3 source extractors bipartite Ramsey graphs and other
3 -source extractors, bi-partite Ramsey graphs, and other explicit constructions GUY KINDLER Institute for Advanced Study Joint work with Boaz barak r. Onen shaltiel Benny sudakov avi wigderson
In this talk: Explicit constructions are very important. Randomness extraction is also important. Our results, concerning explicit constructions of randomness extraction objects. Something about the techniques.
The importance of being explicit ª [Erdos] The probabilistic method: show that a random object has a wanted property with positive probability. ª Example: [Shanon 48] optimal error-correcting codes exist. ª Great result, but no applications. ª A random code is not efficiently computable. ª We need explicit (efficiently computable) codes.
The importance of being explicit ª Why are explicit constructions so hard? ª (Why is the probabilistic method so powerful)? ª My answer: Because the family of explicitly constructible objects is tiny, and also very complex. ª Why be explicit?
explicit construction problems ª Some of the profound problems of computer science are of explicit constructions. Derandomization of BPP. Very difficult Problems! Complexity lower bounds. ª We need new tools/ideas for explicit constructions. ª Great example: SL=L ! ª We have some new tools and ideas!
We’ve Done: Explicit constructions are very important. We need new explicit construction tools. Next: About randomness extraction.
About randomness ª Why Randomness: for algorithms (theory and practice), and cryptography (absolutely essential!). ª Analysis requires unbiased, independent random bits. r r=(100. . 10) alg (m-bit strings) I=(011… 010)
About randomness ª Why Randomness: for algorithms (theory and practice), and cryptography (absolutely essential!). ª Analysis requires unbiased, independent random bits. Analysis says (set of size 2 m) nothing in this case! r alg (m-bit strings) I=(011… 010)
About randomness ª Randomness can be weak because of: Bad sources: clock/keystroke, electrostatic noise, yesterday’s news, process id’s. . Measurement: bits, heat of components, voltage, sound. . Intervention: bit-fixing, temperature, radio waves. . r ª [Goldberg, Wagner `96] Broke netscape’s ssl protocol using its poor random source. alg r alg
Randomness extraction Solution [shannon]: extract m pure bits from n>m imperfect bits. r (m-bit strings) alg I=(011… 010)
Randomness extraction X Solution [shannon]: extract m pure bits from n>m imperfect bits. |X|>>2 m (n-bit strings) We want of ext: |X|>>2 m ext(x)~Um x ext Polnomial time! ext(x) alg (m-bit strings) I=(011… 010)
Randomness extraction X (n-bit strings) We want of ext: |X|>>2 m ext(x)~Um Problem: No such thing! x ext (m-bit strings) alg I=(011… 010)
Seeded extractor X x (n-bit strings) ext (m-bit strings) alg I=(011… 010)
Seeded extractor X x Good for: (n-bit strings) ª Simulating BPP using weak sources. ext Problems: ª Blowup 5 1 4 2 3 (m-bit strings) ª Doesn’t work for cryptography alg I=(011… 010)
k-source extractor X 1 x 1 X 2 x 2 We want: |X 1|, |X 2| “large” ext(x 1, x 2)~Um Such things exist! (n-bit strings) ext (m-bit strings) alg I=(011… 010)
We’ve seen: Wild-grown randomness can be very weak. Extracting pure randomness from one source – not possible. Extraction from two or more independent sources may be possible. next: (Some of) our results – extraction from three sources, and Ramsey construction for two sources.
Entropy and entropy rate k-source extractor for rate : H(X 1) n r(X 1), . . . , r(Xk) r(X 1) Pr[ext(x 1, . . , xk)=0]= Pr[ext(x 1, . . , xk)=1] 0. 1 X 1, |X 1| 2 n X 2, |X 2| 2 n x 1 x 2 k-partite Ramsey for rate : We want: r(X 1), . . . , r(Xk) |X 1|, |X 2| “large” Pr[ext(x 1, . . , xk)=0]>0 , ext(x 1, x 2)~Um Pr[ext(x 1, . . , xk)=1]>0 (n-bits) ext 0/1 (m-bits)
Entropy and entropy rate k-source extractor for rate : r(X 1), . . . , r(Xk) Pr[ext(x 1, . . , xk)=0]= Pr[ext(x 1, . . , xk)=1] 0. 1 X 1, |X 1| 2 n X 2, |X 2| 2 n x 1 x 2 (n-bits) ext 0/1
Results: past and current ª [Chor+Goldreich , Vaz ’ 85]: 2 -source X 1, |X 1| 2 n X 2, |X 2| 2 n extractor for entropy rate >1/2. (n bits) ª Long time: 1/2 -barrier not broken. ext 0/1 ª [BIW ‘ 03] Extractor for constant , with k( )-sources. ª We get: 3 -source extractor, for any constant entropy rate . ª For 2 sources: Bipartite Ramsey for all constant ! Bipartite k-source Ramsey extractorfor forrate : : r(X 1), r(X ), . . . , r(X 2) k) Pr[Ram(x Pr[ext(x 1, . . , x )=0]= , 1, xk 2)=0]>0 Pr[Ram(x Pr[ext(x 1, . . , x )=1] 0. 1 1, xk 2)=1] >0
the bipartite Ramsey problem X 1, |X 1| 2 n X 2, |X 2| 2 n x 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 0 x 2 (n-bits) Ram 0/1 2 n x 2 n Pr[0], Pr[1]>0
the bipartite Ramsey problem X 1, |X 1| 2 n x 1 x 2 X 2, |X 2| 2 n 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 0 x 2 (n-bits) Ram 0/1 2 n x 2 n Pr[0], Pr[1]>0
the bipartite Ramsey problem X 1, |X 1| 2 n x 1 x 2 X 2, |X 2| 2 n 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 0 x 2 (n-bits) Ram 0/1 2 n x 2 n Pr[0], Pr[1]>0
the bipartite Ramsey problem X 1, |X 1| 2 n X 2, |X 2| 2 n x 1 |X X 1 | 2 n 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 0 x 2 (n-bits) Ram 0/1 2 n x 2 n Pr[0], Pr[1]>0
the bipartite Ramsey problem X 1, |X 1| 2 n x 1 |X 1| 2 n X 2, |X 2| 2 n 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 0 x 2 (n-bits) Ram 0/1 2 n x 2 n Pr[0], Pr[1]>0
the bipartite Ramsey problem X 1, |X 1| 2 n x 1 |X 1| 2 n X 2, |X 2| 2 n 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 0 x 2 (n-bits) Ram 0/1 Pr[0], Pr[1]>0 ª [chor+goldreich ’ 85] >1/2. 2 n x 2 n ª We get: any constant .
We have: First extractor with number of sources independent of entropy rate. First explicit bipartite Ramsey graph for rate < ½. Several other results and constructions. next: How to break the ½ barrier with const. number of sources. A 4 -source extractor.
A dream 2 -source extractor |X 1| 2 n x 1 (n-bits) |Y 1| 20. 9 s (s= n-bits) con
A dream 2 -source extractor |X 1 | 2 n |X 2| 2 n x 1 x 2 (n-bits) Problem: 0. 9 s |Y | 2 1 No such thing! con |Y 2| 20. 9 s (s= n-bits) con (s-bits) Vaz (100 bits)
BREAKING THE ½ BARRIER: a somewhere condenser |X 1 |X 2| 2 n x 1 | 2 n x 2 (n-bits) con New! Key component! 1 (3 times s) ( 3=3( ) ) 2 (3 times s) Constant entropy rate. 3 Constant number of bits! Vaz Vaz Vaz (9 times 100)
A 2 -source somewhere extractor |X 1 | 2 n |X 2| 2 n x 1 x 2 (n-bits) con (3 con Somewhere extractor times s) Vaz Vaz Vaz (3 times s) Vaz (9 times 100)
the 4 -source extractor Somewhere extractor (900 bits) Opt 0/1
The BIW lemma [BIW] X 1, X 2, X 3 independent |X 1| 2 m x 1 r(Y)>min{ + 2, 1 } |X 2| 2 m |X 3| 2 m x 2 x 3 (3 x m-bits) x 1 x 2+x 3 Y, |Y| 2 (1+ )m y (m-bits)
Somewhere condenser x |X| 2 n (n-bits) X 1 x 1 X 2 x 2 X 3 x 3 ( 3 x (n/3) ) [Thm] max( r(X 1), r(Xx 2), 1 xr(X 2+x 33), r(X 4) ) min{ + 2/10, 1 } X 4 x 4 (n/3)
Somewhere condenser x |X| 2 n (n-bits) If equality holds then X 1, X 2, X 3 are independent! X 1 x 1 X 2 x 2 X 3 x 3 ( 3 x (n/3) ) If strong enough inequality here, we’re done! |X 1|. |X 2|. |X 3| |X| (r(X 1)+r(X 2)+r(X 3))/3 r(X)
Somewhere condenser x |X| 2 n And by [BIW], [BIW] we’re done! X 1 x 1 X 2 x 2 X 3 (n-bits) x 3 ( 3 x (n/3) ) x 1 x 2+x 3 X 4 x 4 (n/3)
Bipartite Ramsey – the general idea |X 2| 2 n x 1 |X 1| 2 n x 2 (n-bits) Y 1 (n-bits) y 1 Y 2 y 2 Y 3 4 -source extractor 0/1 y 3 Y 4 y 4
Bipartite Ramsey – the general idea |X 2| 2 n x 1 |X 1| 2 n x 2 (n-bits) P-selector Y 1 y 1 Y 2 y 2 Y 3 4 -source extractor 0/1 y 3 Y 4 y 4
IN Summation: Somewhere condenser: key to breaking ½ barrier. Lots of more work to get bipartite Ramsey. Testing entropy, then using it: not a stupid idea. . Open: 2 -source extractor, 17 -source extractor with no “opt” and small error. seeded extractor with polynomially small error.
Other results: Ramsey-coloring for affine subspaces of dim. n. 7 source extractor for linear number of bits. a zero error 2 -source disperser for loglog(n) bits. a 2 -source disperser for linear number of bits.
- Slides: 39