1 CURRENT INJECTION AND VOLTAGE INSERTION ATTACKS AGAINST THE VMG -KLJN SECURE KEY EXCHANGER

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CURRENT INJECTION AND VOLTAGE INSERTION ATTACKS

AGAINST THE VMG-KLJN SECURE KEY EXCHANGER

SHAHRIAR FERDOUS

, CHRISTIANA CHAMON, LASZLO B. KISH

Department of Electrical and Computer Engineering, Texas A&M University, TAMUS 3128,

College Station, TX 77841-3128, USA

ferdous.shahriar@tamu.edu , cschamon@tamu.edu , laszlokish@tamu.edu

Abstract: In this paper, the vulnerability of the Vadai, Mingesz and Gingl (VMG)-

Kirchhoff-Law-Johnson-Noise (KLJN) Key Exchanger (Nature, Science Report 5 (2015)

13653) against two active attacks is demonstrated. The security vulnerability arises from

the fact that the effective driving impedances are different between the HL and LH cases

for the VMG-KLJN scheme; whereas for the ideal KLJN scheme they are same. Two

defense schemes are shown against these attacks but each of them can protect against only

one of the attack types; but not against the two attacks simultaneously. The theoretical

results are confirmed by computer simulations.

Keywords: Information theoretic (unconditional) security; Vadai, Mingesz and Gingl

(VMG)-KLJN scheme; active attacks.

1. Introduction

Sensitive data must be encrypted and protected against any kind of breach or eavesdropping

during secure communications. Information theoretic (unconditional) security [1-7]

provides protection and privacy; irrespective of the computational power, measurement

accuracy and speed of the eavesdropper (Eve). At the heart of secure communications is

the secure key exchange. Robust, unconditional, and hardware-based secure key exchange

is offered by Quantum Key Distribution (QKD) [8-43] and its statistical-physical

competitor, the Kirchhoff-Law-Johnson-Noise (KLJN) secure key exchanger [3-7, 44-96].

Quantum Key Distribution (QKD) is based on Quantum physics, particularly the no-

cloning theorem; as opposed to KLJN scheme, which is based on classical statistical

physics, particularly the Fluctuation Dissipation Theorem [3], Gaussian stochastic process

[57] and thermal equilibrium [96].

Corresponding Author

1.1. The KLJN key exchanger

The core of the KLJN system is shown in Figure 1 [7]. It consists of an information channel

which is a wire line between the two communicating parties Alice & Bob; two switches

and an identical resistor pair, RH and RL, (where RH > RL and RH ≠ RL) at each of the

communicators, Alice and Bob [95]. At the start of the Bit Exchange Period (BEP), each

party (Alice or Bob) can arbitrarily choose from RH and RL, and then connect the selected

resistor to the wire line for the whole BEP. Alice and Bob publicly agree about the

interpretation of the bit value (0 or 1) for the HL (RH at Alice and RL at Bob), and the LH

(RL at Alice and RH at Bob) situations, respectively. The other two situations, HH and LL,

are unimportant because those results will be discarded [3, 5], see below.

Fig. 1. The core of the KLJN secure key exchanger scheme consists of a wire line connection between the two

communicating parties Alice & Bob. The voltage U(t), the current I(t), and their spectra Su(f) and Si(f),

respectively, are measurable by Alice, Bob and Eve. In the private space of Alice and Bob, the voltage generators

ULA(t), UHA(t), ULB(t) and UHB(t) represent the independent thermal (Johnson-Nyquist) noises of the resistors, or

optionally they are external Gaussian noise generators for higher noise temperature. The homogeneous

temperature T in the system guarantees that the LH (Alice RL, Bob RH) and HL (Alice RH, Bob RL) resistor

connections provide identical mean-square voltage and current, and the related spectra, in the wire [3,5,95].

The security of the KLJN system is based on the Kirchhoff’s Loop Law and the

Fluctuation–Dissipation theorem [3, 5-7]. More generally, the unconditional security of the

KLJN scheme is derived from the Second Law of Thermodynamics [3], and it requires

thermal equilibrium (homogeneous temperature) for the system. The key exchanger

utilizes the thermal noise of the resistors that can be emulated by external Gaussian voltage

noise generators, too. Since, in thermal equilibrium Alice and Bob operate at equal noise

temperatures (T), the net power flow between Alice and Bob is zero. Due to the Johnson

formula, the HL or LH pairs provide the same mean-square noise voltage spectra and noise

current spectra because both the parallel and serial resultant resistances are identical in the

two cases.

If, during the BEP, both parties connect the same resistance value HH (RH, RH) or LL (RL,

RL), the situation is not secure, see Figure 2, because then Eve knows the connected

resistance values at two parties. Thus, the HH & LL bit situations are discarded. The only

secure combinations are the HL (RH, RL) and LH (RL, RH) cases [3, 5] because then Eve

does not know the locations of the connected resistances.

Fig. 2. The mean-square value of noise voltage on the wire line versus time during operation [5]. The three

different levels (dotted lines) are at HH, HL/LH and LL. Obviously, of the three levels, only the HL/LH level is

secure against Eve.

With passive eavesdropping, that is, by measuring the wire voltage and current, Eve can

determine the resultant (both parallel and serial) values of the connected resistances by

evaluating the noise voltage and current spectra, see below. However, in the HL or LH

cases, Eve cannot determine which side has RH and which side has RL [3, 7], thus she does

not know if the state is HL or LH, which means she does not know if the key bit value is 0

or 1. For Eve this is an information entropy of 1 bit indicating perfect unconditional

security.

Specifically, the noise spectra Su(f) and Si(f) of the voltage U(t) and current I(t) in the wire,

respectively, are given by the Johnson-Nyquist formulas of thermal noise [3, 5]:

( ) 4S f kTR=

, (1)

() kT

Sf R

, (2)

where k is the Boltzmann constant, and Rp and Rs are the parallel and serial resultant values

of the connected resistors, respectively. In the HL and LH cases the resultant values are:

pLH pHL

RRRR

(3)

sLH sHL L H

R R R R= = +

. (4)

The quantities that Eve can access with passive measurements satisfy the following

equations that, together with Equations (3)-(4), form the pillars of security against passive

attacks against the KLJN scheme [95]:

LH HL

UU=

(5)

LH HL

II=

(6)

LH HL 0PP==

, (7)

where the voltage and current values stand for the effective (RMS) amplitudes in the wire,

and P is the mean power flow between Alice and Bob.

1.2. The Vadai, Mingesz and Gingl (VMG)-KLJN scheme

Vadai, Mingesz and Gingl (VMG) introduced a modified scheme in Nature Scientific

Reports [47] and proposed that, instead of using identical resistor pairs; we can actually

use four arbitrary resistors and still maintain perfectly secure communications (see Figure

3). VMG showed that the four arbitrary resistors require (typically) different noise

temperatures to guarantee that the voltage and current spectra and the net power flow in

the wire are identical for the HL and LH cases. But this implies non-zero mean power flow

between Alice and Bob, since the noise temperatures between the resisters are

inhomogeneous, so the system is not in thermal equilibrium, anymore. So, it looked like

thermal equilibrium was not needed for perfect security, on the contrary of the former

understanding [5, 83].

We started investigating these claims in former papers [95, 96] and concluded that these

claims are incorrect because there are passive attack types that work against the VMG-

KLJN scheme while not against the original KLJN system. In the present paper we are

exploring active attacks against the VMG-KLJN scheme, see Sections 2 and 3.

Fig. 3. The core of the VMG-KLJN secure key exchanger scheme [47, 95]. The four resistors are different and

can be freely chosen (though not totally arbitrarily because of certain unphysical solutions). The voltage

generators ULA(t), UHA(t), ULB(t) and UHB(t) represent the thermal noise of the resistors RLA, RHA, RLB and RHB

respectively. TLA, THA, TLB and THB represent the noise temperature of these resistors. The temperature of one of

the resistors is freely chosen, and the other 3 temperatures depend on the corresponding resistor values and are

given by the VMG Equations (9-11) [47, 95].

For the VMG-KLJN scheme, there are four equations that describe the RMS noise voltages.

In search for their solution, they modified Equation (7) by removing the zero power flow

conditions [47, 95], as follows:

LH HL

PP=

. (8)

Then they used Equations (5) and (6) to obtain the necessary mean-square voltages that

implies the corresponding temperatures, too [47, 95]:

( )

LB HA HB HA HB HB

HB LA HB HB

LA LB LA HA HA LA

R R R R R R

U U kT R B

R R R R R R

+ − −

(9)

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1CURRENTINJECTIONANDVOLTAGEINSERTIONATTACKSAGAINSTTHEVMG-KLJNSECUREKEYEXCHANGERSHAHRIARFERDOUS1,CHRISTIANACHAMON,LASZLOB.KISHDepartmentofElectricalandComputerEngineering,TexasA&MUniversity,TAMUS3128,CollegeStation,TX77841-3128,USAferdous.shahriar@tamu.edu,cschamon@tamu.edu,laszlokish@tamu.eduAbstrac...

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