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Cryptography and Complexity at the Weizmann Institute

Moni Naor

Weizmann Institute of Science

Open Day February 2005

נאור מוני

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Computational Complexity Theory• Study the resources needed to solve computational problems

– Computer time– Computer memory– Communication– Parallelism– Randomness– …

• Identify problems that are infeasible to compute by any reasonable machine

• Taxonomy: classify problems into classes with similar properties wrt the resource requirements– Help find the most efficient algorithm for a problem

A computational problem:

•multiplying two numbers,

•selecting a move in a chess position

•Find the shortest tour visiting all cities

P=NP?

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What is Cryptography?

Traditionally: how to maintain secrecy in communication

Alice and Bob talk while Eve tries to listen

Alice Bob

Eve

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History of Cryptography

• Very ancient occupation Biblical times -

ותתפש תהלת כל הארץששךאיך נלכדה

בגוייםבבלאיך היתה לשמה

• Egyptian Hieroglyphs – Unusual ones

... • Many interesting books and sources, especially about the

Enigma (WW2)

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Modern Times• Up to the mid 70’s - mostly classified military work• Since then - explosive growth

– Commercial applications– Scientific work: tight relationship with Computational Complexity Theory

• Recently - more involved models for more diverse tasks.

How to maintain the secrecy, integrity and functionality in computer and communication system.

Prevalence of the Internet:•Cryptography is in the news (daily!)•Cryptography is relevant to ``everyone” - security and privacy issues for individuals

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Cryptographic Research

Complexity Theory -• Study the resources needed

to solve computational problems – computer time, memory

• Identify problems that are infeasible to compute.

Cryptography -• Find ways to specify security

requirements of systems• Use the computational

infeasibility of problems in order to obtain security.

The development of these two areas is tightly connected!

“A match made in heaven”

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Faculty members in Cryptography and Complexity

• Prof. Uri Feige

• Prof. Oded Goldreich

• Prof. Shafi Goldwasser

• Prof. Moni Naor

• Dr. Omer Reingold

• Prof. Ran Raz

• Prof. Adi Shamir

אורי פייגה

גולדרייך עודד

גולדווסר שפי

נאור מוני

ריינגולד עומר

רן רז

שמיר עדי

One of the most active groups in the world!

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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 m’ he receives is indeed the message m Alice sent.

Alice Bob

Eve

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

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

– Provided Alice (the sender) has a unique public key– Essential to contract signing, e-commerce…– First implementation: Rivest, Shamir and Adleman 1977

• Digital Signatures: last 25 years major effort in– Research

• Notions of security• Computationally efficient constructions

– Technology, Infrastructure (PKI), Commerce, Legal

Turing Award2003

Existential Unforgeability under an adaptive message attack

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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

Alternative: (Plausible) Deniability• If the recipient (or any recipient) could have generated the

conversation himself – or an indistinguishable one

Key concept in cryptography and complexity

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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.

– This property is known as Zero-Knowledge!– An example where zero-knowledge is the ends, not the means!

Proof of security consists of Unforgeability and Deniability

Yet another WIS concept

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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

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Deniable Ring AuthenticationCompleteness: 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

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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 indistinguishable

Source 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

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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

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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 {0,1}n. Send Y=E(PK, mx , r)• P V : Verify that prefix of plaintext is indeed m. If yes - send x.V accepts iff the received x’=x

Is it Unforgeable? Is it Deniable?

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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

There are encryption schemes satisfying the desired requirements

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Simulator for honest receiver

Choose x R {0,1}n.

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

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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 Y

Perfect 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

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Deniable Protocol P has a public key PK of an encryption scheme E.

To authenticate message m:

• V P: Choose xr{0,1}n.

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.

P commits to the value x. Does not want to reveal it yet

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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

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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

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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

...

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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 {0,1}n. 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

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The Ring Authentication ProtocolRing has public keys PK1

, PK2, …, PKn

of encryption scheme E

To authenticate message m with jth decryption key PSj:

V P: Choose x {0,1}n. 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

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Properties of the Scheme

• Works with any good encryption scheme - members of the ring are unwilling participants.

• Fairly efficient scheme:– Need n encryptions n verifications and one decryption

• Can extend the scheme so that convince a verifier that At least k members confirm the message.

• What are the social implications of the existence of ring authentication?

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Summary

• Cryptography and Complexity are very active research areas

• Research activities in the areas range from– providing firm foundations to the construction of methods – providing actual constructions and analysis for specific

needs.

• Many unexpected results...

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Security of the schemeUnforgeability: 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’