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Non-Malleable Extractors New tools and improved constructions
Gil Cohen
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bad seeds for the source ๐
Seeded Extractors Meta definition A randomness extractor is a function that transforms a โweak source of randomnessโ to one that is close to uniform. Definition (NisanZuckermanโ93) all seeds ๐ โ 0,1 ๐ ๐ธ๐ฅ๐ก ๐,๐ ~๐ E๐ฅ๐ก: 0,1 ๐ ร 0,1 ๐ โ 0,1 ๐ is a ๐-seeded extractor if for any random variable ๐ having entropy ๐, ๐ธ๐ฅ๐ก ๐,๐ ~๐ for a typical ๐ โ 0,1 ๐ . ๐ Goal Construct extractors with short seeds that support low entropy and output many bits. bad seeds for the source ๐ [NisanZuckermanโ93,Trevisanโ99, LuReingoldVadhanWigdersonโ03, Guruswami-UmansVadhanโ07, DvirKoppartySarafSudanโ09,โฆ] explicit extractors that output almost all of the entropy, for any entropy bound, using logarithmic seeds.
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Non-Malleable Extractors
Definition (DodisWichsโ09) A seeded extractor is non-malleable if for any source ๐, any good seed ๐ , and any ๐ โฒ โ ๐ , ๐ธ๐ฅ๐ก ๐,๐ ~๐ conditioned on ๐ธ๐ฅ๐ก ๐,๐ โฒ . Natural (and also in retrospective useful) to consider ๐ก-non-malleable extractors [C-RazSegevโ12]. all seeds ๐ โ 0,1 ๐ Applications ๐ธ๐ฅ๐ก ๐,๐ ~๐ ๐ * Privacy amplification [DodisWichsโ09] * Two-source extractors [ChattopadhyayZuckermanโ15] ๐ โฒโฒ ๐ โฒ ๐ธ๐ฅ๐ก ๐,๐ โฒโฒ ๐ธ๐ฅ๐ก ๐,๐ โฒ bad seeds for the source ๐
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Previous Work and Our Contribution
Construction Seed length ๐ Supported entropy ๐ [DodisWichsโ09] (non constructive) log ๐ log log ๐ [LiWooleyZuckermanโ11] ๐ 0.501๐ [C-RazSegevโ11] [LiWooleyZuckermanDodisโ11, Liโ12a] [Liโ12b] 0.499๐ [ChattopadhyayGoyalLiโ15] log 2 ๐ This work - construction I log ๐ โ
log log ๐ This work โ construction II ๐ ๐๐๐๐ฆ log ๐
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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 โStandardโ NM-extractor ๐ก-NM extractor
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Correlation Breakers
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๐ถ๐ต ๐,๐,๐ผ ,๐ถ๐ต ๐ โฒ , ๐ โฒ ,๐ผโฒ ~ ๐,๐ถ๐ต ๐ โฒ , ๐ โฒ ,๐ผโฒ
Correlation Breakers ๐ ๐ผ The Flip-Flop primitive [Cโ15] is a CB for ๐=1. CB By adapting the construction of local correlation breakers [Cโ15], it was shown by [ChattopadhyayGoyal-Liโ15] that a sequential application of the Flip-Flip primitive yields CB for arbitrary advice length. ๐ ๐ CB ๐โฒ ๐โฒ ๐โฒ ๐ ๐ผโฒ Definition A correlation breaker with advice is a function CB: 0,1 โ ร 0,1 ๐ ร 0,1 ๐ โ 0,1 โ such that for any ๐~ ๐ โ , weak-source ๐, and ๐ โฒ ,๐โฒ such that ๐,๐โฒ independent of ๐,๐โฒ , for any fixed ๐-bit strings ๐ผโ ๐ผโฒ ๐ถ๐ต ๐,๐,๐ผ ,๐ถ๐ต ๐ โฒ , ๐ โฒ ,๐ผโฒ ~ ๐,๐ถ๐ต ๐ โฒ , ๐ โฒ ,๐ผโฒ
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2016 * In [C-Schulmanโ16] CB were used to construct multi-source extractors. Main new pseudorandom primitive โ independence preserving mergers (IPM). * In [Cโ16a] 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, [ChattopadhyayLiโ16] 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. * In [Cโ16b] a CB with optimal ฮ log ๐ randomness requirement was obtained. One application is a NM-Ext with near-optimal seed length d= log ๐ + log 1+๐ /๐ , output length 0.49๐, and supported entropy ฮฉ ๐ . Thanks!
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