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Non-Malleable Extractors

This work presents explicit constructions of non-malleable seeded extractors, building on a joint effort by Gil Cohen, Ran Raz, and Gil Segev from the Weizmann Institute. We explore the efficiency and limitations of these extractors, with foundational results attributed to key studies like [DodisWichs09] and [DodisLiWooleyZuckerman11]. Our main results demonstrate unconditionally efficient methods, addressing both strong and small-bias conditions. Additionally, we outline open questions regarding non-malleable extractors with smaller min-entropies and seek applications beyond traditional cryptography.

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Non-Malleable Extractors

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  1. Non-Malleable Extractors Gil Cohen Weizmann Institute Joint work with Ran Razand Gil Segev

  2. Seeded Extractors 0 1 Seeded Extractor

  3. Seeded Extractors 11 00 10 01 01 00 10 11 Seeded Extractor Strong Seeded Extractor

  4. Seeded Extractors 11 10 101 100 000 111 01 No limitation … … 00 Seeded Extractor Small-Bias Set

  5. Non-Malleable Extractors [DodisWichs09] 0 0 1 1 Seeded Extractor Strong Seeded Extractor 0 Non-Malleable Extractor

  6. The Explicit Construction of [DodisLiWooleyZuckerman11] • Conditional efficiency

  7. The Explicit Construction of [DodisLiWooleyZuckerman11] • Conditional efficiency

  8. The Explicit Construction of [DodisLiWooleyZuckerman11] • Conditional efficiency

  9. The Explicit Construction of [DodisLiWooleyZuckerman11] • Conditional efficiency

  10. Main Result • Unconditionally efficient

  11. Main Result • Unconditionally efficient

  12. Main Result • Unconditionally efficient

  13. Main Result • Unconditionally efficient

  14. Explicit Constructions [Li12] Moreover, Bourgain’s extractor is non-malleable.

  15. The Construction

  16. Raz’s Theorem [Raz05] is a

  17. Proof Sketch

  18. Proof Idea

  19. Proof Idea is typically biased (say towards 0).

  20. Proof Idea is typically biased (say towards 0).

  21. Proof Idea

  22. Proof Idea Acyclic Many vertices Average edge weight is large

  23. Proof Idea Acyclic Many vertices Average edge weight is large is typically biased

  24. Proof Idea Small-Bias Set [Raz05] implies that this is also an extractor stands in contradiction!

  25. Open Questions

  26. Open Questions ? Construct a non-malleable extractor for smaller min-entropies, or prove this is hard. ? Waiting for applications to complexity (as apposed to cryptography).

  27. Thank You!

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