Foundations of CryptographyLecture 1

Lecturer: Moni Naor

What is Cryptography? Traditionally: how to maintain secrecy in communication

Alice and Bob talk while Eve tries to listen

AliceBob

Eve

History of Cryptography• Very ancient occupation Biblical times -

ותתפש תהלת כל הארץששךאיך נלכדה

בגוייםבבלאיך היתה לשמה • Many interesting books and sources, especially about

the Enigma– David Kahn, The Codebreakers, 1967– Gaj and Orlowski, Facts and Myths of Enigma: Breaking

Stereotypes Eurocrypt 2003 • Not the subject of this course

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– Major works: Diffie-Hellman, Rivest, Shamir and Adleman (RSA)

• Recently - more involved models for more diverse tasks.

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

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!

The interplay between these areas is the subject of the course

Cryptography and Complexity

Administrivia• Instructor: Moni Naor• Grader: Guy Rothblum • When: Thursday 14:00--16:00

Where: Ziskind 1Home page of the course: www.wisdom.weizmann.ac.il/~naor/COURSE/foundations_of_crypto.html • METHOD OF EVALUATION: around 12 homework assignments and a

final (in class) exam– Homework assignments should be turned in on time (usually two weeks after they

are given)!– Try and do as many problems from each set. – You may (and are encouraged to) discuss the problems with other students, but

the write-up should be individual.

Official Description

Cryptography deals with methods for protecting the privacy, integrity and functionality of computer and communication systems.

The goal of the course is to provide a firm foundation to the construction of such methods.

In particular we will cover topics such as notions of security of a cryptosystem, proof techniques for demonstrating security and cryptographic primitives such as one-way functions and trapdoor permutations

Sources

Books:Oded Goldreich, Foundations of Cryptography• Vol 1, Basic Tools, Cambridge ,2001 • Other volumes in www.wisdom.weizmann.ac.il/~oded/books.html

Web courses• Trevisan and Wagner: www.cs.berkeley.edu/~daw/cs276• Bellare and Rogaway: www.cs.ucsd.edu/users/mihir/cse207/index.html

Three Basic Issues in Cryptography

• Identification• Authentication• Encryption

Example: Identification

• When the time is right, Alice wants to send an `approve’ message to Bob.

• They want to prevent Eve from interfering – Bob should be sure that Alice indeed approves

Alice Bob

Eve

Rigorous Specification of Security

To define security of a system must specify:1. What constitute a failure of the system 2. The power of the adversary

– computational – access to the system– what it means to break the system.

Specification of the Problem

Alice and Bob communicate through a channelBob has two external states {N,YN,Y}Eve completely controls the channelRequirements:• If Alice wants to approve and Eve does not interfere –

Bob moves to state YY• If Alice does not approve, then for any behavior from

Eve, Bob stays in N N • If Alice wants to approve and Eve does interfere - no

requirements from the external state

Can we guarantee the requirements?

• No – when Alice wants to approve she sends (and receives) a finite set of bits on the channel. Eve can guess them.

• To the rescue - probability.– Want that Eve will succeed with low probability.– How low? Related to the string length that Alice

sends…

Example: Identification

Alice Bob

Eve

X X

??

Suppose there is a setup period

• There is a setup where Alice and Bob can agree on a common secret– Eve only controls the channel, does not see the internal state

of Alice and Bob (only external state of Bob)

Simple solution:– Alice and Bob choose a random string X R{0,1}n

– When Alice wants to approve – she sends X– If Bob gets any symbols on channel – compares to X

• If equal moves to YY• If not equal moves permanently to NN

Eve’s probability of success

• If Alice did not send X and Eve put some string X’ on the channel, then– Bob moves to YY only if X=X’

Prob[X=X’] ≤ 2-n

Good news: can make it a small as we wish

• What to do if Alice and Bob cannot agree on a uniformly generated string X?

Less than perfect random variables

• Suppose X is chosen according to some distribution Px cover some set of symbols Γ

• What is Eve’s best strategy?• What is her probability of success

(Shannon) Entropy

Let X be random variable over alphabet Γ with distribution Px

The (Shannon) entropy of X isH(X) = - ∑ x Γ Px (x) log Px (x)

Where we take 0 log 0 to be 0.

Represents how much we can compress X

Examples

• If X=0 (constant) then H(x) = 0– Only case where H(X)=0 when X is constnat– All other cases H(X) >0

• If X {0,1} and Prob[X=0] = p and Prob[X=1]=1-p, then

H(X) = -p log p + (1-p) log (1-p) ≡ H(p)If X {0,1}n and is uniformly distributed, then

H(X) = - ∑ x {0,1}n 1/2n log 1/2n = 2n/2n n = n

Properties of Entropy

• Entropy is bounded H(X) ≤ log | Γ | with equality only if X is uniform over Γ

Does High Entropy Suffice for Identification?

• If Alice and bob agree on X {0,1}n where X has high entropy (say H(X) ≥ n/2 ), what are Eve’s chances of cheating?

• Can be high: say – Prob[X=0n ] = 1/2– For any x{0,1} n-1 Prob[X=x ] = 1/2n

Then H(X) = n/2+1/2But Eve can cheat with probability at least ½ by

guessing that X=0n

Another Notion: Min Entropy

Let X be random variable over alphabet Γ with distribution Px

The min entropy of X isHmin(X) = - log max x Γ Px (x)

The min entropy represents the most likely value of XProperty: Hmin(X) ≤ H(X)Why?

High Min Entropy and Passwords

Claim: if Alice and Bob agree on such that Hmin(X) ≥ m, then the probability that Eve succeeds in

cheating is at most 2-m

Proof: Make Eve deterministic, by picking her best choice, X’ = x’. Prob[X=x’] = Px (x’) ≤ max x Γ Px (x) = 2 –Hmin(X) ≤ 2-m

Conclusion: passwords should be chosen to have high min-entropy!

One-time vs. many times

• This was good for a single identification. What about many identification?

• Later…

A different scenario – now Charlie is involved

• Bob has no proof that Alice indeed identified • If there are two possible verifiers, Bob and

Charlie, they can each pretend to each other to be Alice – Can each have there own string– But, assume that they share the setup phase

• Whatever Bob knows Charlie know• Relevent when they are many of them

The new requirement• If Alice wants to approve and Eve does not

interfere – Bob moves to state YY• If Alice does not approve, then for any behavior

from Eve and Charlie, Bob stays in N N • Similarly if Bob and Charlie are switched

Alice

Bob

Eve

Charlie

Can we achieve the requirements?• Observation: what Bob and Charlie received in the

setup phase might as well be public• Therefore can reduce to the previous scenario (with no

setup)…• To the rescue - complexityAlice should be able to perform something that neither Bob

nor Charlie (nor Eve) can do

Must assume that the parties are not computationally all powerful!

Function and inversions

• We say that a function f is hard to invert if given y= f(x) it is hard to find x’ such that y=f(x’) – x’ need not be equal to x– We will use f-1(y) to denote the set of preimages of y

• To discuss hard must specify a computational model • Use two flavors:

– Concrete– Asymptotic

One-way functions - asymptotics

A function f: {0,1}n → {0,1}n is called a one-way function, if• f is a polynomial-time computable function • for every probabilistic polynomial-time algorithm A, every positive

polynomial p(.), and all sufficiently large n’s

Prob[A[f(x)] f-1(f(x)) ] ≤ 1/p(n) Where x is chosen uniformly in {0,1}n and the probability is also over

the internal coin flips of A

One-way functions – concrete version

A function f: {0,1}n → {0,1}n is called a (t,ε) one-way function, if

• f is a polynomial-time computable function (independent of t)

• for every t-time algorithm A,

Prob[A[f(x)] f-1(f(x)) ] ≤ εWhere x is chosen uniformly in {0,1}n and the probability is

also over the internal coin flips of ACan either think of t and ε as being fixed or as t(n), ε(n)

Complexity Theory and One-way Functions

• Claim: if P=NP then there are no one-way functions

• Proof: for any one-way function f: {0,1}n → {0,1}n consider the language :– Consisting of strings of the form {y, b1, b2…bk} – There is an x {0,1}n s.t. f(x)=y and – The first k bits of x are b1, b2…bk

Lf is NP – guess x and checkIf Lf is P then f is invertable in polynomial time

A few properties and questions concerning one-way functions

• Major open problem: connect the existence of one-way functions and P=NP? question.

• If f is one-to-one it is a called a one-way permutation. In what complexity class does the problem of inverting one-way permutations reside? Homework

• If f is a one-way function, is f’ where f’(x) is f(x) with the last bit chopped a one-way function?

• If f is a one-way function, is fL where fL(x) consists of the first half of the bits of f(x) a one-way function? Homework

• If f is a one way function is g(x) = f(f(x)) necessarily a one-way function? Homework

Solution to the password problem• Assume that

– f: {0,1}n → {0,1}n is a (t,ε) one-way function– Adversaries run times is bounded by t

• Setup phase: Alice chooses x{0,1}n , computes y=f(x) and given Bob and Charlie y

• When Alice wants to approve – she sends X• If Bob gets any symbols on channel – call them z;

compute f(z) and compares to y– If equal moves to state YY– If not equal moves permanently to state NN

Eve’s and Charlie’s probability of success

• If Alice did not send x and Eve (Charlie) put some string x’ on the channel to Bob, then:

– Bob moves to state YY only if f(x’)=y=f(x) – But we know that

Prob[A[f(x)] f-1(f(x)) ] ≤ ε or else we can use Eve to break the one-way function

Good news: if ε can be made as small as we wish, then we have a good scheme.

• Can be used for monitoring• Similar to the Unix password scheme

– f(x) stored in login file– DES used as the one-way function.