1. Randomness (and Pseudorandomness) Avi Wigderson IAS, Princeton

2. Plan of the talk Randomness Prob[T]=Prob[H]=½
HHTHTTTHHHTHTTHTTTHHHHTHTTTTTHHTHH
HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
The amazing utility of randomness -
lots of examples. Are they real?
Pseudorandomness
Deterministic structures which share some properties of random ones
Lots of examples & applications
Surviving weak or no randomness

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

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

5. Efficient (!!) Probabilistic Algorithms
Where are the random bits from?
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 [Luby-Randall-Sinclair]:
Efficient probabilistic
approximate counting algorithm
(“Monte-Carlo” method [von Neumann-Ulam])
Best deterministic algorithm known requires exponential time!
One of numerous examples

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

7. Game Theory – Rational behavior
Chicken game [Aumann]
1 1
0 2 C
A: Aggressive
C: Cautious A
2 0
-3-3 C 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 (random) strategies
(Pr[C] =¾, Pr[A]= ¼)
False for pure (deterministic) strategies
Where are the random bits from?

8. Cryptography & E-commerce
Secrets
Theorem [Shannon] A secret is as good as the entropy in it. (if you pick a 9 digit password randomly, my chances of guessing it is 1/109 )
Public-key encryption (on-line shopping)
Digital signature (identification)
Zero-Knowledge Proofs (enforcing correctness)
………
All require randomness
Where are the random bits from?

9. Gambling
Where are the random bits from?

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

11. Defining
randomness

12. What is random? I toss the coin, you guess how it will landMe
I toss the coin, you guess how it will land
Probability of guessing correctly?
1/2 1
Randomness is in the eye of the beholder
xxx
computational power
Operative, subjective definition!

13. Pseudorandomness
The study of deterministic structures (numbers, graphs, sequences, tables, walks) with some “random-like” properties (properties which most objects have)
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?
Fact: Not every number is normal?
………
Pseudorandom
property

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.
Prove the same for the TSP
(Traveling Salesman Problem)!
P vs NP – A mathematical problem!
In what sense is it different than the others?
Random walks stay close to the origin. Prove the same for the Möbius walk!

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

17. { Möbius’ walk μ(x) = 1 p(x) is even
x integer, p(x) number of distinct prime divisors
0 if x has a square divisor
μ(x) = p(x) is even
-1 p(x) is odd
x =
μ(x)= 1 −1 −1 0 −1 1 − −1 0 −
Theorem [Mertens 1897]: These are equivalent:
- For all N |Σx<N μ(x)| ~ √N
- The Riemann Hypothesis {

18. Coping in a world without perfect randomness

19. Possible worlds Perfect randomness Weak random sources
- Few independent ones
- One weak source
- No randomness F E
Which applications survive?

20. Weak random sources and randomness purification
Applications:
Analyzed on perfect randomness
Statistics, Cryptography, Algorithms,
Game theory, Gambling,……
Unbiased,
independent
Extractor Theory
biased,
dependent
Reality:
Sources of
imperfect
randomness
Stock market
fluctuations
Sun spots
Radioactive
decay
Extractors are designed to purify the weak (biased, correlated) randomness appearant in physical phenomena, and convert them efficiently to the uniform distribution, needed for many applications.
As it turns out, they are useful in other contexts, like error correcting codes, data structures, network design and algorithms.

21. Pseudorandom Tables Random table1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 21 32
111 74 16 66
198
101 43 91 94 25 97
208
148 62
132
185 27 37
127
115
193
137
128 45
209
179
204
124
202 89
212 39 75 26
214
143
129
134
156
224
162
130 96 35
219
125
113 53 69 81 41
109 68 51
140 73
180 87
182
216
142 22
195
206 23
175 88 95
173 33
120 30
221
207
188 36 31 67
163 28
112 79
210 24
197
138
116
171 90
161
222
170 58 55 61
220
117
119
133 64 19
155
186 99
118
151
217
152
108
191
160
215
187 34
184 57 18 82
189
192
177
178 86
203
Random
table
Random
table
Theorem: In a random matrix, every window is “rich”: have many different entries.
rich: k×k windows to have k1.1 distinct entries
Can we construct such matrices deterministically?

22. 1983 Erdos-Szemeredi, 2004 Bourgain-Katz-Tao: + 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 X 33 36 39 42 45 32 40 44 48 52 56 60 35 50 55 65 70 75 54 66 72 78 84 90 49 63 77 91 98
105 64 80 88 96
104
112
120 81 99
108
117
126
135
100
110
130
140
150
121
132
143
154
165
144
156
168
180
169
182
195
196
210
225
Addition
table
Multiplication table
1983 Erdos-Szemeredi, 2004 Bourgain-Katz-Tao:
Every window will be rich in one of these tables!
von Neumann”Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin.”

23. Independent-source extractors
Unbiased,
independent
Applications:
Analyzed on perfect randomness
Statistics, Cryptography, Algorithms,
Game theory, Gambling,……
Extractor
biased,
dependent
Reality:
Independent
weak sources of randomness
Stock market
fluctuations
Sun spots
Radioactive
decay +
Extractors are designed to purify the weak (biased, correlated) randomness appearant in physical phenomena, and convert them efficiently to the uniform distribution, needed for many applications.
As it turns out, they are useful in other contexts, like error correcting codes, data structures, network design and algorithms. X
Theorem [Barak-Impagliazzo-W ’05]:
Extractor for independent sources

24. Single-source extractors
algorithm A
input
output correct with high probability
n unbiased,
independent
n biased,
dependent
algorithm A
input
output correct with high probability??
Randomness in the eye of the beholder
Pragmatic definition
Outputs equal up to 1% (doesn’t matter as we had correctness 99%)
Coarsening of L1 metric – efficient tests
Naïve implementation fails!
Reality: one weak random source

25. Single-source extractors
Probabilistic algorithms with 1 weak random source
Many other applications:
- Hardness of approx
- Derandomization,
- Data structures
Error correction …
algorithm A
input
output correct with high probability
n unbiased,
independent
Algorithm A’
input
output correct whp!!
Run A on each, and
take a majority vote
B, SV, AKS
CW, IZ, NZ,
Ta, Tr, ISW,
………….
LRVW, GUV
Dv, DW,…
n3 biased,
dependent
entropy n2
A perfect
sample
Extractor

26. Deterministic de-randomization
Hardness vs. Randomness
Efficient
algorithm A
input
output correct with high probability
n unbiased,
independent
All efficient probabilistic algorithms have efficient deterministic counterparts
Efficient Algorithm A’
input
output correct always!
Run A on each, and
take a majority vote
BM, Y, …
………….
NW, IW,…
…………
IKW, KI,...
“P ≠ NP”
TSP is
hard
A perfect
sample

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