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Non-Malleable Extractors New tools and improved constructions

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1 Non-Malleable Extractors New tools and improved constructions
Gil Cohen

2 bad seeds for the source ๐‘‹
Seeded Extractors Meta definition A randomness extractor is a function that transforms a โ€œweak source of randomnessโ€ to one that is close to uniform. Definition (NisanZuckermanโ€™93) all seeds ๐‘ โˆˆ 0,1 ๐‘‘ ๐ธ๐‘ฅ๐‘ก ๐‘‹,๐‘  ~๐‘ˆ E๐‘ฅ๐‘ก: 0,1 ๐‘› ร— 0,1 ๐‘‘ โ†’ 0,1 ๐‘š is a ๐’Œ-seeded extractor if for any random variable ๐‘‹ having entropy ๐‘˜, ๐ธ๐‘ฅ๐‘ก ๐‘‹,๐‘  ~๐‘ˆ for a typical ๐‘ โˆˆ 0,1 ๐‘‘ . ๐‘  Goal Construct extractors with short seeds that support low entropy and output many bits. bad seeds for the source ๐‘‹ [NisanZuckermanโ€™93,Trevisanโ€™99, LuReingoldVadhanWigdersonโ€™03, Guruswami-UmansVadhanโ€™07, DvirKoppartySarafSudanโ€™09,โ€ฆ] explicit extractors that output almost all of the entropy, for any entropy bound, using logarithmic seeds.

3 Non-Malleable Extractors
Definition (DodisWichsโ€™09) A seeded extractor is non-malleable if for any source ๐‘‹, any good seed ๐‘ , and any ๐‘  โ€ฒ โ‰ ๐‘ , ๐ธ๐‘ฅ๐‘ก ๐‘‹,๐‘  ~๐‘ˆ conditioned on ๐ธ๐‘ฅ๐‘ก ๐‘‹,๐‘ โ€ฒ . Natural (and also in retrospective useful) to consider ๐‘ก-non-malleable extractors [C-RazSegevโ€™12]. all seeds ๐‘ โˆˆ 0,1 ๐‘‘ Applications ๐ธ๐‘ฅ๐‘ก ๐‘‹,๐‘  ~๐‘ˆ ๐‘  * Privacy amplification [DodisWichsโ€™09] * Two-source extractors [ChattopadhyayZuckermanโ€™15] ๐‘ โ€ฒโ€ฒ ๐‘ โ€ฒ ๐ธ๐‘ฅ๐‘ก ๐‘‹,๐‘ โ€ฒโ€ฒ ๐ธ๐‘ฅ๐‘ก ๐‘‹,๐‘ โ€ฒ bad seeds for the source ๐‘‹

4 Previous Work and Our Contribution
Construction Seed length ๐‘‘ Supported entropy ๐‘˜ [DodisWichsโ€™09] (non constructive) log ๐‘› log log ๐‘› [LiWooleyZuckermanโ€™11] ๐‘› 0.501๐‘› [C-RazSegevโ€™11] [LiWooleyZuckermanDodisโ€™11, Liโ€™12a] [Liโ€™12b] 0.499๐‘› [ChattopadhyayGoyalLiโ€™15] log 2 ๐‘› This work - construction I log ๐‘› โ‹… log log ๐‘› This work โ€“ construction II ๐‘› ๐‘๐‘œ๐‘™๐‘ฆ log ๐‘›

5 Previous Work and Our Contribution
Reducing entropy requirement NM-extractor for entropy ๐‘˜ with seed length ๐‘‘ NM-extractor for entropy ๐‘˜/ ๐‘‘ ฮฉ 1 , with seed length ๐‘‚ ๐‘‘ Increasing output length NM-extractor with ฮฉ 1 output bits and seed length ๐‘‘ NM-extractor with ฮฉ ๐‘˜ output bits and seed length ๐‘‚ ๐‘‘ Increasing ๐’• โ€œStandardโ€ NM-extractor ๐‘ก-NM extractor

6 Correlation Breakers

7 ๐ถ๐ต ๐‘Œ,๐‘‹,๐›ผ ,๐ถ๐ต ๐‘Œ โ€ฒ , ๐‘‹ โ€ฒ ,๐›ผโ€ฒ ~ ๐‘ˆ,๐ถ๐ต ๐‘Œ โ€ฒ , ๐‘‹ โ€ฒ ,๐›ผโ€ฒ
Correlation Breakers ๐‘‹ ๐›ผ The Flip-Flop primitive [Cโ€™15] is a CB for ๐‘Ž=1. CB By adapting the construction of local correlation breakers [Cโ€™15], it was shown by [ChattopadhyayGoyal-Liโ€™15] that a sequential application of the Flip-Flip primitive yields CB for arbitrary advice length. ๐‘Œ ๐‘ CB ๐‘Œโ€ฒ ๐‘โ€ฒ ๐‘‹โ€ฒ ๐‘‹ ๐›ผโ€ฒ Definition A correlation breaker with advice is a function CB: 0,1 โ„“ ร— 0,1 ๐‘› ร— 0,1 ๐‘Ž โ†’ 0,1 โ„“ such that for any ๐‘Œ~ ๐‘ˆ โ„“ , weak-source ๐‘‹, and ๐‘Œ โ€ฒ ,๐‘‹โ€ฒ such that ๐‘Œ,๐‘Œโ€ฒ independent of ๐‘‹,๐‘‹โ€ฒ , for any fixed ๐‘Ž-bit strings ๐›ผโ‰ ๐›ผโ€ฒ ๐ถ๐ต ๐‘Œ,๐‘‹,๐›ผ ,๐ถ๐ต ๐‘Œ โ€ฒ , ๐‘‹ โ€ฒ ,๐›ผโ€ฒ ~ ๐‘ˆ,๐ถ๐ต ๐‘Œ โ€ฒ , ๐‘‹ โ€ฒ ,๐›ผโ€ฒ

8 2016 * In [C-Schulmanโ€™16] CB were used to construct multi-source extractors. Main new pseudorandom primitive โ€“ independence preserving mergers (IPM). * In [Cโ€™16a] NM-Ext with seed length ๐‘‚ log ๐‘› were constructed for any min-entropy ๐‘˜=ฮฉ log ๐‘› . Main idea is to use CB (also) for generating the advice. * Based on IPM, [ChattopadhyayLiโ€™16] obtained incomparable NM-Ext with improved error dependence. ! All these works rely on the Flip-Flop based CB. Circumventing in different ways the fact that its randomness requirement is linear in the advice length ๐‘Ž. * In [Cโ€™16b] a CB with optimal ฮ˜ log ๐‘Ž randomness requirement was obtained. One application is a NM-Ext with near-optimal seed length d= log ๐‘› + log 1+๐‘œ /๐œ– , output length 0.49๐‘˜, and supported entropy ฮฉ ๐‘‘ . Thanks!


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