Extractors with Weak Random Seeds

1
Extractors with Weak Random Seeds

• Ran Raz
• Weizmann Institute

2

• A Weak Source of Randomness
• A random variable XX1,...,Xn
• that is not uniformly distributed
• min-entropy(X) maximal b s.t.
• 8 a 2 0,1n, ProbXa 2-b
• rate ? b/n (min-entropy rate)
• How to extract pure random bits ?

3

• The Story of Extractors
• 1) Seeded Extractors use a small number of truly
  random bits
• 2) Multi-Sources Extractors use several
  independent weak sources
• In this work conclusions about
• both types of extractors

4

• Seeded Extractors NZ
• XX1,...,Xn a weak source with
• min-entropy b
• ZZ1,...,Zd truly random bits
• E 0,1n 0,1d ! 0,1m s.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 ? gt 0,
• 9 E with seed of length dO(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 ? gt0)
• 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 ? gt0)
• 1) O(n) bits from one source of rate 0.5? and
  one source of logarithmic ME (optimal error)
• 2) O(n) bits from one source of rate ? and O(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 ? gt0)
• 1) O(n) bits from one source of rate 0.5? and
  one source of logarithmic ME (optimal error)
• 2) O(n) bits from one source of rate ? and O(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 ? gt0)
• Source 1 (n1,b1) b1/n1 gt 0.5?
• Source 2 (n2,b2) b2 gt 5log(n1)
• and s.t., n1 gt O(log(n2))
• Then, we can extract O(minb1,b2)
• bits that are independent of each
• source separately (optimal error)
• Previously GS,Alo 1 bit when n1n2
• Independently BKSSW O(minb1,b2) bits
  
• when n1n2

12

• Main Idea (for extracting one bit)
• Y1,...,YN 2 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 ?, ? gt0)
• Input 1) A source of rate ? gt 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 ? gt0)
• 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 ? gt0)
• We color the complete bipartite
• 2n2n 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