Extractors: applications and constructions Avi Wigderson IAS, Princeton Randomness

Extractors: original motivation Unbiased, independent Probabilistic algorithms Cryptography Game Theory Applications: Analyzed on perfect randomness biased, dependent Reality: Sources of imperfect randomness Stock market fluctuations Sun spots Radioactive decay Extractor Theory

Running probabilistic algorithms with weak random bits Probabilistic algorithm InputOutput Error prob < δ E XT unbiased, independent biased, dependent

Monte-Carlo algorithms with few random bits Setting: Statistical mechanics model (Ising, Potts, Percolation, Spin Glass,….) Task: Estimate parameters (free entropy, partition function, long-range correlations,…) Algorithm: Sample a random state from Gibbs dist. (Glauber dynamics, Metropolis algorithm,…) State Space {0,1} n n sites

Monte-Carlo algorithms with few random bits Resources of the typical Monte-Carlo algorithm - Space: ~ n -Time: t < poly(n) -Randomness: ~ tn bits [Nisan-Zuckerman] Randomness = space! Deterministically expand n  tn bits, with r t ~ uniform ! State Space {0,1} n any r 1 r 2 r i r t ~ uniform

Certifying randomness  What if the device/detectors are faulty? [Colbeck ‘06, Pioroni et al ‘10, Vidick-Vazirani ‘12,…] Amplification & certification of randomness: QM Algorithm QM device k bits 2 k bits With High Probability: If device good: output ~ uniform If device faulty: rejects No signaling Extractor Insnside

Applications of Extractors Using weak random sources in prob algorithms [B84,SV84,V85,VV85,CG85,V87,CW89,Z90-91] Randomness-efficient error reduction of prob algorithms [Sip88, GZ97, MV99,STV99] Derandomization of space-bounded algorithms [NZ93, INW94, RR99, GW02] Distributed Algorithms [WZ95, Zuc97, RZ98, Ind02]. Hardness of Approximation [Zuc93, Uma99, MU01] Cryptography [CDHKS00, MW00, Lu02 Vad03] Data Structures [Ta02] Coding Theory [TZ01,TZS01] Certifying & expanding randomness [Col09,Pir+09,VV12]

Unifying Role of Extractors Extractors are intimately related to: Hash Functions [ILL89,SZ94,GW94] Expander Graphs [WZ93, RVW00, TUZ01, CRVW02] Samplers [G97, Z97] Pseudorandom Generators [Tre99, …] Error-Correcting Codes [TZ01, TZS01, SU01, U02] Ergodic Theory [Lindenstrauss 07] Exponential sums  Unify the theory of pseudorandomness.

Definitions

Weak random sources Distributions X on {0,1} n with “some” entropy: X=(X 1,X 2,…,X n ) [vN] sources: n coins of unknown fixed bias [SV] sources : Pr[X i+1 =1|X 1 =b 1,…,X i =b i ]  ( δ, 1-δ) [LLS] sources : n coins, some “sticky” ….. [Z] k-sources: H ∞ (X) ≥ k  x Pr[X = x]  2 -k e.g X uniform with support ≥ 2 k k – the entropy in the weak source {0,1} n X

Randomness Extractors (1 st attempt) E XT X k -source of length n m (almost) uniform bits Ext : {0,1} n  {0,1} m Impossible even if k=n-1 and m=1 “weak” random source X k can be e.g n/2, √n, log n,… Ext=0 Ext=1 {0,1} n X m ≤ k

Extractors [Nisan & Zuckerman `93] E XT k -source of length n m bits  -close to uniform d random bits (short) “seed” {0,1} n X {0,1} m Ext i (X) i  {0,1} d Want: efficient Ext, small d, , large m

Explicit & Efficient Extractors Non-constructive & optimal [Sip88,NZ93,RT97]: –Seed length d = log n + O(1). –Output length m = k - O(1). [...B86,SV86,CG87, NZ93, WZ93, GW94, SZ94, SSZ95, Zuc96, Ta96, Ta98, Tre99, RRV99a, RRV99b, ISW00, RSW00, RVW00, TUZ01, TZS01, SU01, LRVW03,…] Explicit constructions [GUV07, DW08] - Seed length d = O(log n) - Output length m =.99k

Running probabilistic algorithms with weak random bits k-source of length n m random bits E XT d random bits Probabilistic algorithm Input (upto  L 1 error) Output Error prob < δ ++ Try all possible 2 d = poly(n) seeds. Take majority vote. Efficient! k=2m

Constructions via the Kakeya Problem

Mergers [Ta96] – very special case d random bits seed Mer X Y m ≥. 99k k k k X,Y  F q k q ~ n 100 X or Y is random X,Y correlated! [LRVW] Mer = aX+bY a,b  F q ( d=2log q ) Major problems in analysis and geometry! Wolf: Smallest set in F q k containing a line in every direction? Kakeya: Smallest set in R 2 cont. a needle in every direction? Besikovich: Smallest set in R 2 has area 0! Dvir: Smallest set in F q k has volume > (cq) k. Polynomial method!

Thanks!