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Extractors with Weak Random Seeds

Extractors with Weak Random Seeds. Ran Raz Weizmann Institute. A Weak Source of Randomness: A random variable X=X 1 ,...,X n that is not uniformly distributed min-entropy( X ) = maximal b s.t. 8 a 2 {0,1} n , Prob[ X=a ] · 2 -b rate:  = b/n (min-entropy rate)

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Extractors with Weak Random Seeds

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  1. Extractors with Weak Random Seeds Ran Raz Weizmann Institute

  2. A Weak Source of Randomness: • A random variableX=X1,...,Xn • that isnotuniformly distributed • min-entropy(X) = maximalbs.t. • 8a 2 {0,1}n,Prob[X=a] · 2-b • rate: = b/n (min-entropy rate) • How to extract pure random bits ?

  3. The Story of Extractors: • 1)Seeded Extractors:use a smallnumber oftrulyrandom bits • 2)Multi-Sources Extractors:use severalindependentweak sources • In this work: conclusions about • both types of extractors

  4. Seeded Extractors [NZ]: • X=X1,...,Xn = a weak source with • min-entropy b • Z=Z1,...,Zd = truly random bits • E: {0,1}n£{0,1}d : !{0,1}ms.t., • E(X,Z)is-close to uniform • Parameters:n,b,d,m, • Explicit Constructions: NZ,Zuc,Ta-Shma, Tre,RRV,ISW,RSW,TUZ,TZS,SU,LRVW,...

  5. Our Result: • 8 seeded extractor E, and 8 > 0, • 9 E’ with seed of length d’=O(d) • and other parameters same as E, • s.t. the seed of E’ can come from • a source of min-entropy rate 0.5+ • That is: Any seeded extractor can • be operated with a seed of rate • arbitrarily close to 0.5

  6. Multi-Sources Extractors: (8  >0) • 1) [SV,Vaz,CG...]: O(n) bits from 2 sources of rate 0.5+(optimal error) • 2) [BIW]: O(n) bits from O(1) sources of rate (optimal error) • 3) [BKSSW]: O(1) bits from 3 sources of rate (constant error)

  7. Our Results: • In all these constructions: • 1) All but one source can be of logarithmic ME (min-entropy) • 2) All sources can be of different lengths

  8. Our Results: (8  >0) • 1) O(n) bits from one source of rate 0.5+andone source of logarithmic ME (optimal error) • 2) O(n) bits from one source of rate andO(1) sources of logarithmic ME (optimal error) • 3) O(n) bits from one source of rate and 2 sources of logarithmic ME (constant error)

  9. Our Results: (8  >0) • 1) O(n) bits from one source of rate 0.5+andone source of logarithmic ME (optimal error) • 2) O(n) bits from one source of rate andO(1) sources of logarithmic ME (optimal error) • 3) O(n) bits from one source of rate and 2 sources of logarithmic ME (constant error) • sources can be of different lengths

  10. Tools: • 1) A new 2-Sources Extractor • 2) A new Condenser • 3) A new Merger • All results are proved by combining • the 3 tools in different ways

  11. Strong 2-Sources Extractor: (8  >0) • Source 1: (n1,b1):b1/n1 > 0.5+ • Source 2: (n2,b2):b2 > 5log(n1) • and s.t.,n1 > O(log(n2)) • Then, we can extract O(min[b1,b2]) • bits that are independent of each • source separately (optimal error) • Previously [GS,Alo]: 1 bit when n1=n2 • Independently [BKSSW]: O(min[b1,b2]) bits • when n1=n2

  12. Main Idea (for extracting one bit): • Y1,...,YN2 {0,1}: random variables • -biased for small linear tests, s.t. • n2 = log2N and Y1,...,YN can be • generated using n1 random bits. • Use source 1 to choose the random • bits and source 2 to choose Yi from • Y1,...,YN • Use the construction of[AGHP]

  13. Strong Condenser: (8 ,  >0) • Input: 1) A source of rate  > 0 • 2) A constant number of • truly random bits • Output:O(n) bits of rate 1- • (for almost all seeds) • (constant error) • Independently [BKSSW]: • O(n) bits of rate 1-for at least one seed

  14. Main Idea: • Use the recent multi-sources • extractors of [BIW]

  15. Strong Merger: (8  >0) • Input: • 1) O(1) sources (not independent), • s.t. one of them is truly random • 2) A constant number of truly • random bits • Output:O(n) bits of rate 1- • (for almost all seeds) • (constant error) • Previously [LRVW]: n bits of rate 0.5

  16. Ramsey Graphs: (8  >0) • We color the complete bipartite • 2n£2n graph with a constant number • of colors s.t.: no monochromatic • sub-graphs of size 2n£ n5 • [BKSSW] color with 2 colors, s.t., • no monochromatic sub-graphs of • size 2n£ 2n

  17. The End

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