The Power and Limitations of Randomness in Computational Complexity, Exams of Cryptography and System Security

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Randomness

A computational complexity view

Avi Wigderson

Institute for Advanced Study

Plan of the talk

•^

Computational complexity

-- efficient algorithms, hard and easy problems,

P vs. NP

-^

The power of randomness

-- in saving time

-^

The weakness of randomness

-- what is randomness? -- the hardness vs. randomness paradigm

-^

The power of randomness

-- in saving space -- to strengthen proofs

Theorem Proving and P vs. NP

Theorem:

If

Theorem proving

is Easy

then

Factoring

is Easy

P vs. NP problem: Formal:

Is Theorem proving Easy?

Informal: Can creativity be efficiently automated?

Theorem proving : Input: a mathematical statement S (

e.g. Riemann’s hypothesis

and an integer n

Task: Find a proof of S (

e.g. in ZF

) of length

≤n

(if exists)

Theorem [Cook-Levin ‘71]:Theorem proving is NP-complete …. Numerous equally hard problems in all sciences

Fundamental question

Is NP

≠P?

How fast can we solve:

• Factoring integers- Theorem proving- Computing the Permanent of a matrix -^

Deciding knottedness of a knot

-^

Solving a set of polynomial equations in finite fields

-^ …….Best known algorithms: exponential time/size.Is exponential time/size necessary for some?Conjecture 1 : YES

Coin Flips and Errors

Algorithms will make decisions using coin flips

(flips are independent and unbiased)

When using coin flips, we’ll guarantee:“task will be achieved, with probability >99%”Why tolerate errors? •^ We tolerate uncertainty in life •^ Here we can reduce error arbitrarily <exp(-n) •^ To compensate – we can do much more…

Number Theory: Primes

Problem 1 [Gauss]: Given x

∈[

n^ , 2

n+

], is x prime?

1975 [Solovey-Strassen, Rabin] : Probabilistic 2002 [Agrawal-Kayal-Saxena]: Deterministic !! Problem 2: Given n, find a prime in [

n^ , 2

n+

]

Algorithm: Pick at random x

, x 1

,…, x 2

1000n

For each x

apply primality test.i^

Prime Number Theorem

Pr [

i x

prime] > .99i^

Analysis: Fourier coefficients

Given (implicitely) a function f:(Z

n ) 2

(e.g. as a formula), and

Find all characters

such that |<f,

Comment : At most 1/

such

Algorithm [Goldreich-Levin ‘89] : …adaptive sampling…

Pr[ success ] >.

[AGS] : Extension to other Abelian groups. Applications: Coding Theory, Complexity Theory

Learning Theory, Game Theory

Geometry: Estimating Volumes

Algorithm [Dyer-Frieze-Kannan ‘91]:Approx counting

≈^

random sampling

Random walk inside K.Rapidly mixing Markov chain.

Analysis: Spectral gap

≈^

isoperimetric inequality

Applications: Statistical Mechanics, Group Theory

Given (implicitly) a convex body K in R

d^ (d large!)

(e.g. by a set of linear inequalities) Estimate

volume (K)

Comment: Computing volume(K) exactly is #P-complete

Hardness vs. Randomness

Theorems [Blum-Micali,Yao,Nisan-Wigderson,

Impagliazzo-Wigderson…] :

If there are natural hard problems Then randomness can be efficiently eliminated. Theorem [Impagliazzo-Wigderson ‘98] NP requires exponential

size

BPP=P

(every probabilistic polynomial-time algorithm has a deterministic counterpart) Theorem [IKW ’04, Impagliazzo-Kabanets ‘05] : Partial converse! Derandomization

Hardness

Computational Pseudo-Randomness

pseudorandom if

for

every efficient

algorithm, for

every input,

output

≈output

efficientdeterministic

none

pseudo-randomgenerator

algorithm

input

output

manyunbiasedindependent

n

algorithm

input

outputmanybiaseddependent

n

few

k ~ c log n

Derandomization

G^

efficient algorithm deterministicpseudo-randomgenerator

input

output^ n k ~ c log n

Deterministic algorithm: -^ Try all possible 2

k=n

c^ “seeds”

-^ Take majority vote^ Pseudorandomness

paradigm:

Can derandomize specific algorithms without assumptions! e.g. Primality Testing & Maze exploration

The Power of Randomness

In other settings…

Probabilistic Proof System

[Goldwasser-Micali-Rackoff, Babai ‘85]

Is a mathematical statement claim true? E.g.claim:

“No integers x, y, z, n>2 satisfy x

n^ +y

n^ = z

n “

claim:

“The Riemann Hypothesis has a 200 page proof” An efficient Verifier V(claim, argument) satisfies:) If claim is true then V(claim, argument) = TRUEfor some argument(in which case claim=theorem, argument=proof)*) If claim is false then V(claim, argument) = FALSEfor every argument

probabilistic

always with probability > 99%

Prover

Remarkable properties ofProbabilistic Proof Systems

• Probabilistically Checkable Proofs (PCPs) - Zero-Knowledge (ZK) proofs