Position-based Quantum Cryptography and Catalytic Computation
Florian Speelman
Abstract:
In this thesis, we present several results along two different lines of research. The first part concerns the study of position-based quantum cryptography, a topic in quantum cryptography. In the second part we introduce a new notion of computation, catalytic computation, and study this new model within complexity theory.
Part I: Position-based quantum cryptography
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By combining quantum mechanics with special relativity theory, new cryptographic tasks can be developed that use the causality constraints of relativity theory in a constructive way. Position-based cryptography is a type of cryptography that wants to use location as a credential, instead of (or in addition to) a secret key – for instance to create a protocol to send messages that can only be read at one specific location.
After earlier proposals, which used only classical information, were shown to be insecure, new schemes for position-based cryptography that used quantum information at first seemed promising. Recent results showed that all such schemes can be broken by attackers that use an exponential amount of quantum resources. This leaves the following question open: Is it possible to create a scheme which is secure under realistic assumptions?
If an attack on a scheme requires more entanglement than the number of particles in the universe, then it surely is secure. Therefore, limiting the attacker’s entanglement is a natural step. Our results will all consider this distinction: schemes that can be attacked using little entanglement are insecure, while schemes for which a coalition of attackers needs large entangled states will be secure.
Chapter 3.
The first chapter on this topic analyzes a family of position-verification schemes that combines a single qubit with classical information. We introduce a new tool to study these schemes, the garden-hose model. In this simple combinatorial model, two parties, Alice and Bob, share ‘pipes’ between them, and they want to compute a function by linking these pipes together with ‘hoses’. By studying garden-hose complexity, we characterize a class of teleportation attacks on the family of schemes, and show a surprising relationship between their security and open problems in computational complexity theory. We prove several smaller results on the new model and additionally introduce natural variants: the randomized garden-hose model, where the players share a random string, and the quantum garden-hose model, where Alice and Bob have access to a pre-shared entangled quantum state.
Chapter 4.
In the next chapter we continue our study of the use of quantum information in position verification, but now our attention turns to a different class of protocols: those that that can be written using a class of small quantum circuits, those with low T-gate complexity. We combine techniques from blind and delegated quantum computation with the new garden-hose model and construct new efficient attacks on these schemes. As an additional application, we present an efficient attack on the Interleaved Product protocol for position verification, recently introduced by Chakraborty and Leverrier.
Chapter 5.
The final chapter on this topic looks at questions that are directly inspired by practical considerations. Positioning protocols will likely use photons as carriers of quantum information, possibly traveling in optical fiber. This is incompatible with current protocols in two ways: a significant fraction of photons are lost in transmission, and the speed of light in fiber is lower than in vacuum. Adapting protocols to deal with these problems opens them up to new attacks. We propose a new protocol for position verification that prevents these attacks and use semidefinite programming to show security of this protocol against attackers that do not share entanglement.
Part II: Catalytic computation
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In the second part of this thesis, we study the notion of a catalytic-space computation. This is a computation that has a small amount of clean space available and is equipped with additional auxiliary space, with the caveat that the additional space is initially in an arbitrary, possibly incompressible, state and must be returned to this state when the computation is finished. The term ‘catalytic’ comes from chemistry, where it refers to a reactant which speeds up a chemical reaction but is not consumed – just like the extra space available to the computation.
Chapter 6.
In this chapter, we show that the extra space adds a surprising amount of power to the model. To obtain this result, we study an algebraic model of computation, called Transparent Programs, a variant of straight-line programs. Within these Transparent Programs, we can adapt a construction by Ben-Or and Cleve to show that it’s possible to compute TC1 circuits using only a logarithmic amount of clean space. Additionally, we present some complexity-theoretical limits on the power of catalytic computation, by showing that computations that use a logarithmic amount of clean memory, can be simulated probabilistically in polynomial time.
Chapter 7.
We continue the study of catalytic computation by translating two foundational results on space-bounded computation to this new setting. First we extend the model to incorporate non-determinism. The Immerman–Szelepcsényi Theorem is an important classic result in complexity theory that shows that the complexity class of problems solvable by non-deterministic log-space Turing machines is closed under complement. We show that non-deterministic catalytic space is also closed under complement, under standard derandomization assumptions. Finally, we present a hierarchy theorem – we show that adding more space enables the catalytic computation to solve strictly more problems.