Complexity Theory

Lecture 10

Lecturer: Moni Naor

RecapLast week:

IP=PSPACEStart with #P in PSPACE

Public vs. Private CoinsIP[k]=AM

Mechanism for showing non NP-Completeness

This Week:Statistical zero-knowledge

AM protocol for VC dimension

Hardness and Randomness

Lack of certificate

Bug or feature?

Disadvantages clear, but:• Advantage: proof remains `property’ of prover and not

automatically shared with verifier• Very important in cryptographic applications

– Zero-knowledge • Many variants• Can be used to transform any protocol designed to work with benign players

into one working with malicious ones – The computational variant is useful for this purpose

• Can be used to obtain (plausible) deniability

Honest verifier perfect zero-knowledge

Perfect or Statistical Zero-KnowledgeAn interactive protocol for proving that x 2 L is honest verifier perfect zero-knowledge if

for all x 2 L: there exists a probabilistic polynomial time simulator S such that S(x) generates an output

identically distributed to the verifier’s view in a real execution:– transcript of (P,V) conversation– the random bits used by the verifier

• assume no erasures, otherwise can get it trivially If the distributions are nearly identical – Statistical Zero-Knowledge

Variation distance is negligible in program size

Example: the protocol for GNI is honest verifier perfect zero-knowledge.The transcript consist of a graph H graph and a bit r

The simulator: given (G0, G1)Pick random c 2R 0,1 and 2R S|V| Output simulated view: transcript hH = (Gc), r=ci random string: hc, i

The distribution is identical to the one generated by an honest verifier Some precautions need to be taken if the verifier might not be following the protocol

Detour: AuthenticationOne of the fundamental tasks of cryptography• Alice (sender) wants to send a message m to

Bob (receiver).• They want to prevent Eve from interfering

– Bob should be sure that the message he receives is indeed the message m Alice sent.

Alice Bob

Eve

Authentication and Non-Repudiation• Key idea of modern cryptography [Diffie-Hellman]:

can make authentication (signatures) transferable to third party - Non-repudiation.

– Essential to contract signing, e-commerce…• Digital Signatures: last 25 years major effort in

– Research• Notions of security• Computationally efficient constructions

– Technology, Infrastructure (PKI), Commerce, Legal

Is non-repudiation always desirable?

Not necessarily so:• Privacy of conversation, no (verifiable) record.

– Do you want everything you ever said to be held against you?

• If Bob pays for the authentication, shouldn't be able to transfer it for free

• Perhaps can gain efficiency

Deniable AuthenticationSetting:• Sender has a public key known to receiver• Want to come up with an (perhaps interactive) authentication

scheme such that the receiver keeps no receipt of conversation.

This means:• Any receiver could have generated the conversation itself.

– There is a simulator that for any message m and verifier V* generates an indistinguishable conversation.

– Similar to Zero-Knowledge!– An example where zero-knowledge is the ends, not the means!

Proof of security consists of Unforgeability and Deniability

Ring Signatures and AuthenticationCan we keep the sender anonymous?Idea: prove that the signer is a member of an ad hoc set

– Other members do not cooperate– Use their `regular’ public-keys

• Encryption – Should be indistinguishable which member of the set is

actually doing the authentication

Bob

Alice? Eve

Deniable Ring Authentication

Completeness: a good sender and receiver complete the authentication on any message m

Unforgeability Existential unforgeable against adaptive chosen message attack for any sequence of messages m1, m2,… mk

Adversarially chosen in an adaptive manner

Even if sender authenticates all of m1, m2,… mk

Probability forger convinces receiver to accept a m m1, m2,… mk

is negligible Properties of an interactive authentication scheme

Deniable Ring AuthenticationDeniability

– For any verifier, for any arbitrary set of keys, some good some bad, there is simulator that can generate computationally indistinguishable conversations.

• A more stringent requirement: statistically indistinguishableSource Hiding:

– For any verifier, for any arbitrary set of keys, some good some bad, the source is computationally indistinguishable among the good keys

• A more stringent requirement: statistically indistinguishable

Source Hiding and Deniability – incomparable

Encryption• Assume a public key encryption scheme E

– Public key PK – knowing PK can encrypt message m • generate Y=E(PK , m, r)

– With corresponding secret key PS, given Y can retrieve m • m =D(PS , Y)

• Encryption process is probabilisticEach message induces a distribution on the ciphertexts

• Security of encryption scheme:– non-malleable against chosen ciphertext attacks in the post-

processing mode.– In particular given Y=E(PK, m, r) hard to generate Y’=E(PK,

m’, r’) for a related message m’• Example of a very malleable scheme: one-time pad

Plaintext

Ciphertext

A Public Key Authentication Protocol

P has a public key PK of an encryption scheme E.To authenticate a message m:• V P : Choose x R 0,1n. Send Y=E(PK, mx , r)• P V : Verify that prefix of plaintext is indeed m. If yes - send x.V accepts if the received x’=x

Is it Unforgeable? Is it Deniable?

Security of the schemeUnforgeability: depends on the strength of E• Sensitive to malleability:

– if given E(PK, mx, r) can generate E(PK, m’x’, r) where m’ is related to m and x’ is related to x then can forge.

• The protocol allows a chosen ciphertext attack on E.– Even of the post-processing kind!

• Can prove that any strategy for existential forgery can be translated into a CCA strategy on E

• Works even against concurrent executions.Deniability: does V retain a receipt??

– It does not retain one for an honest V– Need to prove knowledge of x

Simulator for honest receiver

Choose x R 0,1n. Output: hY=E(PK, mx, r), x, ri

Has exactly the same distribution as a real conversation when the verifier is following the protocolStatistical indistinguishability

Verifier might cheat by checking whether certain ciphertext have as a prefix mNo known concrete way of doing harm this way

Encryption: Implementation

• Under any trapdoor permutation - rather inefficient [DDN].• Cramer & Shoup: Under the Decisional DH assumption

– Requires a few exponentiations.• With Random Oracles: several proposals

– RSA with OAEP - same complexity as vanilla RSA [Crypto’2001]– Can use low exponent RSA/Rabin

• With additional Interaction: J. Katz’s non malleable POKS?

Encryption as Commitment

When the public key PK is fixed and known Y=E(PK, x, r) can be seen as commitment to x

To open x reveal r, the random bits used to create YPerfect binding: from unique decryption

For any Y there are no two different x and x’ and r and r’ s.t. Y=E(PK, x, r) =E(PK, x’, r’)

Secrecy: no information about x is leaked to those not knowing private key PS

Deniable Protocol

P has a public key K of an encryption scheme E.To authenticate message m:

• V P: Choose xr0,1n.

Send Y=E(PK, mx , r)

• P V: Send E(PK, x, t)

• V P: Send x and r - opening Y=E(PK, mx, r)

• P V: Open E(PK, x , t) by sending t.

Security of the scheme

Unforgeability: as before - depends on the strength of E can simulate previous scheme (with access to D(PK , . ))Important property: E(PK, x, t) is a non-malleable commitment

(wrt the encryption) to x.Deniability: can run simulator:• Extract x by running with E(PK, garbage, t) and

rewinding• Expected polynomial time• Need the semantic security of E - it acts as a commitment

scheme

Ring Signatures and AuthenticationWant to keep the sender anonymous by proving

that the signer is a member of an ad hoc set – Other members do not cooperate– Use their `regular’ public-keys– Should be indistinguishable which member of the set

is actually doing the authentication

Bob

Alice? Eve

Ring Authentication Setting

• A ring is an arbitrary set of participants including the authenticator

• Each member i of the ring has a public encryption key PKi

– Only i knows the corresponding secret key PSi

• To run a ring authentication protocol both sides need to know PK1

, PK2, …, PKn

the public keys of the ring members

...

An almost Good Ring Authentication Protocol

Ring has public keys PK1, PK2

, …, PKn of encryption scheme E

To authenticate message m with jth decryption key PSj:

V P: Choose x 0,1n. Send E(PK1

, mx, r1), E(PK2, mx, r2), …, E(PKn

, mx, rn)

P V: Decrypt E(PKj, mx, rj), using PSj

and

Send E(PK1, x, t1), E(PK2

, x, t2), …, E(PKn, x, tn)

V P: open all the E(PKi, mx, ri) by

Send x and r1, r2 ,… rn

P V: Verify consistency and open all E(PKi, x, ti) by

Send t1, t2 ,… tn

Problem: what if not all suffixes (x‘s) are equal

The Ring Authentication Protocol

Ring has public keys PK1, PK2

, …, PKn of encryption scheme E

To authenticate message m with jth decryption key PSj:

V P: Choose x 0,1n. Send E(PK1

, mx, r1), E(PK2, mx, r2), …, E(PK1

, mx, rn)

P V: Decrypt E(PKj, mx, rj), using PSj

and

Send E(PK1, x1, t1), E(PK2

, x2, t2), …, E(PKn, xn, tn)

Where x=x1+x2 + xn

V P: open all the E(PKi, mx, ri) by

Send x and r1, r2 ,… rn

P V: Verify consistency and open all E(PKi, x, ti) by

Send t1, t2 ,… tn and x1, x2 ,…, xn

Complexity of the scheme

Sender: single decryption, n encryptions and n encryption verifications

Receiver: n encryptions and n encryption verifications

Communication Complexity: O(n) public-key encryptions

Security of the scheme

Unforgeability: as before (assuming all keys are well chosen) since

E(PK1, x1, t1), E(PK2

, x2, t2),…,E(PK1, xn, tn)

where x=x1+x2 + xn

is a non-malleable commitment to x

Source Hiding: which key was used (among well chosen keys) is – Computationally indistinguishable during protocol– Statistically indistinguishable after protocol

• If ends successfully

Deniability: Can run simulator `as before’

Promise ProblemsA promise problem L is similar to a language recognition problem except

that there is a set A• if x 2 A then should report correctly whether x 2 L or not • if x 2 A then do not care how algorithm responds

Example: unique sat A=|either is not statisfiable or has a unique satisfying

assumption

If A=0,1n, then this is the usual language recognition problem

O satisfying assignments1 satisfying assignment

Statistical Zero-KnowledgeSo if statistical zero-knowledge is so good, why not use it

all the time?

Definition: L is a promise problemSZK=L|L has a statistical zero-knowledge protocolHVSZK=L|L has an honest verifier statistical zero-

knowledge protocol

Clearly: SZK µ HVSZK since any protocol good against all verifiers is good against honest

ones as well

The surprising world of Statistical Zero-Knowledge• There are complete problems for HVSZK

– Entropy Difference (ED): given two circuits generating distributions D1 and D2 distinguish between the cases

H(D1) ¸ H(D2)+1 and H(D2) ¸ H(D1)+1

Any promise problem in HVSZK problem is Karp reducible to ED– ED has an honest verifier public-coins statistical zero-knowledge protocol in

constant number of rounds– VD(,) is also complete

• Conclusions: – All languages and promise problems in HVSZK have public coins protocols– HVSZK is closed under complementation

• If a promise problem L has a public coins statistical zero-knowledge proof system then it has one with good against general verifier

• Conclusions:– SZK is closed under complementation– SZK 2 AM Å Co-AM

H(D)=x PrD[x] log PrD[x]

Homework

• Show an AM protocol for Entropy Difference when the two distribution are promised also to be flat – For x and x’ such that Pr[x] ≠ 0 and Pr[x’] ≠ 0

we have Pr[x] = Pr[x’]

Is the protocol SZK or HVSZK?

Statistical Zero-KnowledgeTheorem: if a language L has a statistical zero-knowledge proof system,

then L 2 AM Å Co-AM

Conclusion: if interested in zero-knowledge proofs for all languages in NP need to either– Relax notion of proof: argument

• Prover is assumed to be a polynomial time machine– Having access to some secret information

• Such protocols exists assuming a certain kind of commitment exists– Based on one-way permutations

• Another possible relaxation of proof: assume that there are two provers who do not exchange information during the execution of the protocol

– Relax notion of zero-knowledge: computational• There exists a simulator which generates computationally indistinguishable

distributions• Such protocols exists for all of IP assuming one-way function exist

This led to PCP

Sources for Statistical Zero-Knowledge

• Salil Vadhan Thesis:www.eecs.harvard.edu/~salil/papers/phdthesis-abs.html

• Results are due to Goldreich, Sahai and Vadhan• Building on

– Fortnow and Aiello and Hastad: SZK µ AM Å Co-AM– Okamoto: SZK is closed under complementation

VC DimensionLet T=(U,S) be a set system

– Each s 2 S is a subset of ground set U We say that X µ U is shattered by T if

8 X’ µ X there is an s 2 S is such that X’=s Å X

The VC-Dimension of T [Vapnik Chervonenkis]: the maximum cardinality of any set X shattered by a T

The complexity of determining the VC-Dimension of T=(U,S):If (U,S) is given explicitly, then since VC(U,S) · log |S|

Can determine in |U| log |S| timeKnown to be at as hard as solving sat with log2 n variables

If (U,S) is given implicitly, by a circuit C. For j 2 U and 1 · i · |S| the value C(i,j)=1 iff j 2 si and 0 otherwise

The problem of determining VC(C) is complete for 3P

Will see that showing an approximate lower bound on VC is in AM

AM Protocol for 2-approximating the VC Dimension

Conditions for large VC-dimensionSauer’s Lemma: in any set T=(U,S) such that

i=0m (|U|

i ) · |S|we have that m+1 · VC(U,S).

Idea for protocol: Merlin gives the set X that is shattered by the set system encoded in circuit C.

If X is `mostly’ shattered,For more than half of X’ µ X there is an s 2 S is such that X’=s Å X

then there must be a large Y µ X that is fully shattered:

Consider T’=(X,S’) where each s’ 2 S’ is equal to s Å X for some s2S

If |S’| is large - at least ½ 2|X| ¸ i=0|X|/2 (|X|

i ) after removing duplicates, then

Sauer’s Lemma implies that VC(T’) ¸ |X|/2

Large collections have a lower bound on the VC dimension

Lemma is also due to Perles and Shelah

The protocolInput: circuit C encoding set system (U,S) and promise that

1. Either VC(U,S)¸ k • Arthur should always accept if Merlin follows the protocol

2. or VC(U,S)· k/2 • Arthur should reject with reasonable probability

The protocol:

Merlin: send set X µ U that is shattered by the set system encoded in C.

Arthur: Pick a random X’ µ X and send it

Merlin: send index 1 · i · |S| such that X’ = si Å X where si=j|C(i,j)=1

Arthur: verify that X’ = si Å X by running C on all j 2 X

We know that

VC(U,S) · log |S|

Open Problems• Many problems can be formulated as

– For a fixed huge graph• E.g. nodes correspond to permutations

– Given two nodes, what is the length of the shortest path between them

• Is it possible to apply the AM approach to show that various problems of finding distances in huge graphs are not NP-Hard.

– Either exact version or approximateExamples:• Pancake Problem

– Given n pancakes of various sizes in an arbitrary order what is the minimum number of flips to sort them

Genome Rearrangement problems:• Sorting by transpositions• Sorting by translocation

Idea for protocol

Verifier• Pick at random one of the two nodes as a starting

point• Make a random work of certain length and give

prover end pointProver:• Determine the which node is the starting point

References

• See Tzvika Hartman’s PhD thesis (Weizmann Institute)

The diameter of the Pancake Graph Program– Bill Gates and Christos Papadimitriou: Bounds For

Sorting By Prefix Reversal. Discrete Mathematics, vol 27, pp. 47-57, 1979.

– H. Heydari and H. I. Sudborough: On the Diameter of the Pancake Network. Journal of Algorithms, 1997

What’s nextNatural development of Interactive Proofs:• PCP:

Probabilistically Checkable Proofs– The prover sends the a polynomial sized proof – The verifier checks only small parts of itA new characterization of NPCan reduce any problem

• Main applications:– Inapproximability of many optimization problems– Low communication cryptographic protocols

Get back to it towards end of course

Derandomization

A major research question:• How to make the construction of

– Small Sample space `resembling’ large one– Hitting sets

Efficient.

Successful approach: randomness from hardness– (Cryptographic) pseudo-random generators– Complexity oriented pseudo-random generators

Recall: Derandomization ITheorem: any f 2 BPP has a polynomial size

circuitSimulating large sample spaces• Want to find a small collection of strings on

which the PTM behaves as on the large collection – If the PTM errs with probability at most , then

should err on at most + of the small collection• Choose m random strings• For input x event Ax is more than (+) of

the m strings fail the PTM

Pr[Ax] · e-22m < 2-2n

Pr[[x Ax] · x Pr[Ax] < 2n 2-2n=1

Good 1-

Bad

Collection that should resemble probability of success on ALL inputs

Chernoff

Pseudo-random generators

• Would like to stretch a short secret (seed) into a long one• The resulting long string should be usable in any case

where a long string is needed– In particular: cryptographic application as a one-time pad

• Important notion: IndistinguishabilityTwo probability distributions that cannot be distinguished– Statistical indistinguishability: distances between probability

distributions– New notion: computational indistinguishability

Computational Indistinguishability

Definition: two sequences of distributions Dn and D’n on 0,1n are computationally indistinguishable iffor every polynomial p(n) for every probabilistic polynomial time

adversary A for sufficiently large nIf A receives input y 0,1n and tries to decide whether y was

generated by Dn or D’n then

|Prob[A=‘0’ | Dn ] - Prob[A=‘0’ | D’n ] | < 1/p(n)

Without restriction on probabilistic polynomial tests: equivalent to variation distance being negligible∑β 0,1n |Prob[ Dn = β] - Prob[ D’n = β]| < 1/p(n)

advantage

Pseudo-random generatorsDefinition: a function g:0,1* → 0,1* is said to be a

(cryptographic) pseudo-random generator if• It is polynomial time computable • It stretches the input |g(x)|>|x|

– denote by ℓ(n) the length of the output on inputs of length n• If the input (seed) is random, then the output is indistinguishable from

randomFor any probabilistic polynomial time adversary A that receives input y of length ℓ(n)

and tries to decide whether y= g(x) or is a random string from 0,1ℓ(n) for any polynomial p(n) and sufficiently large n

|Prob[A=`rand’| y=g(x)] - Prob[A=`rand’| yR 0,1ℓ(n)] | < 1/p(n)

Want to use the output a pseudo-random generator whenever long random strings are used

Anyone who considers arithmetical methods of producing random numbers is, of course, in a state of sin. J. von Neumann

Pseudo-random generatorsDefinition: a function g:0,1* → 0,1* is said to be a

(cryptographic) pseudo-random generator if• It is polynomial time computable • It stretches the input g(x)|>|x|

– denote by ℓ(n) the length of the output on inputs of length n• If the input is random the output is indistinguishable from random

For any probabilistic polynomial time adversary A that receives input y of length ℓ(n) and tries to decide whether y= g(x) or is a random string from 0,1ℓ(n) for any polynomial p(n) and sufficiently large n

|Prob[A=`rand’| y=g(x)] - Prob[A=`rand’| yR 0,1ℓ(n)] | < 1/p(n)

Important issues: • Why is the adversary bounded by polynomial time?• Why is the indistinguishability not perfect?

Pseudo-Random Generators and Derandomization

All possible strings of length k

A pseudo-random generator mapping k bits to n bits strings

Any input should see roughly the same fraction of accept and rejects

The result is a derandomization of a BPP algorithm by taking majority

Complexity of Derandomization

• Need to go over all 2k possible input string• Need to compute the pseudo-random generator on

those points

• The generator has to be secure against non-uniform distinguishers:– The actual distinguisher is the combination of the

algorithm and the input• If we want it to work for all inputs we get the non-uniformity

Construction of pseudo-random generatorsrandomness from hardness

• Idea: for any given a one-way function there must be a hard decision problem hidden there

• If balanced enough: looks random • Such a problem is a hardcore predicate• Possibilities:

– Last bit– First bit– Inner product

Hardcore PredicateDefinition: let f:0,1* → 0,1* be a function. We say that h:

0,1* → 0,1 is a hardcore predicate for f if • It is polynomial time computable • For any probabilistic polynomial time adversary A that receives input

y=f(x) and tries to compute h(x) for any polynomial p(n) and sufficiently large n

|Prob[A(y)=h(x)] - 1/2| < 1/p(n)where the probability is over the choice y and the random coins of A

• Sources of hardcoreness: – not enough information about x

• not of interest for generating pseudo-randomness– enough information about x but hard to compute it

Single bit expansion

• Let f:0,1n → 0,1n be a one-way permutation

• Let h:0,1n → 0,1 be a hardcore predicate for f

Consider g:0,1n → 0,1n+1 whereg(x)=(f(x), h(x))

Claim: g is a pseudo-random generatorProof: can use a distinguisher for g to guess h(x)

f(x), h(x)) f(x), 1-h(x))

From single bit expansion to many bit expansion

• Can make r and f(m)(x) public – But not any other internal state

• Can make m as large as needed

x f(x) h(x,r)

OutputInternal Configuration

r

f(2)(x)

f(3)(x)

Input

h(f(x),r)

h(f (2)(x),r)

h(f (m-1)(x),r)f(m)(x)

Two important techniques for showing pseudo-randomness

• Hybrid argument

• Next-bit prediction and pseudo-randomness

Hybrid argument

To prove that two distributions D and D’ are indistinguishable: • suggest a collection of distributions

D= D0, D1,… Dk =D’ such that

If D and D’ can be distinguished, there is a pair Di and Di+1 that can be distinguished.Advantage ε in distinguishing between D and D’ means advantage

ε/k between some Di and Di+1

Use such a distinguisher to derive a contradiction

Next-bit TestDefinition: a function g:0,1* → 0,1* is said to pass the next bit

test if• It is polynomial time computable • It stretches the input |g(x)|>|x|

– denote by ℓ(n) the length of the output on inputs of length n• If the input (seed) is random, then the output passes the next-bit test

For any prefix 0≤ i< ℓ(n), for any probabilistic polynomial time adversary A that receives the first i bits of y= g(x) and tries to guess the next bit, or any polynomial p(n) and sufficiently large n

|Prob[A(yi,y2,…, yi) = yi+1] – 1/2 | < 1/p(n)

Theorem: a function g:0,1* → 0,1* passes the next bit test ifand only if it is a pseudo-random generator

Complexity Oriented Pseudo-Random Generators

Cryptography• Only crude upper bound

on the time of the `user’– Distinguisher

• The generator has less computational power than the distinguisher

Derandomization• when derandomizing an

algorithm you have a much better idea about the resources– In particular know the run time

• The generator has more computational power– May be from a higher

complexity class Quantifier order switch

Ideas for getting better pseudo-random generators for derandomization

• The generator need not be so efficient

• For derandomizing – a parallel algorithm generator may be more sequential

• Example: to derandomize AC0 circuits can compute parities– A low memory algorithm may use more space

In particular we can depart from the one-way function assumption• Easy one-way hard the other

• The (in)distinguishing probability need not be so small– We are are going to take a majority at the end

Parameters of a complexity oriented pseudo-random generator

All functions of n• Seed length t• Output length m • Running time nc

• fooling circuits of size s • error ε Any circuit family Cn of size s(n) that tries to distinguish

outputs of the generator from random strings in 0,1m(n) has at most ε(n) advantage

Hardness Assumption: Unapproximable FunctionsDefinition: E = [k DTIME(2kn)

Definition: A family of functions f = fn fn:0,1n

0,1is said to be s(n)-unapproximable if for every family of circuits Cn of size s(n):

Prx[Cn(x) = fn(x)] ≤ ½ + 1/s(n).

s(n) is both the circuit size and the bound on the advantage

Example: if g is a one-way permutation and h is a hardcore function strong against s(n)- adversaries, then

f(y)=h(g-1(y)) is s(n)-unapproximable

Average hardness notion

One bit expansion Assumption: f = fn is

– s(n)-unapproximable, for s(n) = 2Ω(n)

– in E

Claim: G = Gn:Gn(y) = yflog n(y)

is a single bit expansion generator family

Proof: suppose not, then • There exists a predictor that computes flog n with probability

better than ½ + 1/s(log n) on a random input• As in the single bit expansion using hardcore functions

Parametersseed length t = log noutput length m=log n + 1 fooling circuits of size s nδ running time nc

error ε =1/nδ < 1/m

Getting Many Bits Simultaneously Try outputting many evaluations of f on various parts of the seed:

Let bi(y) be a projection of y and consider

G(y) = f(b1(y))f(b2(y))…f(bm(y))

• Seems that a predictor must evaluate f(bi(y)) to predict ith bit

• But: predictor might use correlations without having to compute f

• Turns out: sufficient to decorrelate the bi(y)’s in a pairwise manner

If |y|=t and S µ 1...t we denote by y|S the sequence of bits of y whose index is in S

Nearly-Disjoint SubsetsDefinition: a collection of subsets

S1,S2,…,Sm 1…t

is an (h, a)-design if:– Fixed size: for all i, |Si| = h

– Small intersection: for all i ≠ j, |Si Å Sj| ≤ a

1...t

S1

S2

S3

Each of size h

Parameters: (m,t,h,a)Each intersection of size · a

Nearly-Disjoint Subsets

Lemma: for every ε > 0 and m < n can construct in poly(n) time a collection of subsets S1,S2,…,Sm 1…t

which is a (h, a)-design with the following parameters:• h = log n, • a = εlog n design • t is O(log n).

Both a proof of existence and a sequential construction

Nearly-Disjoint SubsetsProof: construction in a greedy manner

repeat m times:– pick a random (h=log n)-subset of 1…t– set t = O(log n) so that:

• expected overlap with a fixed Si is ½εlog n

• probability overlap with Si is larger than εlog n is at most 1/m– Can get by picking a single element independently from t’ buckets of size ½ εFor Si event Ai is: intersection is larger than (1/2+)t’

Pr[Ai] · e-22t’ < 2-log m

– Union bound: some h-subset has the desired small overlap with all the Si picked so far

– find the good h-subset by exhaustive search

Other construction of designs

• Based on error correcting codes

• Simple construction: based on polynomials

The NW generator Nisan-Wigderson

Need:• f E that is s(n)-unapproximable, for s(n) = 2δn

• A collection S1,…,Sm 1…t

which is an (h,a) design with h=log n, a = δlog n/3 and t = O(log n)

Gn(y)=flog n(y|S1)flog n(y|S2

)…flog n(y|Sm)

010100101111101010111001010

flog n:

seed yA subset Si

The NW generator

Theorem: G=Gn is a complexity oriented pseudo-random generator with:

– seed length t = O(log n)– output length m = nδ/3

– running time nc for some constant c– fooling circuits of size s = m– error ε = 1/m

The NW generator

• Proof:– assume G=Gn does not -pass a statistical test C =

Cm of size s:|Prx[C(x) = 1] – Pry[C( Gn(y) ) = 1]| > ε

– can transform this distinguisher into a predictor A of size s’ = s + O(m):

Pry[A(Gn(y)1…i-1) = Gn(y)i] > ½ + ε/m

• just as in the next-bit test using a hybrid argument

Proof of the NW generator

Pry[A(Gn(y)1…i-1) = Gn(y)i] > ½ + ε/m

– fix bits outside of Si to preserve advantage:

Pry’[A(Gn(y’)1…i-1) = Gn(y’)i] > ½ + ε/m and is the assignment to 1…t\Si maximizing the advantage of A

Gn(y)=flog n(y|S1)flog n(y|S2

)…flog n(y|Sm)

010100101111101010111001010

flog n:

Y’ Si

Proof of the NW generator

– Gn(y’)i is exactly flog n(y’)– for j ≠ i, as y’ varies, y’|Sj

varies over only 2a values!

• From the small intersection property• To compute Gn(y’)j need only a lookup table of size 2a

hard-wire (m-1) tables of 2a values to provide all Gn(y’)1…i-1

Gn(y)=flog n(y|S1)flog n(y|S2

)…flog n(y|Sm)

010100101111101010111001010

flog n:

y ’ Si

The Circuit for computing fCan construct a small circuit for approximating f from A

A

output flog n(y’)

y’

Properties

• size: s + O(m) + (m-1)2a

< s(log n) = nδ

• advantage

ε/m=1/m2 > 1/s(log n) = n-δ

hardwired tables

1 2 3 i-1

References

• Theory of cryptographic Pseudo-randomness developed by:– Blum and Micali, next bit test, 1982– Computational indistinguishability, Yao, 1982,

• The NW generator:– Nisan and Wigderson, Hardness vs. Randomness,

JCSS, 1994. – Some of the slides on the topic: Chris Umans’ course

Lecture 9