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) How to extract pure random bits ?

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

Seeded Extractors [NZ]: X=X 1,...,X n = a weak source with min-entropy b Z=Z 1,...,Z d = truly random bits E: {0,1} n £ {0,1} d : ! {0,1} m 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,...

Our Result: 8 seeded extractor E, and 8  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

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)

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

Our Results: ( 8  >0) 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)

Our Results: ( 8  >0) 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

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

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

Main Idea (for extracting one bit): Y 1,...,Y N 2 {0,1}: random variables  -biased for small linear tests, s.t. n 2 = log 2 N and Y 1,...,Y N can be generated using n 1 random bits. Use source 1 to choose the random bits and source 2 to choose Y i from Y 1,...,Y N Use the construction of [AGHP]

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

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

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

Ramsey Graphs: ( 8  >0) We color the complete bipartite 2 n £ 2 n graph with a constant number of colors s.t.: no monochromatic sub-graphs of size 2  n £ n 5 [BKSSW] color with 2 colors, s.t., no monochromatic sub-graphs of size 2  n £ 2  n

The End