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Explicit Constructions of Two-Source and Affine Non-Malleable Extractors for Small Entropy


Core Concepts
This paper presents explicit constructions of two-source and affine non-malleable extractors that match the state-of-the-art constructions of standard extractors for small entropy.
Abstract
The paper focuses on constructing explicit two-source and affine non-malleable extractors for small entropy. The main results include: Two-source and affine non-malleable extractors (over F2) for sources on n bits with min-entropy k ≥ log^C n and polynomially small error, matching the parameters of standard extractors. Two-source and affine non-malleable extractors (over F2) for sources on n bits with min-entropy k = O(log n) and constant error, also matching the parameters of standard extractors. The authors significantly improve the entropy requirement compared to previous non-malleable extractor constructions. The key technical ideas include: Taking the parity of two resilient functions to obtain a non-malleable extractor. Generating non-oblivious bit-fixing (NOBF) sources from the inputs and their tampered counterparts. Handling the general case using a convex combination of subsources analysis. The improved affine non-malleable extractors also give strong lower bounds for a certain kind of read-once linear branching programs, matching the previously best-known average-case hardness results.
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Key Insights Distilled From

by Xin Li,Yan Z... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.17013.pdf
Two-Source and Affine Non-Malleable Extractors for Small Entropy

Deeper Inquiries

How can the error of the non-malleable extractors be further improved without first improving the error of standard two-source and affine extractors for small entropy

To improve the error of non-malleable extractors without enhancing the error of standard two-source and affine extractors for small entropy, one approach could involve refining the analysis of the correlation breakers used in the construction of non-malleable extractors. By optimizing the parameters and techniques within the correlation breakers, it may be possible to reduce the error without directly impacting the error of the standard extractors. Additionally, exploring novel methods for generating advice strings or enhancing the efficiency of the advice generators could contribute to lowering the error of non-malleable extractors independently of standard extractors.

Are there other applications of the techniques used in this paper beyond non-malleable extractors and read-once linear branching programs

The techniques and concepts utilized in the construction of non-malleable extractors and their applications to read-once linear branching programs have broader implications beyond these specific areas. Some potential applications could include: Cryptography: The methods for generating non-malleable extractors could be adapted for enhancing the security and resilience of cryptographic protocols, especially in scenarios where adversaries attempt to tamper with inputs or outputs. Data Privacy: The principles of non-malleable extractors could be employed in data privacy mechanisms to ensure that sensitive information remains secure even in the presence of malicious actors attempting to manipulate the data. Error Correction: The techniques for constructing non-malleable extractors could be valuable in error correction codes and fault-tolerant systems, where maintaining data integrity is crucial. Distributed Computing: The concepts of non-malleable extractors could find applications in distributed systems to prevent unauthorized alterations to data during communication or computation processes.

Is there a deeper connection between the constructions of non-malleable extractors and standard extractors that can be exploited

There is indeed a profound connection between the constructions of non-malleable extractors and standard extractors that can be leveraged for further advancements in both areas. Some potential avenues for exploring this connection include: Analyzing Common Components: By dissecting the shared components and techniques used in both non-malleable and standard extractors, researchers can identify synergies and optimize these elements to improve the efficiency and error rates of both types of extractors. Unified Framework: Developing a unified framework that integrates the principles of non-malleable extractors with those of standard extractors could lead to more robust and versatile randomness extraction techniques that are applicable across a wider range of scenarios. Cross-Pollination of Ideas: Encouraging collaboration and knowledge exchange between researchers working on non-malleable extractors and standard extractors can foster innovation and the discovery of novel approaches that bridge the gap between these two domains.
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