resistances R , and R ,, U (t ) U (t ) LBn H L HAn LAn respectively. U(t) and I(t) are the measured noise voltage and the current in the wire that are used to , and are the (thermal) noise voltage generators for the related resistances U (t ) U (t ) R HBn LBn H evaluate the power density spectra S ( f ) and S ( f ), respectively. and , respectively. U (t ) and I (t ) are the measured noise voltage and the current in the wire On Former Attacks Against the KLJN Secure Key Distribution that are used to evaluate the power density spectra S ( f ) and S ( f ) , respectively. u i Several attacks have been proposed but no attack has been able to compromise the unconditional security of the KLJN scheme because each known attack can efficiently be nullified by a corresponding On Former Attacks Against the KLJN Secure Key Distribution defense scheme. Several attacks have been proposed but no attack has been able to compromise the The attacks can be categorized into two classes: unconditional security of the KLJN scheme because each known attack can efficiently be nullified by (i) Passive attacks that utilize the non-ideal or parasitic features in a practical KLJN system for a corresponding defense scheme. information leaks. Non-zero wire resistance (see [37,38]) poses the greatest known threat, and the The attacks can be categorized into two classes: most efficient attack is power balance measurement (Second Law Attack) [39]. An efficient defense is (i) Passive attacks that utilize the non-ideal or parasitic features in a practical KLJN system for based on a proper temperature-offset [39,40]. Temperature-inaccuracies [41] and resistance-inaccuracies [42] information leaks. Non-zero wire resistance (see [37,38]) poses the greatest known threat, and the can also cause information leaks. On the other hand, these inaccuracies can compensate for each most efficient attack is power balance measurement (Second Law Attack) [39]. An efficient defense is other [43] if used in a creative way. Non-zero cable capacitance [44] or cable inductance can also yield based on a proper temperature-offset [39,40]. Temperature-inaccuracies [41] and information leaks that can be fixed by specific designs including the proper choice of frequency range resistance-inaccuracies [42] can also cause information leaks. On the other hand, these inaccuracies and privacy amplification. Transients can also be utilized for attack [45] but there are various means of can compensate for each other [43] if used in a creative way. Non-zero cable capacitance [44] or cable defense against these [46,47]. The newest KLJN system, the random-resistor-random-temperature inductance can also yield information leaks that can be fixed by specific designs including the proper KLJN (RRRT-KLJN) scheme [48], is robust against the above vulnerabilities, or at least, no known attack exists against it yet. (ii) Active attacks, where Eve either modifies the information channel or she injects an extra current into it. Current injection attacks [30,49] and man-in-the-middle attacks [50] are examples which have been explored [56]. Due to the current and voltage comparison [50] feature and its more advanced cable-modeling version [49], active attacks are, so far, the least efficient attacks against the KLJN scheme. Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 10 choice of frequency range and privacy amplification. Transients can also be utilized for attack [45] but there are various means of defense against these [46,47]. The newest KLJN system, the random-resistor-random-temperature KLJN (RRRT-KLJN) scheme [48], is robust against the above vulnerabilities, or at least, no known attack exists against it yet. (ii) Active attacks, where Eve either modifies the information channel or she injects an extra current into it. Current injection attacks [30,49] and man-in-the-middle attacks [50] are examples which have been explored [56]. Due to the current and voltage comparison [50] feature and its more Appl. Sci. 2019, 9, 666 4 of 10 advanced cable-modeling version [49], active attacks are, so far, the least efficient attacks against the KLJN scheme. (iii) Flawed attacks. There are some proposed attack methods that are based on misconceptions (iii) Flawed attacks. There are some proposed attack methods that are based on misconceptions and and they they do do not not work. See thei work. See their r br brief ief s summary ummary and and c criticism riticism in, in, fo for r ex example, ample, p papers apers [ [5 51 1– –55 55] ] and and t the he book [56]. book [56]. 2. The New Attack Scheme Utilizing Deterministic Loop Currents 2. The New Attack Scheme Utilizing Deterministic Loop Currents 2.1. The Situation that Eve Utilizes for the Attack 2.1. The Situation that Eve Utilizes for the Attack In practical KLJN systems, in order to save a wire, the common end of the resistors (see Figure 1) In practical KLJN systems, in order to save a wire, the common end of the resistors (see Figure is often connected to the ground. In practical situations there is often an imbalance, a voltage 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 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] or electromagnetic interference (EMI) [53]. This potential information leak was pointed out in [53] as as a potential source of information leaks in the case of significant cable resistance. However, it was a potential source of information leaks in the case of significant cable resistance. However, it was not not realized in [53] that information leaks can exist even at zero cable resistance. The present study is 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 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 different units in instruments [33,34]). For EMI-induced ground loops, our DC-limited study is only first step in addressing a more general situation (which should be investigated in future works). 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, 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 consideration was given to situations where during the bit exchange period, the relative change in in the parasitic voltage is small. For the sake of simplicity but without the limitation of generality, the parasitic voltage is small. For the sake of simplicity but without the limitation of generality, we we assume that the imperfection is represented by a positive DC voltage generator located at Alice’s assume that the imperfection is represented by a positive DC voltage generator located at Alice’s end end (see Figure 2). (see Figure 2). Due to Kerckhoffs’s principle of security, that is, the assumption that the enemy knows everything Due to Kerckhoffs's principle of security, that is, the assumption that the enemy knows except the momentary key, we must assume that Eve knows the polarity and value of this DC voltage everything except the momentary key, we must assume that Eve knows the polarity and value of (if she does not know it at first, she will be able to extract it via long-time averaging). The direction of this DC voltage (if she does not know it at first, she will be able to extract it via long-time averaging). the current I(t) is assumed to point from Alice to Bob. The voltage U(t) and current I(t) in the wire The direction of the current I(t) is assumed to point from Alice to Bob. The voltage U(t) and current contain the sum of a DC component and an AC (stochastic, that is, noise) component. I(t) in the wire contain the sum of a DC component and an AC (stochastic, that is, noise) component. Figure 2. The KLJN system with ground loop voltage. Here U ∈{U ;U } and Figure 2. The KLJN system with ground loop voltage. Here U 2 fU ; U g and U 2 fU ; U g An LAn HAn An Bn LAn L HA Bn n HBn are the voltage noises belonging to the randomly chosen resistors, R &R 2 fR ; R g, belonging to U ∈{U ;U } are the voltage noises belonging to the rando A B mlyL chosen H resistors, Bn LBn HBn Alice and Bob, respectively. U is the ground loop DC voltage source and U(t) and I(t) are the DC R & R ∈{R ; R }, belonging to Alice and Bob, respectively. U is the ground loop DC voltage A B L H DC voltage and current on the wire, respectively. source and U (t ) and I (t ) are the voltage and current on the wire, respectively. Let us analyze the resulting voltages and currents. The current in the wire is Let us analyze the resulting voltages and currents. The current in the wire is I(t) = I + I (t) (3) DC n where I is its DC component DC DC I = (4) DC R + R A B and I (t) is its AC (noise) component n Appl. Sci. 2019, 9, 666 5 of 10 U (t) U (t) An Bn I (t) = (5) R + R A B in which U and U , with U 2 fU ; U g and U 2 fU ; U g, are the voltage noise An Bn An LAn HAn Bn LBn HBn sources of the chosen resistors, R and R , respectively. A B The voltage on the wire is U(t) = I(t)R + U (t). (6) B Bn From Equations (3) and (6) we obtain U(t) = U + U (t) = I R + I (t)R + U (t) (7) DCw ACw DC B n B Bn where U and U (t) represent the DC and AC voltage components in the wire, respectively. DCw ACw The DC component can be written as DC U = I R = R . (8) DCw DC B B R + R A B The DC component is different during Alice’s and Bob’s LH and HL bit situations of secure bit exchange, which yields information leaks. In the LH situation, that is, when R = R and R = R , A L B H the DC component of the voltage on the wire is U  U = U (9) DCw LH DC R + R H L and, in the HL bit situation, U  U = U . (10) DCw HL DC R + R H L Note that as we have been assuming that in the given KLJN setup R > R , in this H L particular situation U < U . (11) HL LH For later usage, we evaluate the average of U and U and call this quantity the threshold LH HL voltage, U , where th U + U U LH HL DC U = = . (12) th 2 2 The effective (RMS) amplitude U of the noise voltage on the wire is identical in both the LH ACw and HL cases: R R L H U = 4kTB . (13) ACw W R + R L H Note that the voltage and current noises in the wire follow a normal distribution since the addition of normally distributed signals results in a signal that has normal (Gaussian) distribution with a corresponding mean (see Equation (10)) and variance. For an illustration of the information leak, see Figure 3. The DC component, that is, the mean value of the resulting (AC + DC) Gaussian depends on the bit situation during the secure key exchange. This dependence poses as a source of information for Eve about the secret key. This feature will be exploited below for the new attack scheme. 2.2. The Attack Scheme The attack consists of three steps: measurement, evaluation, and guessing. (i) Measurement: During a single secure bit exchange, Eve measures N independent samples of the wire voltage. Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 10 R R LH Uk = 4TB . (13) ACw W R + R LH Note that the voltage and current noises in the wire follow a normal distribution since the addition of normally distributed signals results in a signal that has normal (Gaussian) distribution with a corresponding mean (see Equation (10)) and variance. For an illustration of the information leak, see Figure 3. The DC component, that is, the mean value of the resulting (AC + DC) Gaussian depends on the bit situation during the secure key exchange. This dependence poses as a source of information for Eve about the secret key. This feature will be exploited below for the new attack scheme. 2.2. The Attack Scheme The attack consists of three steps: measurement, evaluation, and guessing. (i) Measurement: During a single secure bit exchange, Eve measures N independent samples of the wire voltage. Appl. Sci. 2019, 9, 666 6 of 10 (ii) Evaluation: She evaluates the fraction γ of these N samples that are above U , which is th (ii) Evaluation: She evaluates the fraction of these N N samples that are above U , which is th (14) γ = g = (14) N N where is the number of samples that are above U . th + 0.5 < γ γ < 0.5 (iii) Guessing (based on Equations (9)–(14)): For and , Eve's guesses are the where N is the number of samples that are above U . th and bit situations, respectively. For γ = 0.5 her decision is undetermined and carries no LH(iii) Guessing HL (based on Equations (9)–(14)): For 0.5 < g and g < 0.5, Eve’s guesses are the LH and HL bit situations, respectively. For g = 0.5 her decision is undetermined and carries no useful information. useful information. (iv) Eve's correct guessing probability p is given as (iv) Eve’s correct guessing probability p is given as cor p = lim (15) cor n →∞ tot p = lim (15) tot n !¥ n tot tot where n is the total number of guess bits and n is the number of correctly guessed bits. The tot cor where n is the total number of guess bits and n is the number of correctly guessed bits. tot cor situation p = 0.5 indicates perfect security against Eve's attack. The situation p = 0.5 indicates perfect security against Eve’s attack. In the next section, we demonstrate the attack method via computer simulation. In the next section, we demonstrate the attack method via computer simulation. Figure 3. Eve’s threshold scheme to guess the bit situation LH versus HL. Figure 3. Eve’s threshold scheme to guess the bit situation LH versus HL. 3. Simulation Results 3. Simulation Results To test Eve’s correct guessing probability p for the LH situation, we assumed that Alice and Bob selected R = 1 kW and R = 10 kW. During these experiments, the DC voltage was kept at a constant L H level of 0.1 V (see Figures 2 and 3). To generate noise, we used the white Gaussian noise function (wgn) from the Matlab communication system toolbox to test the success statistics of the attack scheme while varying the temperature. The effective bandwidth D f and the range of temperatures were 1 MHz and 8 18 10 < T < 10 K, respectively. At lower temperatures p was 1, within the statistical inaccuracy of simulations; at the high-temperature limit it converged to 0.5. The duration of the secure bit exchange period was characterized by the number N of independent noise samples used during the exchange of the particular bit. We tested secure key length M = 700 bits at different bit exchange durations represented by sample/bit numbers N = 1000, 500, and 200, respectively. Figure 4 shows Eve’s correct guessing probability (p) of a key bit versus temperature. With temperature approaching infinity, the effective noise voltage on the wire also approaches infinity and the Gaussian density function is symmetrically distributed around the threshold voltage U . Thus, the probabilities of finding the noise amplitude th above or below U are identical (0.5) Therefore, Eve’s correct guessing probability represents the th perfect security limit, p = 0.5. Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 10 To test Eve's correct guessing probability p for the LH situation, we assumed that Alice and Bob selected and . During these experiments, the DC voltage was kept at a R = 1 kΩ R = 10 kΩ L H constant level of 0.1 V (see Figures 2 and 3). To generate noise, we used the white Gaussian noise function (wgn) from the Matlab communication system toolbox to test the success statistics of the Δ f attack scheme while varying the temperature. The effective bandwidth and the range of 8 18 10 < T < 10 K temperatures were 1 MHz and , respectively. At lower temperatures p was 1, within the statistical inaccuracy of simulations; at the high-temperature limit it converged to 0.5. The duration of the secure bit exchange period was characterized by the number N of independent noise samples used during the exchange of the particular bit. We tested secure key length M = 700 bits at different bit exchange durations represented by sample/bit numbers N = 1000, 500, and 200, respectively. Figure 4 shows Eve's correct guessing probability ( p ) of a key bit versus temperature. With temperature approaching infinity, the effective noise voltage on the wire also approaches infinity and the Gaussian density function is symmetrically distributed around the threshold voltage . Thus, the probabilities of finding the th noise amplitude above or below are identical (0.5) Therefore, Eve's correct guessing probability th represents the perfect security limit, p = 0.5. Appl. Sci. 2019, 9, 666 7 of 10 Δ f Figure 4. Eve's correct guessing probability (p) of key bits versus temperatures at bandwidth Figure 4. Eve’s correct guessing probability (p) of key bits versus temperatures at bandwidth D f equals 10 Hz, for key length 700 bits, and duration/bit (number of samples/bit) 200, 500, and 1000, equals 10 Hz, for key length 700 bits, and duration/bit (number of samples/bit) 200, 500, and 1000, respectively. The limit p = 0.5 stands for perfect security. respectively. The limit p = 0.5 stands for perfect security. The observed dependence can be interpreted by the behavior of the error function (see also The observed dependence can be interpreted by the behavior of the error function (see also Equations (8) and (12)) Equations (8) and (12)) U U th DCw    U − U pfU(t)  U g = 0.5 1 er f p (16) th th DCw   pU t ≥ U = 0.5 1 − erf () U 2 (16) {}   eff th   U 2   eff   where U(t) is the instantaneous voltage amplitude in the wire and the error function is where Ut is the instantaneous voltage amplitude in the wire and the error function is () er f (x) = p exp y dy. (17) p 2 erf ( x) = exp − y dy. (17) () − x The noise in the KLJN scheme is a bandlimited white noise, and thus, in accordance with the Johnson formula, the effective noise voltage scales as U µ T D f (18) eff Therefore, when temperature T is converging towards infinity, p converges to the perfect security limit of 0.5 (see Figure 4). 4. Some of the Possible Defense Techniques Against the Attack Based on the considerations above, the impact of the attack can be eliminated by various means. The most natural ways are: (i) Cancelling the effect of the DC-voltage sources. For example, Bob can use a variable DC source that compensates for its effect. Similarly, eliminating ground loops is also beneficial. (ii) Alice and Bob can increase the effective temperature, that is, the amplitudes of their noise generators (see Equation (18) and Figure 4). (iii) Alice and Bob can increase the bandwidth to increase the effective value of the noise (see Equations (18) and (20)). However, the bandwidth must stay below the wave limit [54] to avoid information leaks due to reflection, and thus the applicability of this tool is strongly limited. Appl. Sci. 2019, 9, 666 8 of 10 5. Conclusions The KLJN secure key exchange scheme is a statistical physical system that offers unconditional (information-theoretic) security. For a detailed survey and its history, see the recent book [56]. In this paper a novel attack against the KLJN protocol is shown which has revealed that uses a frequently occurring parasitic feature, namely the imbalance of voltages between the ground points at the two ends. We showed that, in the DC limit, such parasite voltages and currents could cause information leaks. The present study is directly relevant for DC current-based ground loops (for example, during secure communication between different units in instruments [33,34]). The attack was demonstrated via computer simulation and proper defense protocols were shown to eliminate the information leak. For AC-type ground loops, our DC-limited study is only a first step in addressing a more general situation (which should be investigated in future works). Author Contributions: M.Y.M. and L.B.K. conceived and designed the studies; M.Y.M. performed the computer simulations; M.Y.M. and L.B.K. analyzed the data; M.Y.M. and L.B.K. wrote the paper. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. References 1. Diffie, W.; Hellman, M. New Directions in Cryptography. IEEE Trans. Inf. Theory 1976, 22, 644–654. [CrossRef] 2. Delfs, H.; Knebl, H. Introduction to Cryptography; Springer: Berlin/Heidelberg, Germany, 2002; Volume 2. 3. Wiesner, S.J. Conjugate Coding. Sigact News 1983, 1, 78–88. [CrossRef] 4. Bennett, C.H.; Brassard, G. Quantum cryptography: Public key distribution and coin tossing. Proc. IEEE Int. Conf. Comput. Syst. Signal Process. 1984, 175, 8. [CrossRef] 5. Wootters, W.; Zurek, W. A Single Quantum Cannot be Cloned. Nature 1982, 299, 802–803. [CrossRef] 6. Yuen, H. Security of quantum key distribution. IEEE Access 2016, 4, 724–749. [CrossRef] 7. 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Totally Secure Classical Communication Utilizing Johnson (-Like) Noise and Kirchhoff’s Law. Phys. Lett. A 2006, 352, 178–182. [CrossRef] 31. Kish, L.B. Methods of Using Existing Wire Lines (power lines, phone lines, internet lines) for Totally Secure Classical Communication Utilizing Kirchhoff’s Law and Johnson-like Noise. arXiv 2006, arXiv:physics/0610014. 32. Gonzalez, E.; Kish, L.B.; Balog, R.; Enjeti, P. Information theoretically secure, enhanced Johnson noise based key distribution over the smart grid with switched filters. PLoS ONE 2013, 8, e70206. [CrossRef] 33. Kish, L.B.; Entesari, K.; Granqvist, C.G.; Kwan, C. Unconditionally secure credit/debit card chip scheme and physical unclonable function. Fluct. Noise. Lett. 2017, 16, 1750002. [CrossRef] 34. Kish, L.B.; Kwan, C. Physical Uncloneable Function Hardware Keys Utilizing Kirchhoff-Law-Johnson-Noise Secure Key Exchange and Noise-Based Logic. Fluct. Noise Lett. 2013, 12, 1350018. [CrossRef] 35. 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PLoS ONE 2013, 8, e81810. [CrossRef] 53. Chen, H.P.; Kish, L.B.; Granqvist, C.G. On the Cracking Scheme in the Paper a Directional Coupler Attack against the Kish Key Distribution System by Gunn, Allison and Abbott. Metrol. Meas. Syst. 2014, 21, 389–400. [CrossRef] 54. Chen, H.P.; Kish, L.B.; Granqvist, C.G.; Schmera, G. Do Electromagnetic Waves Exist in a Short Cable at Low Frequencies? What Does Physics Say? Fluct. Noise Lett. 2014, 13, 1450016. [CrossRef] 55. Kish, L.B.; Gingl, Z.; Mingesz, R.; Vadai, G.; Smulko, J.; Granqvist, C.G. Analysis of an attenuator artifact in an experimental attack by Gunn-Allison-Abbott against the Kirchhoff-law-Johnson-noise (KLJN) secure key exchange system. Fluct. Noise Lett. 2015, 14, 1550011. [CrossRef] 56. Kish, L.B. The Kish Cypher. The Story of KLJN for Unconditional Security. World Sci. 2017. [CrossRef] © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute

A Static-loop-current Attack Against the Kirchhoff-Law-Johnson-Noise (KLJN) Secure Key Exchange System

Applied Sciences , Volume 9 (4) – Feb 15, 2019

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applied sciences Article A Static-loop-current Attack Against the Kirchhoff-Law-Johnson-Noise (KLJN) Secure Key Exchange System Mutaz Y. Melhem * and Laszlo B. Kish Department of Electrical and Computer Engineering, Texas A & M University, 3128 TAMU, College Station, TX 77843, USA; laszlokish@tamu.edu * Correspondence: yar111@tamu.edu; Tel.: +1-979-847-9071 Received: 10 October 2018; Accepted: 12 February 2019; Published: 15 February 2019 Abstract: In this study, a new attack against the Kirchhoff-Law-Johnson-Noise (KLJN) key distribution system is explored. The attack is based on utilizing a parasitic voltage-source in the loop. Relevant situations often exist in the low-frequency limit in practical systems, especially when the communication is over a distance, or between different units within an instrument, due to a ground loop and/or electromagnetic interference (EMI). Our present study investigates the DC ground loop situation when no AC or EMI effects are present. Surprisingly, the usual current/voltage comparison-based defense method that exposes active attacks or parasitic features (such as wire resistance allowing information leaks) does not function here. The attack is successfully demonstrated and proposed defense methods against the attack are shown. Keywords: KLJN key exchange; unconditional security; ground loop vulnerability; passive attack 1. Introduction 1.1. On Secure Communications Communications systems, standards, and technologies have been developed since ancient times. Today we have the internet, Internet-of-Things (IoT), operating fourth generation wireless networks (LTE), and the expected fifth generation wireless networks. An important requirement of any communication paradigm between these devices is to accomplish secure communication, i.e., to protect the privacy and integrity of users’ data that is transferred over the network. To achieve the security of transferred data which can contain sensitive information (e.g., bank account credentials, social security number, etc.) it is of utmost importance to defend against attacks. These attacks might be launched by an eavesdropper (Eve) who has access to the information channel between the communicating parties A (Alice) and B (Bob). The attack is passive if it eavesdrops without disturbing the channel. The attack is active (invasive) if Eve disturbs or changes the channel, such as with a man-in-the-middle attack. In the present paper, we introduce a new passive attack against the Kirchhoff-Law-Johnson-Noise (KLJN) secure key exchange scheme. 1.1.1. Secure Key Exchange Secure communication systems employ ciphers to encrypt messages (plaintext) and to decrypt encrypted messages (cyphertext). While the creation of a secure and efficient cipher is a complex problem, this problem may be solved simply. Ciphers operate with secure keys that form a momentary shared secret between Alice and Bob. Sharing (exchanging) the key securely is the difficult task. A communicator system cannot be more secure than its key. The security of the key exchange can be conditional or information-theoretic (unconditional). Appl. Sci. 2019, 9, 666; doi:10.3390/app9040666 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 666 2 of 10 1.1.2. Conditional Security Conditionally secure key exchange systems are the ones used generally nowadays. They are software protocols installed at Alice and Bob. Such algorithms utilize computational complexity and achieve only (computationally) conditional security (see e.g., [1,2]). The system is temporarily secure provided the adversary has limited computational resources. A major goal of quantum computer developments is to crack these types of key exchange systems (e.g., the Shor algorithm). From an information-theoretic point of view, security is non-existent because Eve has all the information to crack the encryption, but she needs a long time to do that unless she has a quantum computer or a yet-to-be-discovered classical algorithm that can do the job in a short time. The security is not future-proof. 1.1.3. Unconditional (Information-Theoretic) Security In order to achieve unconditional (information-theoretic) security at the key exchange, proper laws of physics with a special hardware are utilized. Two major classes of physics-based schemes have emerged for unconditional security: (i) Quantum key distribution (QKD) [3,4] concepts assume single photons and utilize quantum physics. The underlying laws of physics are Heisenberg’s uncertainty principle and the related quantum no-cloning theorem [5]. Even though there are serious debates about the actual level of unconditional security a practical QKD can offer (see e.g., [6–29]), most scientists agree that QKD is unique in its offering information-theoretic security via (a dark) optical fiber and also through air at night, provided the visibility is good. (ii) The Kirchhoff-Law-Johnson-Noise key distribution method that is based on the statistical physical features of the thermal noise of resistors [30–56]. The related law of physics is the fluctuation-dissipation-theorem (FDT). Some of its advantages are: It works via wire connections including power, phone, and internet lines, which can be used as information channels [31,32] to connect all homes and other establishments. It can be integrated on a chip, which implies excellent robustness, low price, and applicability in bankcards, computers, instruments, and physical unclonable function (PUF) hardware keys [33,34]. Its low price allows general applications such as unconditional security for the control of autonomous vehicular networks [35,36]. 1.2. On the KLJN Secure Key Distribution The KLJN scheme [30–52] utilizes the thermal noise of resistors (or the emulation of that by a specific hardware). In the core scheme Alice and Bob have two identical pairs of resistors, R and R L H (R < R ), respectively (see Figure 1). L H The key exchange protocol of a single secure bit is as follows: Alice and Bob randomly pick one of their resistors (R or R ), connect it to the wire channel, and keep them there during the bit exchange L H period while they execute voltage and/or current measurements to learn the resistor value at the other end (see below). The noise voltage generators shown in Figure 1 with each resistor can be the resistors’ own thermal noise, or an external noise generator emulating a much higher, common noise-temperature that is publicly agreed. The power density spectra of the voltage and current in the channel are given by the Johnson-Nyquist formulas [11] 4kTR R A B S ( f ) = , (1) R + R A B 4kT S ( f ) = (2) R + R A B where k is the Boltzmann’s constant, T is the common temperature, and R and R are the actually A B connected resistances at Alice’s and Bob’s ends, respectively, with R , R 2 fR , R g. After the A B L H measurement and spectral analysis, Equations (1) and (2) have two unknown variables, namely, Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 10 4kT (2) S ( f ) = R + R A B where k is the Boltzmann’s constant, T is the common temperature, and R and R are the actually connected resistances at Alice’s and Bob’s ends, respectively, with R , R ∈ R , R . After {} A B L H the measurement and spectral analysis, Equations (1) and (2) have two unknown variables, namely, the values of R and R , and thus Eve can find the values of the connected resistors, but not Appl. Sci. 2019, 9, 666 3 of 10 necessarily their locations, by solving these equations. We can represent the four different situations of the connected resistors ( R and/or R ) at the values of R and R , and thus Eve can find the values of the connected resistors, but L not necessarily H A B Alice's and Bob's ends by the indices of the connected resistors, LL, LH, HL, and HH, respectively. their locations, by solving these equations. As all the resistors have the same (noise) temperature, the ideal system is in thermal equilibrium, We can represent the four different situations of the connected resistors (R and/or R ) at L H where the second law of thermodynamics guarantees zero net power-flow. Hence, Eve cannot use Alice’s and Bob’s ends by the indices of the connected resistors, LL, LH, HL, and HH, respectively. the evaluation of power flow to determine the locations of the momentarily connected resistors As all the resistors have the same (noise) temperature, the ideal system is in thermal equilibrium, unless they have the same resistance values. On the other hand, Alice and Bob can determine the where the second law of thermodynamics guarantees zero net power-flow. Hence, Eve cannot use the connected resistor values by using Equations (1) or (2) since they know the value of their own evaluation of power flow to determine the locations of the momentarily connected resistors unless they connected resistors. have the same resist When ance R values. = R , whi On the ch ha other ppens a hand, t 5Alice 0% of and the bi Bob t exchange a can determine ttempts, the resul the connected ts A B resistor values by using Equations (1) or (2) since they know the value of their own connected resistors. are discarded. When R = R , which happens at 50% of the bit exchange attempts, the results are discarded. A B Figure 1. The core of the Kirchhoff-Law-Johnson-Noise (KLJN) system. U (t), U (t), U (t), HAn LAn HBn and Figure 1. U (tThe ) ar core e the of (thermal) the Kirchhoff-Law noise voltage -Johnson-Noise generators for(KLJ theN) related system. resistances R , and R ,, U (t ) U (t ) LBn H L HAn LAn respectively. U(t) and I(t) are the measured noise voltage and the current in the wire that are used to , and are the (thermal) noise voltage generators for the related resistances U (t ) U (t ) R HBn LBn H evaluate the power density spectra S ( f ) and S ( f ), respectively. and , respectively. U (t ) and I (t ) are the measured noise voltage and the current in the wire On Former Attacks Against the KLJN Secure Key Distribution that are used to evaluate the power density spectra S ( f ) and S ( f ) , respectively. u i Several attacks have been proposed but no attack has been able to compromise the unconditional security of the KLJN scheme because each known attack can efficiently be nullified by a corresponding On Former Attacks Against the KLJN Secure Key Distribution defense scheme. Several attacks have been proposed but no attack has been able to compromise the The attacks can be categorized into two classes: unconditional security of the KLJN scheme because each known attack can efficiently be nullified by (i) Passive attacks that utilize the non-ideal or parasitic features in a practical KLJN system for a corresponding defense scheme. information leaks. Non-zero wire resistance (see [37,38]) poses the greatest known threat, and the The attacks can be categorized into two classes: most efficient attack is power balance measurement (Second Law Attack) [39]. An efficient defense is (i) Passive attacks that utilize the non-ideal or parasitic features in a practical KLJN system for based on a proper temperature-offset [39,40]. Temperature-inaccuracies [41] and resistance-inaccuracies [42] information leaks. Non-zero wire resistance (see [37,38]) poses the greatest known threat, and the can also cause information leaks. On the other hand, these inaccuracies can compensate for each most efficient attack is power balance measurement (Second Law Attack) [39]. An efficient defense is other [43] if used in a creative way. Non-zero cable capacitance [44] or cable inductance can also yield based on a proper temperature-offset [39,40]. Temperature-inaccuracies [41] and information leaks that can be fixed by specific designs including the proper choice of frequency range resistance-inaccuracies [42] can also cause information leaks. On the other hand, these inaccuracies and privacy amplification. Transients can also be utilized for attack [45] but there are various means of can compensate for each other [43] if used in a creative way. Non-zero cable capacitance [44] or cable defense against these [46,47]. The newest KLJN system, the random-resistor-random-temperature inductance can also yield information leaks that can be fixed by specific designs including the proper KLJN (RRRT-KLJN) scheme [48], is robust against the above vulnerabilities, or at least, no known attack exists against it yet. (ii) Active attacks, where Eve either modifies the information channel or she injects an extra current into it. Current injection attacks [30,49] and man-in-the-middle attacks [50] are examples which have been explored [56]. Due to the current and voltage comparison [50] feature and its more advanced cable-modeling version [49], active attacks are, so far, the least efficient attacks against the KLJN scheme. Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 10 choice of frequency range and privacy amplification. Transients can also be utilized for attack [45] but there are various means of defense against these [46,47]. The newest KLJN system, the random-resistor-random-temperature KLJN (RRRT-KLJN) scheme [48], is robust against the above vulnerabilities, or at least, no known attack exists against it yet. (ii) Active attacks, where Eve either modifies the information channel or she injects an extra current into it. Current injection attacks [30,49] and man-in-the-middle attacks [50] are examples which have been explored [56]. Due to the current and voltage comparison [50] feature and its more Appl. Sci. 2019, 9, 666 4 of 10 advanced cable-modeling version [49], active attacks are, so far, the least efficient attacks against the KLJN scheme. (iii) Flawed attacks. There are some proposed attack methods that are based on misconceptions (iii) Flawed attacks. There are some proposed attack methods that are based on misconceptions and and they they do do not not work. See thei work. See their r br brief ief s summary ummary and and c criticism riticism in, in, fo for r ex example, ample, p papers apers [ [5 51 1– –55 55] ] and and t the he book [56]. book [56]. 2. The New Attack Scheme Utilizing Deterministic Loop Currents 2. The New Attack Scheme Utilizing Deterministic Loop Currents 2.1. The Situation that Eve Utilizes for the Attack 2.1. The Situation that Eve Utilizes for the Attack In practical KLJN systems, in order to save a wire, the common end of the resistors (see Figure 1) In practical KLJN systems, in order to save a wire, the common end of the resistors (see Figure is often connected to the ground. In practical situations there is often an imbalance, a voltage 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 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] or electromagnetic interference (EMI) [53]. This potential information leak was pointed out in [53] as as a potential source of information leaks in the case of significant cable resistance. However, it was a potential source of information leaks in the case of significant cable resistance. However, it was not not realized in [53] that information leaks can exist even at zero cable resistance. The present study is 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 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 different units in instruments [33,34]). For EMI-induced ground loops, our DC-limited study is only first step in addressing a more general situation (which should be investigated in future works). 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, 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 consideration was given to situations where during the bit exchange period, the relative change in in the parasitic voltage is small. For the sake of simplicity but without the limitation of generality, the parasitic voltage is small. For the sake of simplicity but without the limitation of generality, we we assume that the imperfection is represented by a positive DC voltage generator located at Alice’s assume that the imperfection is represented by a positive DC voltage generator located at Alice’s end end (see Figure 2). (see Figure 2). Due to Kerckhoffs’s principle of security, that is, the assumption that the enemy knows everything Due to Kerckhoffs's principle of security, that is, the assumption that the enemy knows except the momentary key, we must assume that Eve knows the polarity and value of this DC voltage everything except the momentary key, we must assume that Eve knows the polarity and value of (if she does not know it at first, she will be able to extract it via long-time averaging). The direction of this DC voltage (if she does not know it at first, she will be able to extract it via long-time averaging). the current I(t) is assumed to point from Alice to Bob. The voltage U(t) and current I(t) in the wire The direction of the current I(t) is assumed to point from Alice to Bob. The voltage U(t) and current contain the sum of a DC component and an AC (stochastic, that is, noise) component. I(t) in the wire contain the sum of a DC component and an AC (stochastic, that is, noise) component. Figure 2. The KLJN system with ground loop voltage. Here U ∈{U ;U } and Figure 2. The KLJN system with ground loop voltage. Here U 2 fU ; U g and U 2 fU ; U g An LAn HAn An Bn LAn L HA Bn n HBn are the voltage noises belonging to the randomly chosen resistors, R &R 2 fR ; R g, belonging to U ∈{U ;U } are the voltage noises belonging to the rando A B mlyL chosen H resistors, Bn LBn HBn Alice and Bob, respectively. U is the ground loop DC voltage source and U(t) and I(t) are the DC R & R ∈{R ; R }, belonging to Alice and Bob, respectively. U is the ground loop DC voltage A B L H DC voltage and current on the wire, respectively. source and U (t ) and I (t ) are the voltage and current on the wire, respectively. Let us analyze the resulting voltages and currents. The current in the wire is Let us analyze the resulting voltages and currents. The current in the wire is I(t) = I + I (t) (3) DC n where I is its DC component DC DC I = (4) DC R + R A B and I (t) is its AC (noise) component n Appl. Sci. 2019, 9, 666 5 of 10 U (t) U (t) An Bn I (t) = (5) R + R A B in which U and U , with U 2 fU ; U g and U 2 fU ; U g, are the voltage noise An Bn An LAn HAn Bn LBn HBn sources of the chosen resistors, R and R , respectively. A B The voltage on the wire is U(t) = I(t)R + U (t). (6) B Bn From Equations (3) and (6) we obtain U(t) = U + U (t) = I R + I (t)R + U (t) (7) DCw ACw DC B n B Bn where U and U (t) represent the DC and AC voltage components in the wire, respectively. DCw ACw The DC component can be written as DC U = I R = R . (8) DCw DC B B R + R A B The DC component is different during Alice’s and Bob’s LH and HL bit situations of secure bit exchange, which yields information leaks. In the LH situation, that is, when R = R and R = R , A L B H the DC component of the voltage on the wire is U  U = U (9) DCw LH DC R + R H L and, in the HL bit situation, U  U = U . (10) DCw HL DC R + R H L Note that as we have been assuming that in the given KLJN setup R > R , in this H L particular situation U < U . (11) HL LH For later usage, we evaluate the average of U and U and call this quantity the threshold LH HL voltage, U , where th U + U U LH HL DC U = = . (12) th 2 2 The effective (RMS) amplitude U of the noise voltage on the wire is identical in both the LH ACw and HL cases: R R L H U = 4kTB . (13) ACw W R + R L H Note that the voltage and current noises in the wire follow a normal distribution since the addition of normally distributed signals results in a signal that has normal (Gaussian) distribution with a corresponding mean (see Equation (10)) and variance. For an illustration of the information leak, see Figure 3. The DC component, that is, the mean value of the resulting (AC + DC) Gaussian depends on the bit situation during the secure key exchange. This dependence poses as a source of information for Eve about the secret key. This feature will be exploited below for the new attack scheme. 2.2. The Attack Scheme The attack consists of three steps: measurement, evaluation, and guessing. (i) Measurement: During a single secure bit exchange, Eve measures N independent samples of the wire voltage. Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 10 R R LH Uk = 4TB . (13) ACw W R + R LH Note that the voltage and current noises in the wire follow a normal distribution since the addition of normally distributed signals results in a signal that has normal (Gaussian) distribution with a corresponding mean (see Equation (10)) and variance. For an illustration of the information leak, see Figure 3. The DC component, that is, the mean value of the resulting (AC + DC) Gaussian depends on the bit situation during the secure key exchange. This dependence poses as a source of information for Eve about the secret key. This feature will be exploited below for the new attack scheme. 2.2. The Attack Scheme The attack consists of three steps: measurement, evaluation, and guessing. (i) Measurement: During a single secure bit exchange, Eve measures N independent samples of the wire voltage. Appl. Sci. 2019, 9, 666 6 of 10 (ii) Evaluation: She evaluates the fraction γ of these N samples that are above U , which is th (ii) Evaluation: She evaluates the fraction of these N N samples that are above U , which is th (14) γ = g = (14) N N where is the number of samples that are above U . th + 0.5 < γ γ < 0.5 (iii) Guessing (based on Equations (9)–(14)): For and , Eve's guesses are the where N is the number of samples that are above U . th and bit situations, respectively. For γ = 0.5 her decision is undetermined and carries no LH(iii) Guessing HL (based on Equations (9)–(14)): For 0.5 < g and g < 0.5, Eve’s guesses are the LH and HL bit situations, respectively. For g = 0.5 her decision is undetermined and carries no useful information. useful information. (iv) Eve's correct guessing probability p is given as (iv) Eve’s correct guessing probability p is given as cor p = lim (15) cor n →∞ tot p = lim (15) tot n !¥ n tot tot where n is the total number of guess bits and n is the number of correctly guessed bits. The tot cor where n is the total number of guess bits and n is the number of correctly guessed bits. tot cor situation p = 0.5 indicates perfect security against Eve's attack. The situation p = 0.5 indicates perfect security against Eve’s attack. In the next section, we demonstrate the attack method via computer simulation. In the next section, we demonstrate the attack method via computer simulation. Figure 3. Eve’s threshold scheme to guess the bit situation LH versus HL. Figure 3. Eve’s threshold scheme to guess the bit situation LH versus HL. 3. Simulation Results 3. Simulation Results To test Eve’s correct guessing probability p for the LH situation, we assumed that Alice and Bob selected R = 1 kW and R = 10 kW. During these experiments, the DC voltage was kept at a constant L H level of 0.1 V (see Figures 2 and 3). To generate noise, we used the white Gaussian noise function (wgn) from the Matlab communication system toolbox to test the success statistics of the attack scheme while varying the temperature. The effective bandwidth D f and the range of temperatures were 1 MHz and 8 18 10 < T < 10 K, respectively. At lower temperatures p was 1, within the statistical inaccuracy of simulations; at the high-temperature limit it converged to 0.5. The duration of the secure bit exchange period was characterized by the number N of independent noise samples used during the exchange of the particular bit. We tested secure key length M = 700 bits at different bit exchange durations represented by sample/bit numbers N = 1000, 500, and 200, respectively. Figure 4 shows Eve’s correct guessing probability (p) of a key bit versus temperature. With temperature approaching infinity, the effective noise voltage on the wire also approaches infinity and the Gaussian density function is symmetrically distributed around the threshold voltage U . Thus, the probabilities of finding the noise amplitude th above or below U are identical (0.5) Therefore, Eve’s correct guessing probability represents the th perfect security limit, p = 0.5. Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 10 To test Eve's correct guessing probability p for the LH situation, we assumed that Alice and Bob selected and . During these experiments, the DC voltage was kept at a R = 1 kΩ R = 10 kΩ L H constant level of 0.1 V (see Figures 2 and 3). To generate noise, we used the white Gaussian noise function (wgn) from the Matlab communication system toolbox to test the success statistics of the Δ f attack scheme while varying the temperature. The effective bandwidth and the range of 8 18 10 < T < 10 K temperatures were 1 MHz and , respectively. At lower temperatures p was 1, within the statistical inaccuracy of simulations; at the high-temperature limit it converged to 0.5. The duration of the secure bit exchange period was characterized by the number N of independent noise samples used during the exchange of the particular bit. We tested secure key length M = 700 bits at different bit exchange durations represented by sample/bit numbers N = 1000, 500, and 200, respectively. Figure 4 shows Eve's correct guessing probability ( p ) of a key bit versus temperature. With temperature approaching infinity, the effective noise voltage on the wire also approaches infinity and the Gaussian density function is symmetrically distributed around the threshold voltage . Thus, the probabilities of finding the th noise amplitude above or below are identical (0.5) Therefore, Eve's correct guessing probability th represents the perfect security limit, p = 0.5. Appl. Sci. 2019, 9, 666 7 of 10 Δ f Figure 4. Eve's correct guessing probability (p) of key bits versus temperatures at bandwidth Figure 4. Eve’s correct guessing probability (p) of key bits versus temperatures at bandwidth D f equals 10 Hz, for key length 700 bits, and duration/bit (number of samples/bit) 200, 500, and 1000, equals 10 Hz, for key length 700 bits, and duration/bit (number of samples/bit) 200, 500, and 1000, respectively. The limit p = 0.5 stands for perfect security. respectively. The limit p = 0.5 stands for perfect security. The observed dependence can be interpreted by the behavior of the error function (see also The observed dependence can be interpreted by the behavior of the error function (see also Equations (8) and (12)) Equations (8) and (12)) U U th DCw    U − U pfU(t)  U g = 0.5 1 er f p (16) th th DCw   pU t ≥ U = 0.5 1 − erf () U 2 (16) {}   eff th   U 2   eff   where U(t) is the instantaneous voltage amplitude in the wire and the error function is where Ut is the instantaneous voltage amplitude in the wire and the error function is () er f (x) = p exp y dy. (17) p 2 erf ( x) = exp − y dy. (17) () − x The noise in the KLJN scheme is a bandlimited white noise, and thus, in accordance with the Johnson formula, the effective noise voltage scales as U µ T D f (18) eff Therefore, when temperature T is converging towards infinity, p converges to the perfect security limit of 0.5 (see Figure 4). 4. Some of the Possible Defense Techniques Against the Attack Based on the considerations above, the impact of the attack can be eliminated by various means. The most natural ways are: (i) Cancelling the effect of the DC-voltage sources. For example, Bob can use a variable DC source that compensates for its effect. Similarly, eliminating ground loops is also beneficial. (ii) Alice and Bob can increase the effective temperature, that is, the amplitudes of their noise generators (see Equation (18) and Figure 4). (iii) Alice and Bob can increase the bandwidth to increase the effective value of the noise (see Equations (18) and (20)). However, the bandwidth must stay below the wave limit [54] to avoid information leaks due to reflection, and thus the applicability of this tool is strongly limited. Appl. Sci. 2019, 9, 666 8 of 10 5. Conclusions The KLJN secure key exchange scheme is a statistical physical system that offers unconditional (information-theoretic) security. For a detailed survey and its history, see the recent book [56]. In this paper a novel attack against the KLJN protocol is shown which has revealed that uses a frequently occurring parasitic feature, namely the imbalance of voltages between the ground points at the two ends. We showed that, in the DC limit, such parasite voltages and currents could cause information leaks. The present study is directly relevant for DC current-based ground loops (for example, during secure communication between different units in instruments [33,34]). The attack was demonstrated via computer simulation and proper defense protocols were shown to eliminate the information leak. For AC-type ground loops, our DC-limited study is only a first step in addressing a more general situation (which should be investigated in future works). 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Published: Feb 15, 2019

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