Classical Quantum Key Exchange…

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This is a long post. If you don’t read all of it, please read the last sentence. Kish’s scheme depends on the behavior of individual atoms and electrons. Classical electrodynamics does not deal with point charges or individual particles, and does not predict the Johnson-Nyquist noise. The scheme is not so classical as it might appear.

Regarding Matthew Skala’s comments here and on his web site

See especially the new preprint It mentions that another paper is also in preparation which will address practical issues.

Skala is concerned that Eve can record voltage and current as function of time at high bandwidth and at several positions along the wire. This would then help determine which end has the larger resistor, because of propagation delays. Note that Eve does not have to inject any current into the wire.

for a derivation of the Nyquist relation. See also
(Laszlo: perhaps you would contribute to that article, which is a little sparse?)

The average squared noise voltage <V^2> across a resistor is directly proprotional to bandwidth delta nu (Hz). Higher bandwidth gives higher <V^2>. But the wire has a limited bandwidth. Also, in the practical case described in the new preprint, there are low-pass filters at each end of the wire limiting the bandwidth of the noise. Thus, Eve can measure the voltage at high bandwidth, but the low-pass filters will smooth out the signal Eve was hoping to observe.

Information leakage due to taps at each end of a wire which has finite resistance is to be avoided by choosing suitable resistor values such that there will not be enough time to determine the position of the larger resistor before the end of the clock period.

Bollinger’s concerns that this classical method lacks something which is present in the quantum approach:

In my opinion, the difference between “classical” and “quantum” physics is much overrated. As Einstein noted, “there is, strictly speaking, today, no such thing as a classical field-theory” (A.E. Philosopher Scientist, P.A. Schilpp, ed., vol 2 p. 675). The work of Maxwell, Boltzmann, Gibbs, and others on electrodynamics and statistical mechanics is quite different from Newtonian mechanics. But regardless of how we define a classical theory, the problem here arises because people suppose they know for sure that there is a “classical domain” and a “quantum domain” and that experiments in the classical domain somehow cannot benefit from quantum effects.

In “quantum cryptography,” the actual physical state of the photon (e.g. its polarization) cannot be known before it is measured, and even then, only the measurement result is known, not the full state. Because a single photon (or a pair of entangled photons) is involved, we are supposed to believe that this is fundamentally different from the situation where one or another macroscopic resistor is switched into a circuit. But, with the resistor, there are also many things which cannot be known even after the measurement. We know nothing about the microscopic environment of each electron in the resistor, and after we measure a noise voltage, we know only the average squared voltage. Just as measuring photon polarization tells us almost nothing about the full state of the photon, measuring the voltage tells us very little about the microscopic state of the resistor.

Look at the Sonoda derivation of the Nyquist relation. A resistor is modeled as containing N electrons distributed along length L. Each electron has thermal kinetic energy in the x direction of 1/2 kT, and there is a local electric field which accelerates it randomly. All we know about these fields is that they average to zero, that they are uncorrelated, and that they must maintain the average kinetic energy 1/2 kT for each electron. From this we derive the Johnson noise.

One may consider a theory classical if it doesn’t involve Planck’s constant, but I suggest that the appearance of Boltzmann’s constant also makes the theory non-classical. Individual particles and point sources are non-classical.

In quantum cryptography, detection of the evesdropper occurs when Alice and Bob learn they had their polarizers set at equal angles but did not observe compatible results. (By the way, in QC, the transmitted key is used only after Alice and Bob determine that it was not overheard, so the fact that evesdropping is not noticed instantly is not a problem). In Kish’s scheme, evesdropping is detected when Alice and Bob find that they are measuring significantly different voltages and currents.

The point is, in both schemes, the evesdropper cannot measure the physical property of interest without disturbing it. There is nothing more magical about polarization of a single photon than there is about the voltage and current arising from a pair of resistors at opposite ends of a wire. To detect the polarization of a passing photon, you have to insert a polarizer into the fiber. To determine the location of the larger resistor, you have to inject current into the wire. The low-pass filters and the finite clock period serve the same purpose as the photon number: there is not enough information available to a passive evesdropper to detect the randomly chosen polarization or resistance setting, but Alice and Bob know their own instrument setting, and hence have enough additional information to establish a shared secret.


One Response to “Classical Quantum Key Exchange…”

  1. archimerged Says:

    I posted this comment on Schneier’s blog after the above comment: I want to note that there is another use of “classical” e.g. in Landau and Lifshitz _Statistical Physics_, where a gas with Boltzmann statistics is classical while a gas with Fermi or Bose statistics is quantum. To me, classical corresponds to those 19th century physicists who did not believe in atoms. Anyway, what I’m trying to say in the previous post is that some amazing things can arise even when using Boltzmann statistics, which do not arise from the continuous charge distributions considered in classical electrodynamics. So we shouldn’t be so surprised if it is possible to securely exchange keys over a wire using Johnson noise.

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