Non-Malleable Extractors

Gil CohenWeizmann Institute

Joint work withRan Raz and Gil Segev

2𝑛

2𝑑

2𝑘

Seeded Extractors

10

Seeded Extractor

Seeded Extractors

2𝑛

2𝑑

000111 10 2𝑘

Seeded Extractor

2𝑑

Strong SeededExtractor

2𝑘00

0110

11

Seeded Extractors

2𝑛

2𝑑

000111

2𝑘

Seeded Extractor

2𝑑

000

101

100

111

…… No limitation

Small-Bias Set

10

Non-Malleable Extractors [DodisWichs09]

Seeded Extractor Strong SeededExtractor

2𝑑

2𝑘

Non-MalleableExtractor

0

1

0

0

1

𝐴 (𝑠 ) 𝑠

The Explicit Construction of [DodisLiWooleyZuckerman11]

• Conditional efficiency

The Explicit Construction of [DodisLiWooleyZuckerman11]

• Conditional efficiency

The Explicit Construction of [DodisLiWooleyZuckerman11]

• Conditional efficiency

The Explicit Construction of [DodisLiWooleyZuckerman11]

• Conditional efficiency

Main Result

• Unconditionally efficient

Main Result

• Unconditionally efficient

Main Result

• Unconditionally efficient

Main Result

• Unconditionally efficient

Explicit Constructions [Li12]

Moreover, Bourgain’s extractor is non-malleable.

The Construction

Raz’s Theorem [Raz05]

is a

Proof Sketch

Proof Idea

𝑊

𝑠 𝐴 (𝑠 )𝑍1𝑍 2⋯

Proof Idea

𝑊

𝑠 𝐴 (𝑠 )

𝐸𝑥𝑡 (𝑊 ;𝑠 )𝐸𝑥𝑡 (𝑊 ; 𝐴 (𝑠 ) )

is typically biased (say towards 0).

Proof Idea

𝑊

𝑠 𝐴 (𝑠 )

𝐸𝑥𝑡 (𝑊 ;𝑠 )𝐸𝑥𝑡 (𝑊 ; 𝐴 (𝑠 ) )

is typically biased (say towards 0).

Proof Idea

𝑠 𝐴 (𝑠 )

𝑠𝐴 (𝑠 )

𝑏𝑖𝑎𝑠 (𝑌𝑠 )

Proof Idea

𝑠 𝐴 (𝑠 )

𝑠𝐴 (𝑠 )

𝑏𝑖𝑎𝑠 (𝑌𝑠 )Acyclic

Many verticesAverage edge weight is large

Proof Idea

𝑠 𝐴 (𝑠 )

𝑠𝐴 (𝑠 )

𝑏𝑖𝑎𝑠 (𝑌𝑠 )Acyclic

Many verticesAverage edge weight is large

Proof Idea

⋯ 𝑍 𝑠⊕ 𝑍𝐴 (𝑠 )⋯

Small-Bias Set

[Raz05] implies that this is also an

extractor𝑊

stands in contradiction!

𝑌 𝑠=E 𝑥𝑡 (𝑊 ; 𝑠 )⊕𝐸𝑥𝑡 (𝑊 ; 𝐴 (𝑠) )

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).

Open Questions

Thank You!