1. Randomness & Pseudorandomness Avi Wigderson IAS, Princeton

2. Plan of the talk Randomness HHTHTTTHHHTHTTHTTTHHHHTHTTTTTHHTHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH The amazing utility of randomness Lots of examples Pseudorandomness Deterministic structures which share some properties of random ones Lots of examples & applications

3. The remarkable utility of randomness Where are the random bits from?

4. Fast Information Acquisition Theorem: With probability >.99 % in sample = % in population  2% independent of population size Deterministically, need to ask the whole population! Population: 280 million, voting blue or red Sample: 2,000 Where are the random bits from?

5. Efficient (!!) Probabilistic Algorithms Given a region in space, how many domino tilings does it have? Monomer-Dimer problem. Captures thermodynamic properties of matter (free energy, phase transitions,…) Theorem [Randall-Sinclair]: Efficient probabilistic approximate counting algorithm (“Monte-Carlo” method [von Neumann-Ulam]) Best deterministic algorithm known requires exponential time! One of numerous examples Where are the random bits from?

6. Distributed computation The dining philosophers problem - All have identical programs - Each needs 2 forks to eat - Each needs to eat sometime Captures resource allocation and sharing in asynchronous systems Theorem [Dijkstra]: No deterministic solution Theorem [Lehman-Rabin]: A probabilistic program works (symmetry breaking) Where are the random bits from?

7. Game Theory – Rational behavior 1 10 20 2 2 02 0 -3-3 C CA A Nash Equilibrium: No player has an incentive to change its strategy given the opponent’s strategy. Theorem [Nash]: Every game has an equilibrium in mixed strategies (Pr[C] =¾, Pr[A]= ¼) False for pure strategies Where are the random bits from? Chicken game [Aumann] A: Aggressive C: Chicken

8. Cryptography & E-commerce Secrets Theorem [Shannon] A secret is as good as the entropy in it. ( if you pick a 10 digit password randomly, my chances of guessing it is 1/10 10 ) Public-key encryption Digital signature Zero-Knowledge Proofs ……… All require randomness Where are the random bits from?

9. Gambling Where are the random bits from?

10. Radiactive decay Atmospheric noise Photons measurement Where are the random bits from?

11. Defining randomness

12. What is random? Randomness is in the eye of the beholder -xxx computational power Operative, subjective definition! Probability of guessing correctly? 1/2 1

13. Pseudorandomness The study of deterministic structures with some random-like properties. Different properties/tests  Different motivations & applications Mathematics: Study of random-like properties in natural structures Computer Science: Efficient construction of structures with random-like properties Match made in heaven: Generalization & unification of problems, techniques & results

14. Normal Numbers -Every digit (e.g. 7) occurs 1/10 th of the time, -Every pair (e.g. 54) occurs 1/100 th of the time, -Every triple (eg 666) occurs 1/1000 th of the time,… in every base! Theorem[Borel]: A random real number is normal Open: Is π normal? Are √2, e normal? 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 ………

15. Major problems of Math & CS are about Pseudorandomness Clay Millennium Problems - $1M each Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations P vs. NP Poincaré Conjecture Riemann Hypothesis Yang-Mills Theory Random functions are hard to compute. Find a natural function which is hard! Random walks stay close to the origin. Prove the same for the Möbius walk!

16. P vs. NP 1 01 100 X2X2 X3X3 V V V V V X1X1 V f Theorem [Shannon]: Random function on n variables require exponential(n) gates P vs. NP: Exhibit a hard function! e.g. do Traveling-Salesperson, Clique, Trunk-Packing,… require exponential(n) gates? X 1 X 2 X 3 f 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 0 1 1

17. Riemann Hypothesis & the drunkard’s walk Start: 0 Each step - Up: +1 Down: -1 Randomly. Almost surely, after N steps distance from 0 is ~√N position time

18. x integer, p(x) number of distinct prime divisors 0 if x has a square divisor μ(x) = 1 p(x) is even -1 p(x) is odd x = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 μ (x)= 1 −1 −1 0 −1 1 −1 0 0 1 −1 0 −1 1 1 0 Theorem [Mertens 1897]: For all N | Σ x<N μ (x)| ~ √N is equivalent to the Riemann Hypothesis Möbius’ walk {

19. Possible worlds - Perfect randomness - Weak random sources - Several independent ones - One weak source - No randomness F E

20. Weak random sources and randomness purification Unbiased, independent Applications: Analyzed on perfect randomness biased, dependent Reality: Sources of imperfect randomness Stock market fluctuations Sun spots Radioactive decay Extractor Theory Statistics, Cryptography, Algorithms, Game theory, Gambling,……

21. 123456789101112131415 1213211174516566198101439119425 2976620814862132185273712774115193137128 34520917920412410202892123975266214143 41291134458156224141621309614335219125 5113536981411096813021511407318087134 6182216142221952062322175886691273995 717333261203022133692520718836311267 811116317928112792101952162419739138116171 9901611718879272221701309458556175220 10117119133206641915527941869911815111361 11161112128124109217161521087191222160215 121614345187208152155130216341931845514257 13197531181951203910914382872101173189 14192531245717111317712815564817818118209 15101637952661401178614824203256822 Pseudorandom Tables Random table Theorem: In a random matrix, every window will have many different entries. Want: k×k windows to have k 1+ ε distinct entries Can we construct such matrices deterministically? Random table

22. +123456789101112131415 12345678910111213141516 234567891011121314151617 3456789101112131415161718 45678910111213141516171819 567891011121314151617181920 6789101112131415161718192021 78910111213141516171819202122 891011121314151617181920212223 9101112131415161718192021222324 10111213141516171819202122232425 11121314151617181920212223242526 12131415161718192021222324252627 13141516171819202122232425262728 14151617181920212223242526272829 15161718192021222324252627282930 X123456789101112131415 1123456789101112131415 224681012141618202224262830 3369121518212427303336394245 44812162024283236404448525660 551015202530354045505560657075 661218243036424854606672788490 7714212835424956637077849198105 881624324048566472808896104112120 9918273645546372819099108117126135 10 2030405060708090100110120130140150 11 2233445566778899110121132143154165 12 24364860728496108120132144156168180 13 263952657891104117130143156169182195 14 284256708498112126140154168182196210 15 3045607590105120135150165180195210225 Addition table Multiplication table von Neumann ”Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin.” 1983 Erdos-Szemeredi, 2004 Bourgain-Katz-Tao: Every window will have many distinct in one of these tables!

23. Independent-source extractors biased, dependent Reality: Independent weak sources of randomness Stock market fluctuations Sun spots Radioactive decay Extractor Theorem [Barak-Impagliazzo-W ’05]: Extractor for independent sources + X Unbiased, independent Applications: Analyzed on perfect randomness Statistics, Cryptography, Algorithms, Game theory, Gambling,……

24. Single-source extractors algorithm A input output correct with high probability n unbiased, independent Reality: one weak random source n biased, dependent algorithm A input output correct with high probability?? Naïve implementation fails!

25. Single-source extractors algorithm A input output correct with high probability n unbiased, independent Algorithm A ’ input output correct whp!! n 3 biased, dependent entropy n 2 Probabilistic algorithms with 1 weak random source Extractor Run A on each, and take a majority vote Many other applications: - Hardness of approx - Derandomization, - Data structures - Error correction - … A perfect sample Bl, SV, NZ, …………. Ta, Tr, ISW, …………. LRVW, GUV Dv, DW,…

26. Deterministic de-randomization Efficient algorithm A input output correct with high probability n unbiased, independent Efficient Algorithm A ’ input output correct always! Hardness vs. Randomness Clique is hard Run A on each, and take a majority vote BM, Y, … …………. NW, IW,… ………… IKW, KI,... 0010001 1111101 1100000 0111110 1101010 1100010 0000000 0011001 0101101 1000001 A perfect sample All efficient probabilistic algorithms have efficient deterministic counterparts

27. Summary - Randomness is in the eye of the beholder: A pragmatic, subjective definition - Pseudorandomness “tests”  “applications” Capture many basic problems and areas in Math & CS - Applications of randomness survive in a world without perfect (or any) randomness - Pseudorandom objects often find uses beyond their intended application (expanders, extractors,…)

28. Thank you! The best throw of a die is to throw the die away