Quantum Crypto Part 2

This is the second in the Quantum Cryptography series of posts. This post covers some of the theoretical underpinnings of QC, including some of the earliest protocols and a metric for the error rate of a quantum channel.

The BB84 Protocol [1]

The BB84 protocol is widely accepted as the first quantum cryptography protocol published. Its name comes from the initials of the authors of the paper that first explained it (Bennett and Brassard), concatenated with the year (1984) that the paper was published in. The protocol relies on two fundamental tenets of quantum theory for its security; the uncertainty theorem and the no-cloning theorem. The uncertainty theorem extended from conjugate variables to conjugate bases implies that only one bit of information to be derived from measurement on a single qubit [2]. More concretely, consider a qubit that has one of the four following possible values:

The true value of the qubit takes four possibilities, which means that at least two bits of information are required to distinguish between the possible values. First, a choice of basis must be made between the computational basis $\left(\ket{0}, \ket{1}\right)$ vs. the Hadamard basis $\left(\ket{+},\ket{-}\right)$. The second bit comes from a measurement made in the correct basis which then tells us which of the basis states the qubit is in. Suppose we take a qubit that is in the computational basis and measure it in the Hadamard basis. Since the two bases are conjugate [2], the result we get will be completely random; we $\ket{0}$ half the time and $\ket{1}$ the other half. The same holds true if we take a qubit in the Hadamard basis and measure it in the computational basis. Therefore, choosing the wrong basis for measurement completely destroys the information held by the qubit. This introduces another possibility by which the information held by a qubit could be extracted; we simply clone the qubit, and perform a measurement in the computational basis on one copy, and a measurement in the Hadamard basis on the other copy. This is impossible thanks to the no-cloning theorem, and completes the intuition behind the foundations of the security of the BB84 protocol.

Alice begins by generating two random bit strings of length $n$, $a$ and $b$. $a$ will form the basis for the shared secret, while $b$ will decide which base to use in order to encode the current bit in $a$. We index the strings from left to right in the binary representation, so that $a_1$ is the leftmost bit in the representation of $a$. Now, Alice sets up her machinery so as to enable her to transmit one of the four qubits below:

For every index $i$ beginning at $1$ and ending at $n$, Alice transmits the first basis vector if $a_i$ is zero and the second basis vector otherwise. Furthermore, she transmits in the computational basis if $b_i$ is zero and the Hadamard basis if $b_i$ is one. Put together, the numbers formed by the bits $b_ia_i$ are precisely the index of the qubit to use to encode the secret. An example transmission, with a random short $a$ and $b$, along with the corresponding series of qubits is shown below:

Bob, on the other end of the transmission, generates his own random bit string $c$ of length $n$. As he receives Alice’s $i’th$ transmission, he measures it in the computational basis if $c_i$ is zero and the Hadamard basis otherwise. In this manner, he destroys the information in the qubit roughly half the time while accurately recovering it the other half of the time. The amount of information Bob recovers is further limited by transmission errors or inefficiencies in his detector.

The next phase of the protocol occurs on a public communications channel that protects against active eavesdropping. As the name implies, the channel is completely open to passive eavesdropping, implying that the adversary Eve knows everything that is sent over the public channel. However, we assume that it is impossible for an adversary to insert arbitrary messages or modify existing messages on the channel. To see that this assumption is reasonable, consider the case where the public channel is a medium such as a newspaper, and the information conveyed is simply present in the form of an advertisement in the classifieds section addressed to the receiver. For every qubit that is not lost in transmission, Bob reports, over the public channel, the base used to measure the value of that qubit. Alice replies with a confirmation every time Bob reports the correct base, and a refutation when he does not. Both Alice and Bob now know which qubits were transmitted accurately and thus end up with a shared secret consisting of exactly those bits. Since both Alice and Bob choose the qubit basis independently and at random, there is a fifty percent chance that they choose the same basis for qubit $i$, and so, ignoring transmission errors, about half the qubits transmitted are correctly received by Bob, yielding a secret of length about $n/2$. This process is known as sifting, and the generated key is commonly called a sifted key, owing to obvious parallels between the physical process of sifting physical mixtures to extract useful components.

Bennett and Brassard suggest sacrificing a third of the derived key bits in order to guarantee security against eavesdropping in the following manner: Alice and Bob publicly compare a third of the derived key bits, keeping the remaining bits as valid key material only if they all agree. To see why this works, consider the scenario with an eavesdropper intercepting all of the $n$ qubits sent. Suppose Alice and Bob agree on the basis used for $m$ out of the $n$ qubits sent. For each of the $m$ qubits, either Eve chose the wrong basis in which to intercept that qubit, or she chose the correct basis. If she chose the correct basis, both Alice and Bob’s value for the qubit’s state will agree. If she chose the wrong basis, her first measurement destroyed the state of the sent qubit, which means that the qubit she sent Bob has a random value in the wrong basis. Since Bob used the same basis as Alice to measure it, he used a basis conjugate to the basis Eve used, which means that the value he measures is also random, and therefore matches Alice’s value with probability $1/2$. Since Eve has $1/2$ probability of choosing the right basis, the probability of both Alice and Bob agreeing on the value of an arbitrary qubit from the $m$ they chose the same basis for, is:

Here, the first term accounts for Eve choosing the correct basis while the second term accounts for the times Bob randomly gets the correct value even after Eve chooses the wrong basis. However, Eve only has information about the key if she guesses the basis correctly and if Bob measures the basis correctly (she doesn’t know the value of the qubit if she guesses incorrectly, and the qubit isn’t used in the key if Bob guesses incorrectly), which happens $1/4$ of the time. Therefore, intercepting $n$ qubits gives Eve at most $n/4$ bits of information about the key, while disrupting $1/4$ of the bits Alice and Bob agree on, which is to say that Alice and Bob have different valued key bits for $1/4$ of the bits that Bob used the same basis as Alice did, as a result of eavesdropping. Therefore, for each derived key bit, the probability that tampering goes undetected is $3/4$, which means that comparing a third of the key bits gives you a probability of $P_f$ that Eve succeeds in going undetected, where

Alice and Bob can choose $n$ and the fraction of $m$ to sacrifice to be as large as required to satisfy an upper bound for the probability that Eve gets away with eavesdropping undetected.

In the general case, if Eve intercepts a fraction $\lambda$ of transmitted qubits, the probability of one of the $m$ key qubits being incorrect is $\lambda / 4$, where the factor $\lambda$ is the probability of the chosen qubit to be intercepted, and $1/4$ is the probability that Alice’s and Bob’s values for the qubit don’t match, given that it is intercepted by $Eve$. Therefore, if a fraction $\eta$ of the key bits are sacrificed to check for eavesdropping, the probability of a false negative (where eavesdropping goes undetected) is simply $P_f’$, where

Given that all goes well, at the end of the transmission, Alice and Bob share a secret key of length $(1-\eta)m$ which can be used to symmetrically encrypt further communications, e.g. using a one time pad. This assumes that there are is no noise in the transmission. The case of noisy transmissions can be analyzed using Quantum Error Correction Codes, and yields a similar security guarantee. This high level overview of the protocol shows how Alice and Bob can come to share some classical information with a high degree of correlation. Accounting for transmission errors and the fact that Eve obtains some information about the shared secret, we see that this transmission is not perfect. The work of Bennett, Brassard and Robert ([3]) outlines a process they term Privacy Amplification, whereby classical discussion on the public channel can be used to reduce the amount of information Eve has about the shared secret, as well as to correct transmission errors. It is this process that completes the key distribution, leaving Alice and Bob with a perfectly correlated key with a high probability and reducing the information leaked to Eve by an arbitrary amount.

SARG04 [4]

The preceding discussion of the BB84 protocol has focused on the single photon picture. Single photon production and detection for quantum key distribution remain technologically difficult, although it is the focus of several ongoing research efforts with varying degrees of success, with InGaAs based avalanche photodiodes being at the forefront [5]. Most practical implementations of BB84 use weak laser pulses in which Alice has encoded the bit to be sent, which introduces the possibility of circumventing the guarantee provided by the no-cloning theorem; Eve simply diverts some of the photons while allowing the rest to proceed to Bob. Such an attack is known as a photon-number splitting attack (PNS). If we consider an Eve to be constrained only by the laws of physics, it is possible for her to store the diverted photons in a quantum memory construct and wait until the sifting phase, when Alice and Bob reveal the used bases, to measure the stored photons in the correct base, obtaining full information about them and precluding any process by which Bob and Alice can distill completely secret keys. Moreover, Eve does not introduce any detectable errors in this manner, resulting in insidiously successful leakage of information.

The extreme weakness of the protocol against this attack is due to the orthogonality of paired states into which the bits are encoded; if Eve knows the basis used, all she has to do is perform the appropriate measurement to distinguish between the states, and therefore obtain the transmitted bit. Intuitively, this can be circumvented by using non-orthogonal states. Although the states are not orthogonal, one can construct a measurement that distinguishes between the two states, with the caveat that an inconclusive measurement is sometimes obtained. In effect, a greater portion of the transmitted qubits must be satisfied during sifting. For concreteness, a simple such protocol (from the original paper by Scarani et al.) is outlined here.

Alice randomly sends one of the four states $\ket{\pm x}$ or $\ket{\pm z}$ to Bob, who randomly measures either in the $x$ basis using $\sigma_x$ or the $z$ basis using $\sigma_z$. In this protocol, successful transmission of $\ket{\pm x}$ corresponds to a 0, while successful transmission of a $\ket{\pm z}$ corresponds to a 1. During the sifting procedure, instead of revealing the bases, Alice announces one of the four pairs of non-orthogonal states out of every possible combination of sent photons, ${\ket{\pm x}} \times {\ket{\pm z}}$, where the product is the cartesian product over the possible states. This has the effect of limiting the possible valid measurements Bob can make in the following manner: Suppose Alice announces $\ket{+x}, \ket{+z}$. If Bob measured $\sigma_x$ and obtains the result +1, he must discard the qubit, as such a result would have been possible by measurement using $\sigma_z$ as well. Similarly, if he measures +1 using $\sigma_z$, he must discard the qubit, as it would have been possible to obtain the same measurement using $\sigma_x$, and therefore he cannot distinguish between the two possibilities. However, if he measures $\sigma_z$ and obtains a result -1, he knows that the qubit sent must have been in the state $\sigma_x$, as it would have been impossible to obtain -1 using the $\sigma_z$ measurement, given that Alice announces that she sent either $\ket{+x}$ or $\ket{+z}$. Therefore, such a measurement allows Bob to keep the qubit, and add a classical bit 1 to his sifted key. By symmetry, this leaves Bob with a fourth of the raw key material after sifting, compared to the $1/2$ fraction obtained in classical BB84.

[6] A comparison of secret key rate for BB84 (red) vs. SARG04 (blue)

However, this results in a protocol that protects from PNS attacks by significantly reducing the amount of information that Eve can obtain about diverted photons. Scarani’s original paper [4] provides a proof of this fact, calculating the information that Eve obtains in a storage attack to be about $0.4$ bits per pulse at an attenuation rate that allows her to keep one photon out of every pulse, compared to complete leakage of information in the case of a storage attack on BB84.

Quantum Bit Error Rate [7]

A variety of environmental factors make it so that not all of the photons transmitted by Alice are detected correctly by Bob. In practice, some measure of reliability is required in order to characterize the degree of losses induced by the environment. The quantum bit error rate (QBER) provides this measure. It is defined, quite simply, as the ratio of wrong bits to the total number of bits received, and is normally on the order of a few percent. Note that this definition strictly only applies to the BB84 protocol, and must be modified slightly for other protocols.

Clearly, the sifted rate is half the raw rate at which Bob detects incoming photons, corresponding to the approximately one-half likelihood that Alice and Bob pick compatible bases. The raw rate has four major contributions; the pulse rate $f$, the mean number of photons per pulse, $\mu$, the probability $t_{link}$ of the photons arriving at the analyzer, and the probability $\eta$ of the photon being detected, giving us that:

Certain phase-coding setups introduce an additional factor $q$, typically $1$ or $\frac{1}{2}$, which accounts for non-interfering path combinations, which results in the following modification:

Analyzing the error rate yields three contributions; photons that end up in the wrong detector due to imperfections in the interferometer, detector dark counts resulting from environmental light that is not adequately filtered out, and uncorrelated photons due to imperfect photon sources.

The first, $R_{opt}$, is simply the sifted-key rate multiplied by the probability that the photon goes to the wrong detector, $p_{opt}$. This is clear, as the photon must have been transmitted successfully in order for it to reach a detector, which is essentially the same as the condition required for it to be part of the sifted key.

The second, $R_{det}$, is independent of the bit rate, and only contributes to the error rate if the dark count occurs during the time when a photon is expected. Furthermore, the dark count occurs half the time when Alice and Bob choose incompatible bases (in which case it does not contribute to the error, as the corresponding detection is eliminated during sifting) and an additional 50% chance of occurring in the correct detector. If $p_{dark}$ is the probability of registering a dark count per time window, we have the following expression:

The third factor arises due to imperfect photon sources, where two photons in different pairs arrive in the same time window, but not necessarily in the same state. This is only relevant in systems using entangled photons. If $p_{acc}$ is the probability of finding a second photon within the time window belonging to a different pair than the first photon detected, where the factors of $\frac{1}{2}$ are due to the same reasons outlined before.

Therefore, the QBER is simply the ratio from earlier, with the respective rates expanded into their separate contributions:

The first contribution is entirely distance independent, and measures the quality of the optical set up used. Different QC setups have differing contributions and corresponding countermeasures to improve $QBER_{opt}$, and this measure provides a single tool to evaluate the quality of the optical set up alone.

The second contribution is distance dependent, increasing with distance as the probability of a photon arriving at the detector $t_{link}$ goes down with increasing distance while the dark-count rate does not fluctuate significantly. Therefore, issues with range are solely a function of detector noise; better detectors allow for QC over much higher distances.

It is possible to calculate the maximum range of a QC system given these parameters, where greater distances result in intolerably high values of the QBER. Gisin[7] finds the maximum range to be about 100km, practical maxima being closer to 50km owing to a variety of factors.

Bibliography

1. C. H. Bennett and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing,” Proc. IEEE Int. Conf. Comput. Syst. Signal Process. Bangalore. p. 175, 1984.
2. S. Wiesner, “Conjugate Coding,” ACM SIGACT News, vol. 15, no. 1, pp. 78–88, 1983.
3. C. H. Bennett, G. Brassard, and J.-M. Robert, “Privacy Amplification by Public Discussion,” SIAM J. Comput., vol. 17, no. 2, pp. 210–229, 1988.
4. V. Scarani, A. Acin, G. Ribordy, and N. Gisin, “Quantum cryptography protocols robust against photon number splitting attacks for weak laser pulses implementations,” vol. 1, no. 2, pp. 0–4, 2002.
5. L. C. Comandar et al., “Room temperature single-photon detectors for high bit rate quantum key distribution,” Appl. Phys. Lett., vol. 104, no. 2, 2014.
6. Y.-C. Jeong, Y.-S. Kim, and Y.-H. Kim, “An experimental comparison of BB84 and SARG04 quantum key distribution protocols,” Laser Phys. Lett., vol. 11, no. 9, p. 095201, 2014.
7. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys., vol. 74, no. 1, pp. 145–195, 2002.