Quantum And Relativistic Protocols For Secure Multi-Party Computation

TL;DR

The thesis investigates secure two-party cryptography under physics-based models, showing that quantum and relativistic effects can unlock secure primitives beyond classical limits. It introduces a non-relativistic quantum protocol for strong coin tossing achieving optimal known bias, and develops variable bias coin tossing with unconditional and cheat-evident security, plus relativistic protocols that extend security guarantees. It also demonstrates broad impossibility results for secure two-party classical computations within a quantum-relativistic framework, and proposes untrusted-device schemes for private randomness expansion using GHZ-type tests. The work highlights the interplay between information-theoretic security, nonlocal correlations, and spacetime constraints, offering foundational insights and practical avenues for cryptography under minimal assumptions.

Abstract

After a general introduction, the thesis is divided into four parts. In the first, we discuss the task of coin tossing, principally in order to highlight the effect different physical theories have on security in a straightforward manner, but, also, to introduce a new protocol for non-relativistic strong coin tossing. This protocol matches the security of the best protocol known to date while using a conceptually different approach to achieve the task. In the second part variable bias coin tossing is introduced. This is a variant of coin tossing in which one party secretly chooses one of two biased coins to toss. It is shown that this can be achieved with unconditional security for a specified range of biases, and with cheat-evident security for any bias. We also discuss two further protocols which are conjectured to be unconditionally secure for any bias. The third section looks at other two-party secure computations for which, prior to our work, protocols and no-go theorems were unknown. We introduce a general model for such computations, and show that, within this model, a wide range of functions are impossible to compute securely. We give explicit cheating attacks for such functions. In the final chapter we discuss the task of expanding a private random string, while dropping the usual assumption that the protocol's user trusts her devices. Instead we assume that all quantum devices are supplied by an arbitrarily malicious adversary. We give two protocols that we conjecture securely perform this task. The first allows a private random string to be expanded by a finite amount, while the second generates an arbitrarily large expansion of such a string.

Figures (9)

• Figure 1: Sequence of operations for the implementation of a measurement in the $z$ basis on the first part of a state followed by a two-qubit unitary dependent on the outcome in the case (a) where the measurement is performed explicitly, and (b) where the measurement is kept at the quantum level until the end. In the latter case an ancillary system indexed by $A$ has been introduced, and the unitary operation is now controlled on this system. Note that the end result is the same in both cases.
• Figure 2: Schematic showing information reconciliation. The release of $S=f(X,R)$ reduces Bob's uncertainty on Alice's string, $X$, to a negligible amount.
• Figure 3: Schematic showing privacy amplification of string $X$ to form $S$ using a $universal_2$ hash function.
• Figure 4: Schematic of a non-relativistic protocol between two parties. A represents Alice's systems, B represents Bob's systems, and C is the channel. Alice and Bob alternately perform unitaries as the protocol proceeds.
• Figure 5: Schematic of the setup for a relativistic protocol with two separated sites.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.3
• Theorem 1.4
• Definition 1.4
• Definition 1.5
• Definition 1.6