Quantum-proof multi-source randomness extractors in the Markov model (1510.06743v2)

Abstract: Randomness extractors, widely used in classical and quantum cryptography and other fields of computer science, e.g., derandomization, are functions which generate almost uniform randomness from weak sources of randomness. In the quantum setting one must take into account the quantum side information held by an adversary which might be used to break the security of the extractor. In the case of seeded extractors the presence of quantum side information has been extensively studied. For multi-source extractors one can easily see that high conditional min-entropy is not sufficient to guarantee security against arbitrary side information, even in the classical case. Hence, the interesting question is under which models of (both quantum and classical) side information multi-source extractors remain secure. In this work we suggest a natural model of side information, which we call the Markov model, and prove that any multi-source extractor remains secure in the presence of quantum side information of this type (albeit with weaker parameters). This improves on previous results in which more restricted models were considered and the security of only some types of extractors was shown.

• The paper presents a new Markov model ensuring conditional independence between sources and quantum side information, thereby maintaining extractor security.
• It adapts classical randomness extraction methods with operator space theory, achieving quantum security with controlled parameter degradation.
• The work reinforces cryptographic protocols and quantum randomness amplification, paving the way for practical applications in quantum security.

Quantum-Proof Multi-Source Randomness Extractors in the Markov Model

This paper by Rotem Arnon-Friedman, Christopher Portmann, and Volkher B. Scholz explores the construction and implications of quantum-proof randomness extractors, particularly focusing on multi-source scenarios within the Markov model. The work addresses the challenge of ensuring security for randomness extractors when side information, potentially quantum in nature, is available to an adversary. The authors propose a novel model, termed the "Markov model," under which these extractors can achieve security despite the presence of such side information.

Security Context

Randomness extractors are essential in numerous applications, from cryptography to derandomization processes in computer science. The main goal is to create nearly uniform random bits from sources that are only weakly random. In classical contexts, seeded and multi-source extractors have been developed, but the introduction of quantum side information requires rethinking these constructs.

In situations involving adversaries with quantum capabilities, simply knowing the conditional min-entropy of a source is insufficient for ensuring the security of an extractor. Specifically, the authors highlight that multi-source extractors might inherently fail when arbitrary side information is considered. Thus, identifying models under which these extractors remain secure is crucial.

The Markov Model

The Markov model proposed by the authors assumes a specific relationship between the sources and side information, embodying conditional independence concepts. A multi-source extractor within this model must ensure that the sources and any potential adversarial side information form a Markov chain. This structural assumption — that the side information does not introduce dependencies between the sources — enables extractors to maintain their security guarantees.

Main Contributions

1. Markov Model for Extractors: The authors present a new definition of a multi-source extractor within the Markov model where the sources and side information are conditionally independent. The model extends the existing classical understanding to quantum settings.
2. Security Proofs: They provide theoretical proofs demonstrating that any multi-source extractor secure in the classical setting can be adapted to quantum settings, albeit with potential parameter degradation. Specifically, they show that for quantum side information structured in the Markov model, multi-source extractors maintain security with a controlled increase in error terms.
3. Technical Methodology: Leveraging recent developments in operator space theory and quantum information, the authors adapt classical randomness extraction techniques to cope with quantum side information. The proof techniques developed therein are novel, showcasing the potential for extending classical independence assumptions to quantum frameworks.
4. Applications and Implications: By establishing security for extractors in scenarios with quantum side information structured as a Markov chain, the research paves the way for more robust cryptographic protocols and reinforces the foundational aspects of quantum cryptography. Moreover, it provides a framework that could be relevant for quantum randomness amplification processes, which are essential in ensuring true randomness generation in quantum devices.

Future Directions

The findings in this paper suggest intriguing possibilities for further research. Analyzing extractors that maintain efficiency and security under more general quantum side information conditions remains a significant open field. Additionally, exploring the practical implementations of these extractors in quantum computing and communications, especially in device-independent scenarios, offers promising pathways for applied advancements. Furthermore, questions about the tightness of the bounds on error rates and the exact overhead required to maintain security in quantum settings are ripe for exploration.

In summary, the paper is a substantial contribution to the field of quantum randomness extraction, efficiently extending classical methodologies into the quantum domain while ensuring robust security in adversarial conditions. The novel Markov model presents an elegant solution to address the security challenges posed by quantum side information, providing a foundation for both theoretical exploration and practical application in quantum technologies.