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\begin{document}
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\chapter{Physical Security in Quantum Key Distribution}
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\newpage

\section{Cryptography in the Age of Quantum Computers}

For a decade or two now, Quantum Computing has been creating a buzz that nobody in Computer Science and adjacent fields
could evade. Originating in the 1980ies as a highly academic fusion applying concepts from Computer Science in Quantum Physics,
% FIXME citation
its concepts have long found their way into popular science articles. Quantum Computing encompasses a model of
computation that is fundamentally different from the \emph{classical}\footnote{
    In Quantum Computing, the term \emph{classical} is used as the complement of \emph{quantum}, and refers to the
    digital computers we know and (sometimes) love. This terminology stems from the distinction between classical and
    quantum physics.}
digital circuits that underly all of modern computing. While at first this might seem like a step backwards into the era
of early 1900s analog computing,
% FIXME citation
the capabilites of a future quantum computer promise to far outpace those of contemporary classical computers. Key to
this improved processing capability is a property called \emph{Quantum Parallelism}. What this refers to is the fact
that a quantum computer's internal state can simultaneously represent a multitude of states of a classical, digital
computer, and the quantum computer can operate on all those states at once using a single quantum operation.

Applying Quantum Parallelism to practical problems is far more complicated than, e.g., translating a digital circuit
solving some equation to a quantum circuit, but for certain problems we already know \emph{quantum algorithms} that
for large inputs solve these problems much faster than any classical computer ever could. Two of these algorithms, one
by Shor % FIXME citation
and one by Grover % FIXME citation
are what caused most of the buzz around the field of quantum computing, because they spell trouble for a large part of
modern cryptography.

Besides the computational speed-up promised by Quantum Parallelism, there is one more interesting aspect of Quantum
Computing where it radically deviates from classical computing. The reason modern cryptography exists is that when we
transmit (or store!) classical information through some channel (or storage!) that we do not control, there is nothing
we can do to prevent an attacker from reading this information. Even with cryptography we cannot prevent this, but
cryptography gives us tools to very effectively make whatever information the attacker is able to read useless to them.

A basic principle of Quantum Physics is the \emph{No-Cloning Theorem}, which states that it is impossible to create an
identical, independent copy of an arbitrary, unknown quantum state. % FIXME citation
An implication of this theorem is that when we encode classical information into quantum states in just the right way,
we can make it so that an attacker atttempting to eavesdrop on our quantum information can only actually read this
information by destroying it in the process. This property can be exploited to replace a number of classical asymmetric
primitives in interactive settings, % FIXME citation, check if interactive only
the most popular application of which is replacing an asymmetric Diffie-Hellman key exchange % FIXME citation
with a quantum process called Quantum Key Distribution that yields much of the same properties.

In the past decades, the field of cryptography has been fundamentally shaped by the development of Quantum Computing and
Quantum Key Distribution. However, the popular conception that all of today's cryptography will be broken and that we
have to start from scratch is not accurate. Quantum Computing poses an unique threat to modern cryptography, and Quantum
Key Distribution is a promising new tool, but the practical implications of both are much more subtle than how they may
be portrayed. In the remainder of this chapter, we will look into the practical implications of these quantum
technologies, and we will come to two major conclusions: First, that while the underlying cryptographic primitives will
change, apart from some minor engineering issues cryptography as a whole will remain largely the same. Second, that
while Quantum Key Distribution is hailed as a revolution for network security, its practical advantages will remain far
short of how it is usually conceptualized, and hardware security will assume a pivotal role in the practical security of
Quantum Key Distribution systems that is a stark departure from its relative irrelevance in today's applied
cryptography.

Building on these conclusions, we will end this chapter with a study of a use case that illustrates a practical design
for a secure network employing Quantum Key Distribution. Relying on both established classical and quantum primitives
with known security properties we will elaborate how one can construct a large-scale network from those primitives
that provides practical security to its users that goes beyond the (surprisingly limited) extents of quantum security
proofs.

\subsection{Computational Assumptions and Information\Hyphdash Theoretic Security}
\label{qc_comp_assum}

In the past paragraphs we have briefly mentioned that Quantum Computing provides a significant speed-up that can be
applied to solve many cryptographic problems fast enough for it to become a problem, but we have not elaborated on what
that means in practice. In this section, we will attempt to provide concrete numbers to quantify the threat that both
Shor's and Grover's algorithm pose to modern cryptography.

Shor's algorithm allows for the factorization of large numbers in polynomial time on a quantum
computer, a problem whose hardness (or the hardness of variants of which) is the foundation for the vast majority of
today's asymmetric cryptography.

While Shor's algorithm attacks the foundations of most modern asymmetric cryptography, Grover's algorithm can be applied
to hash functionss and symmetric cryptography. Fundamentally, Grover's algorithm is a search algorithm that allows a
quantum computer to find one target entry out of an \emph{unstructured} list of $N$ source entries in
$\mathcal{O}\left(\sqrt{N}\right)$ time instead of the $\mathcal{O}\left(N\right)$ time that a classical computer would
require for an exhaustive search. Applied to cryptography, we model the key space of a symmetric cipher as the
unstructured list that is input to the algorithm, and set it to search for the key that results in the successful
decryption of a given ciphtertext.

An important nuance applying these algorithms to cryptography is that while both provide significant speed-ups over
classical computers, the speed-up of Shor's algorithm is exponential and effectively breaks most modern asymmetric
cryptography as it erases the asymmetric nature of the underlying mathematical problem's computational complexity. That
is, for an asymmetric cryptosystem susceptible to Shor's algorithm, there is no set of parameters that is large enough
to be safe.

In contrast to this, while Grover's algorithm radically speeds up the breaking of a symmetric cryptosystem, this
speed-up is only quadratic. In practice this means that it halves the security level % FIXME definition, citation of sec. lvl
of a given symmetric cipher. While this is bad news for applications that parameterize these symmetric primitives to a
security level at the lower end of what is considered secure today, the advantage provided by Grover's algorithm can
easily be compensated by doubling key size. Longer key sizes require more storage or bandwidth for the additional bits
and result in slightly slower operation of the cipher, but this additional cost is easily manageable even without any
improvement in today's hardware.

\section{The Practical Security Implications of Quantum Computing}
\label{qc-practical-implications}

Given that as of yet, noone has claimed to have a quantum computer powerful enough to pose a threat to current
cryptographic protocols, one may ask the fair question why the possible future development of such a machine would be
consequential for today's cryptographic practice. The answer to this question lies in \emph{Store-Now-Decrypt-Later}
attacks. In such attacks, the attacker records all data transmitted between a cryptographic protocol's parties. The
security of any key exchange protocol rests on a computational hardness assumption about some particular problem. When
this assumption falls, for example because of a powerful quantum computer becoming available, the attacker can then
retroactively break the security of those stored protocol instances and decrypt all traffic.

Modern cryptographic protocols such as TLS or the Signal messenger's key ratchet are designed with facilities to provide
some degree of protection against key compromise called \emph{(Perfect) Forward Secrecy}. Forward Secrecy means that a
compromise of keys at one protocol step will not break the secrecy of past protocol steps. Forward Secrecy is achieved
by repeatedly mixing fresh key material called \emph{Ephemeral Keys} into the protocol's secret state. For a
post-quantum attacker, this implies that to decrypt a run of a forward-secret cryptographic protocol, the quantum
algorithm breaking the protocol's computational assumption must be run a number of times, but this results only in a
linear increase of both protocol and attack complexity, which turns out to no advantage for the defender.

Store-Now-Decrypt-Later attacks are considered a serious threat today based on the stark discrepancy between the
capacity of today's inexpensive storage media, and the comparatively tiny bandwidth of cryptographic protocols in
applications such as \emph{End-To-End-Encrypted} text messaging. A single hard drive can conceivably store years of a
person's encrypted digital communications.

There has been ongoing work on quantum secure cryptographic algorithms, and standardization of several such algorithms
is progressing. However, in the time frame of cryptosystems, these algorithms are still rather young and the recent
discovery of a catastrophic key recovery attack against the Supersingular Isogeny Diffie-Hellman protocol
(SIDH)\cite{hazay_efficient_2023} illustrates the risk in the use of immature cryptographic primitives. Thus,
recommendations on the concrete steps that should be taken today to mitigate Store-Now-Decrypt-Later attacks vary. For
instance, Google's under its threat model as laid out in \textcite{schmieg_blog_2024} recommends a list of quantum
secure counterparts to classically secure cryptographic algorithms, but recognizes the relative immaturity of these
quantum secure algorithms and consequently recommends \emph{Hybrid Deployment}, where a young, quantum secure algorithm
is paired with a mature classically secure algorithm such that \emph{both} algorithms would have to be broken to
compromise the composite protocol's security. Given that quantum secure public key cryptography tends to have both a
much larger key and/or ciphertext size and worse performance compared to state-of-the-art Elliptic Curve-based key
exchange or signature algorithms, pairing it with a classically secure alternative incurs only a negligible overhead in
key storage, network communication and computation costs.

% FIXME TODO research some more policies.

\section{The Physics of Quantum Computing}

\section{Quantum Key Distribution}

As we discussed in Section \ref{qc_comp_assum}, quantum computers promise novel attacks on many contemporary
cryptographic systems. At the same time, quantum technology also promises new cryptographic primitives that support
security guarantees beyond what can be realized with the best classical computers. The core of this nascent field of
Quantum Cryptography is a set of methods that are collectively called Quantum Key Distribution.

Informally speaking, a Quantum Key Distribution system is a system that distributes a secret key between two\footnote{
    Although the key distribution problem can conceptually be framed for any number $n\ge 2$ of parties, practical
    treatment is almost always limited to the two-party case. In case of QKD, problem instances for $n > 2$ parties can
    trivially be reduced to $(n^2 - n)/2$ invocations of the two-party protocol, combined with any
    information-theoretically secure secret sharing scheme.
} parties such that after a successful execution of the protocol, each of the two parties holds a copy of a randomly
generated secret key, and the probability that an attacker was able to extract some portion of the key during the
protocol's execution can be bounded to some negligible $\epsilon$ by each of the parties.

Quantum Key Distribution provides a similar service as cryptographic key exchange protocols such as the classic
Diffie-Hellman key exchange provide. The core difference between QKD and cryptographic key exchange protocols is that
QKD provides information-theoretic security based on the No-Cloning Theorem, where cryptographic protocols provide only
computational security based on the computational hardness assumption underlying some public-key cryptosystem.

QKD is attractive in that it gives practically useful security guarantees without relying on any computational hardness
assumptions. This way, QKD would remain secure even in a scenario where a hybrid deployment of a classically secure but
mature algorithm paired with a quantum secure but young algorithm as discussed in Section
\label{qc-practical-implications} poses too much of a risk---a scenario where both large quantum computers arrive and a
flaw in the quantum secure algorithm is found. Note that here, because we assume we have large quantum computers, the
possibility of a flaw in the quantum secure algorithm extends beyond mathematical flaws leading to practical attacks
with classical computers, and includes novel quantum algorithms.

\subsection{The Technical Implementation of QKD}

On the technical level, QKD must be distinguished from general Quantum Computing. While QKD systems employ the
No-Cloning Theorem and sometimes quantum entanglement in their operation, the scope of their quantum operations is very
limited. QKD systems always operate on photons, while general quantum computers use a variety of physical
implementations for their qubits that include photons and squeezed light, but extend over atom nuclei, trapped ions,
various aspects of currents in superconducters into phonons\cite{berrios_high_2012}.

\subsubsection{Practical Challenges}
% FIXME I don't like this paragraph.
The central challenge in general quantum computers is extending the lifetime of the quantum state encoding a qubit.
Quantum states are extremely sensitive to disturbances, and despite the best efforts to shield their quantum states
against external influence, their lifetime is still inconveniently short compared to the timescales required for quantum
computation, resulting in significant amounts of noise in the output of quantum algorithms run on contemporary quantum
computers. Quantum Key Distribution systems use photons and only perform a handful of operations on each photonic state
between generation and measurement, with the vast majority of the state's lifetime spent in transit between the two
endpoints of the QKD protocol.

While QKD systems are easy to build and operationally robust compared to general quantum computers, at their core they
still exchange information through quantum states that physically need to transit the distance from one endpoint to the
other. For classical computer networks, bridging distances of several hundred kilometers is no big challenge. Using
appropriate high-power transceivers, a single optical link can already bridge upwards of 100km. % FIXME cite
Longer ranges can easily be achieved by either logically chaining multiple links, or by using optical amplifiers.

In contrast, the quantum states at the core of QKD systems must necessarily be ``weak''. A single quantum state on the
wire on average must consist of approximately a single photon. If the system's quantum states consisted of more than one
photon carrying the same information, this would enable a \emph{Photon Number Splitting Attack}, in which an attacker
extracts one of the state's photons for later analysis, and forwards the remaining photons to the receiver. The attacker
can then later measure the captured photon to extract the same information that the receiver measured.

The practical implication of this is that the optical brightness of a QKD system is directly proportional to the rate
at which the system can prepare, and later measure the individual quantum states. With today's electronics, rates up to
a few GHz are feasible. Alas, this brightness limit interacts poorly with the reality of optical communication,
especially through fibers. Even modern, high-quality fiber-optic cables have attenuation in the order of 0.5 dB/km,
which corresponds to roughly half of the signal being lost every 5 km. In classical optical networks, this can be
compensated by increasing transmit power--i.e. packing more photons into each bit--or by optically amplifying the signal
partway through the fiber. In QKD systems however, the signal cannot be amplified, and the system's bit rate
exponentially decreases with distance due to absorption. Some QKD systems can reach ranges of several hundred kilometer,
but the useable data rate (here called \emph{key rate}) of these systems usually is in the kilobits per second or worse.

QKD signals cannot be amplified because their security rests on the fact that each transmitted quantum state on average
only contains on the order of one photon each. Security rests on the No-Cloning Theorem, which implies that not just
attackers, but even the system's operators are unable to duplicate the quantum state in flight without destroying it.

When transmitted over a fiber, there are multiple effects that degrade the quantum-optical signal of a QKD system. We
can coarsely classify these degrading effects into two categories: \emph{Decoherence}, and \emph{Absorption}.
Decoherence effects result in the quantum state being changed in transit, which depending on the QKD implementation may
mean destroying information contained within the state such as by disturbing the pulse's polarization, or destruction of
entanglement between the in-flight state and another local state. In an optical channel affected by such decoherence
effects, a quantum state enters the channel, and subsequently exits it at the other end changed. In contrast, absorption
means the quantum state is not ever leaving the channel.

In practice, absorption limits the length of an individual fiber run, as it becomes problematic at short distances.
Decoherence is less relevant for the distance limitation, and mostly limits which fiber-optic technologies can be
utilized in the first place. Due to decoherence, QKD systems usually use Single-Mode (SM) fiber over Multi-Mode (MM)
fiber, and makes it more difficult to utilize Wavelength Division Multiplexing (xWDM) to send multiple either quantum or
classical optical signals through a single fiber.
% FIXME go more into the details on xWDM, elaborate on decoherence mechanisms, especially crosstalk in the context of
% xWDM.

% FIXME CV-QKD

\subsubsection{Relaying}

The No-Cloning Theorem prevents us from using conventional optical amplifiers to extend the range of a single continuous
QKD link. What remains as ways to extend the range of a QKD link are \emph{relaying} methods, where one QKD link is
terminated at the relay, and another is started, with the relay proxying information between the two. We can separate
relay implementations into two broad categories.

% FIXME mention that one MDI-QKD range doubling hack
\begin{description}
    \item[Classical relays] encompass the trivial implementation of a relay, where the QKD link is formed by simply
        stitching two QKD links together by connecting one link's receiver to the other link's transmitter. The key
        characteristic of classical relays is that inside the relay, the link's cryptographic payload information is
        handled in its classical plaintext form. Classical relays are practically feasible, but because they must handle
        the payload in plaintext form, they are security-critical.

    \item[Quantum relays] are relays that forward the QKD payload information from one link to the other in the quantum
        realm, without translating it to classical information and back. QKD relays are currently not practically
        feasible, but if they become available in the future, they would allow range extension without compromising the
        QKD link's security as the same tamper-detecting properties that the QKD links provide can be extended to cover
        the quantum forwarding process inside the relay.
\end{description}

\section{Quantum Networking}

So far we have focused on the range limitation of a single QKD link with classical relays as the only practical solution
at this point in time. Quantum Networks naturally follow from a relay-assisted QKD link, if we consider a type of
``relay'' that is connected to more than two links. Just like switches and routers can be meshed to construct complex
topologies in classical wide-area networks (WANs), such multi-fanout relays, or \emph{routers} can be used to provide
QKD services over complex network topologies.

\section{Securing QKD Networks with Inertial HSMs}

As we discussed above, when it comes down to practical, end-to-end security properties, Quantum Key Distribution
removes trust in the hardness of particular mathematical problems (good!), but increases trust in the physical
integrity of the transceivers of the QKD link (bad!). In scenarios where the communicating parties are all located
within physical proximity, in QKD meaning within at most a few hundred kilometers from each other depending on secret
key rate requirements, this added trust is of no consequence because the communcating parties' hardware must be trusted
in either QKD-assisted or purely classical setups. However, this trust requirement becomes a burden as soon as at least
one party is too far away (or higher secret key rates are required), as now physically trusted relays become necessary.

Extrapolating to practical deployments, we can make two predictions. First, as QKD only solves key distribution, but the
actual data transfer still happens through normal off-the-shelf telecommunications components in QKD networks, there is
no reason for a practical QKD setup to \emph{not} also use classical cryptography as an additional layer for defense in
depth,
% FIXME citation on defense in depth, and on this hybrid scenario
meaning the QKD setup will at worst degrade to the same security a purely classical system would provide, never less.

The second prediction we can make is that any practical QKD network will have to use trusted relays to bridge large
distances. While in certain specialized applications such as the proposed financial QKD network in Switzerland
% FIXME citation
smaller, isolated networks are conceivable, in every telecommunication system from the telegraph through the telephone
system and up to the internet it has been shown conclusively that there is a real demand for a unified, global
interconnected network. % FIXME citation on historic networks

In this section, we will outline a solution that provides practical, end-to-end security in large-scale QKD networks by
delegating the hardware trust issue of QKD relays to Inertial Hardware Security Modules. The primary design challenges
we will address are the systems' overall envelope design, optical passthroughs, and matching the cryptographic
assumptions behind the IHSM's heartbeat and alarm subsystem to those of the QKD application.

\section{Outlook}

\newpage
\printbibliography[heading=bibintoc]

\appendix

\end{document}
