Geometrical Optics Restricted Eavesdropping Analysis of Secret Key Distillation and its Application to Satellite-to-Satellite Free Space Links

Abstract

The quantum cryptography theoretically aims to promise unconditional information security in the physical layer against an omnipotent eavesdropper. However, such a conventional assumption of an all-powerful eavesdropper can be too strict for certain practical implementation scenarios and can be relaxed considerably. This dissertation reviews the relevant theories on secret key rate bounds, proposes a geometrical optics restricted eavesdropping model and studies secret-key distillation across a lossy and noisy quantum wiretap channel between Alice and Bob, with a separately parameterized realistically lossy quantum channel to the eavesdropper Eve. We show that under such restricted eavesdropping, the key rates achievable can exceed the secret-key-distillation capacity against an unrestricted eavesdropper in the quantum wiretap channel. Furthermore, we show upper bounds on the key rates based on the relative entropy of entanglement. Then we apply this model to the realistic secret key distillation over a satellite-to-satellite free space optics channel starting with a straightforward case where we assume a limited-sized aperture eavesdropper (Eve) in the same plane of the legitimate receiver (Bob) and determine the secret key rate (SKR) lower bounds correspondingly. We first study the input power dependency without assumptions on Bob’s detection scheme before optimizing the input power to determine lower bounds as functions of transmission distances, center frequency or Eve aperture radius. Then we calculate analytical expressions regarding the SKR lower bound and upper bound as transmission distance goes to infinity. Then we study one of Bob’s possible corresponding defense strategies in this realistic application model of secret key distillation over satellite-to-satellite free space channel in which we impose a reasonable restriction on the eavesdropper by setting an exclusion zone around the legitimate receiver. We first study the case where the eavesdropper’s aperture size is unlimited, so her power is only restricted by the exclusion zone. After that we limit Eve’s aperture to a finite size and study the case when her aperture is in the same plane of Bob, investigating how an exclusion zone can help improve security. Correspondingly, we determine the secret key rate lower bounds as well as upper bounds. We show that by putting reasonable restrictions on the eavesdropper through the realistic assumptions of an inaccessible exclusion zone, we can increase the key rate in comparison to those without and do so with relatively lower transmission frequency. In the end, we study the secret key distillation over a satellite-to-satellite free space optics channel in which we assume that the eavesdropper’s limited sized aperture can be dynamically positioned to gain advantages over the communication parties, and we determine the achievable key rate lower and upper bounds with respect to different scenarios. We first study the case where Eve is behind Bob, and we prove that the optimal eavesdropping strategy for her in long-distance transmission case is to place her aperture on the beam transmission axis and set Bob-to-Eve distance equal to Alice-to-Bob distance. We also show that the achievable key rate would be characterized by a Bessel function integral related to Eve’s position in a short-distance transmission case. We then investigate the case where Eve is before Bob and show similar results with Eve’s and Bob’s roles exchanged. For our analyses we also incorporate specific discrete variable (DV) and continuous variable (CV) protocols for comparison.