RANDOMNESS AND PSEUDORANDOMNESSOmer Reingold, Microsoft Research and Weizmann

Randomness and Pseudorandomness

When Randomness is Useful When Randomness can be reduced or

eliminated – derandomization Basic Tool: Pseudorandomness

An object is pseudorandom if it “looks random” (indistinguishable from uniform), though it is not.

Expander Graphs

Randomness In Computation (1)

Distributed computing (breaking symmetry)

Cryptography: Secrets, Semantic Security, …

Sampling, Simulations, …

Can’t live without you

Randomness In Computation (2)

Communication Complexity (e.g., equality)

Routing (on the cube [Valiant]) - drastically reduces congestion

You change my world

Randomness In Computation (3)

In algorithms – useful design tool, but many times can derandomize (e.g., PRIMES in P). Is it always the case?

BPP=P means that every randomized algorithm can be derandomized with only polynomial increase in time

RL=L means that every randomized algorithm can be derandomized with only a constant factor increase in memory

Do I really need you?

In Distributed Computing

Byzantine Agreement

Deterministic

Randomized

Synchronoust failures

t+1 rounds O(1)

Asynchronous

impossible possible

Dining Philosophers: breaking symmetry

Attack Now

Don’t Attack

Randomness Saves Communication

Original File

Copy

=?

Deterministic: need to send the entire file! Randomness in the Sky: O(1) bits (or log in 1/error) Private Randomness: Logarithmic number of bits (derandomization).

In Cryptography

Private Keys: no randomness - no secrets and no identitiesEncryption: two encryptions of same message with same key need to be differentRandomized (interactive) Proofs: Give rise to wonderful new notions: Zero-Knowledge, PCPs, …

Random Walks and Markov Chains

When in doubt, flip a coin: Explore graph: minimal memory Page Rank: stationary

distribution of Markov Chains Sampling vs. Approx counting.

Estimating size of Web Simulations of Physical Systems …

Shake Your Input Communication network (n-dimensional cube)

Every deterministic routing scheme will incur exponentially busy links (in worse case)Valiant: To send a message from x y, select node z at random, send x z y.Now: O(1) expected load for every edge

Another example – randomized quicksort Smoothed Analysis: small perturbations, big

impact

In Private Data AnalysisHide Presence/Absence of Any IndividualHow many people in the database have the BC1 gene? Add random noise to true answer distributed as Lap(/) More questions? More privacy? Need more noise.

0 2 3 4--2-3-4

ratio bounded

Randomness and Pseudorandomness

When Randomness is Useful When Randomness can be reduced or

eliminated – derandomization Basic Tool: Pseudorandomness

An object is pseudorandom if it “looks random” (indistinguishable from uniform), though it is not.

Expander Graphs

Cryptography: Good Pseudorandom Generators are Crucial

With them, we have one-time pad (and more):

ciphertext = plaintext K = 01101011

E Dplaintext data:00001111

plaintext data: (ciphertext K) = 00001111

short key K0: 110

Without, keys are bad, algorithms are worthless (theoretical & practical)

derived key K:01100100

Data Structures & Hash Functions

Linear Probing:

Alice

Mike

Bob

Mary

Bob

F(Bob)

If F is random then insertion time and query time are O(1) (in expectation).

But where do you store a random function ?!? Derandomize!

Heuristic: use SHA1, MD4, … Recently (2007): 5-wise independent

functions are sufficient* Similar considerations all over: bloom

filters, cuckoo hashing, bit-vectors, …

Weak Sources & Randomness Extractors

Available random bits are biased and correlated Von Neumann sources:

Randomness Extractors produce randomness from general weak sources, many other applications

b1 b2 … bi … are i.i.d. 0/1 variables and bi =1 with some probability p < 1 then translate

01 1

10 0

Algorithms: Can Randomness Save Time or Memory?

Conjecture - No* (*moderate overheads may still apply)

Examples of derandomization:

Holdouts: Identity testing, approximation algorithms, …

Primality Testing in Polynomial Time

Graph Connectivity logarithmic Memory

(Bipartite) Expander Graphs

|(S)| A |S|

(A > 1)

S, |S| K

Important: every (not too large) set expands.

D

N N

(Bipartite) Expander Graphs

|(S)| A |S|

(A > 1)

S, |S| K

Main goal: minimize D (i.e. constant D)

• Degree 3 random graphs are expanders! [Pin73]

D

N N

(Bipartite) Expander Graphs

|(S)| A |S|

(A > 1)

S, |S| K

Also: maximize A. Trivial upper bound: A D

even A ≲ D-1 Random graphs: AD-1

D

N N

Applications of Expanders

These “innocent” looking objects are intimately related to various fundamental problems:

Network design (fault tolerance), Sorting networks, Complexity and proof theory, Derandomization, Error correcting codes, Cryptography, Ramsey theory And more ...

Non-blocking Network with On-line Path Selection [ALM]

N (Inputs)

N (Outputs)

Depth O(log N), size O(N log N), bounded degree.

Allows connection between input nodes and output nodes using vertex disjoint paths.

Non-blocking Network with On-line Path Selection [ALM]

N (Inputs)

N (Outputs)

Every request for connection (or disconnection) is satisfied in O(log N) bit steps:

• On line. Handles many requests in parallel.

The Network

“Lossless”

Expander

N (Inputs) N (outputs)

Slightly Unbalanced, “Lossless” Expanders

|(S)| 0.9 D |S|

S, |S| K

D

N M= N

0< 1 is an arbitrary constant D is constant & K= (M/D) = (N/D). [CRVW 02]: such expanders (with D = polylog(1/))

Property 1: A Very Strong Unique Neighbor Property

S, |S| K, |(S)| 0.9 D |S|

S

Non Unique neighbor

S has 0.8 D |S| unique neighbors !

Unique neighbor of S

Using Unique Neighbors for Distributed Routing

Task: match S to its neighbors (|S| K)

S

Step I: match S to its unique neighbors.

S`

Continue recursively with unmatched vertices S’.

Reminder: The Network

Adding new paths: think of vertices used by previous paths as faulty.

Property 2: Incredibly Fault Tolerant

S, |S| K, |(S)| 0.9 D |S|

Remains a lossless expander even if adversary removes (0.7 D) edges from each vertex.

Simple Expander Codes [G63,Z71,ZP76,T81,SS96]

M= N (Parity Checks)

Linear code; Rate 1 – M/N = (1 - ).Minimum distance K. Relative distance K/N= ( / D) = / polylog (1/). For small beats the Zyablov bound and is quite

close to the Gilbert-Varshamov bound of / log (1/).

N (Variables)

1

100

1

++

+

+

0

Error set B, |B| K/2

Algorithm: At each phase, flip every variable that “sees” a majority of 1’s (i.e, unsatisfied constraints).

Simple Decoding Algorithm in Linear Time (& log n parallel phases)

[SS 96]

M= N (Constraints)N (Variables)

++

+

+1

100

1|Flip\B| |B|/4|B\Flip| |B|/4 |Bnew| |B|/2

|(B)| > .9 D |B|

|(B)Sat|< .2 D|B|0

10

0

11

0

Random Walk on Expanders [AKS 87]

...x0

x1

x2 xi

xi converges to uniform fast (for arbitrary x0).

For a random x0: the sequence x0, x1, x2 . . . has interesting “random-like” properties.

Thanks