Transcript Identification, Entropy and One

Theoretical Cryptography
Lecture 1: Introduction, Standard Model of
Cryptography, Identification, One-way functions
Lecturer: Moni Naor
Weizmann Institute of Science
What is Cryptography?
Traditionally: how to maintain secrecy in communication
Alice and Bob talk while Eve tries to listen
Bob
Alice
Eve
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History of Cryptography
• Very ancient occupation
Biblical times ;‫הָ אָ ֶרץ‬-‫ ו ִַׁת ָתפֵ ׂש ְת ִהלַׁ ת כָל‬,‫אֵ יְך נִ ְלכְ ָדה ֵש ַׁשְך‬
.‫ בַׁ ּגֹויִם‬,‫ְתה ְל ַׁש ָמה בָ בֶ ל‬
ָ ‫אֵ יְך הָ י‬
• David Kahn, The Codebreakers, 1967
Atbash
• Egyptian
• Gaj Hieroglyphs
and Orlowski, Facts and Myths of Enigma: ‫אתבש‬
– Unusual
ones Stereotypes Eurocrypt 2003
Breaking
...
Not the subject of this course!
• 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
– Exception: Shannon, Turing
• Since then - explosive growth
Focus of this course
– 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.
Prevalence of the Internet:
•Cryptography is in the news (daily!)
•Cryptography is relevant to ``everyone” - security and privacy issues
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for individuals
Computational Complexity Theory
• Study the resources needed to solve computational
problems
–
–
–
–
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Computer time
Computer memory
Communication
Parallelism
Randomness
…
A computational problem:
•multiplying two numbers,
•selecting a move in a chess position
•Find the shortest tour visiting all cities
• Identify problems that are infeasible to compute by any
reasonable machine
• Taxonomy: classify problems into classes with similar
properties wrt the resource requirements
P=NP?
– Help find the most efficient algorithm for a problem
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The Traveling Salesman problem
Find the shortest tour visiting all cities
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The Traveling Salesman problem
Find the shortest tour visiting all cities
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Factoring numbers
• Given two large (prime) numbers, producing the
product – an `easy’ computational problem
• Given the product of two large prime numbers,
finding them: a computationally difficult problem
– Not quite exponential time, but still mot achieved for
thousand bit numbers
Current record: RSA 768
– Great progress since first considered for cryptography 35
years ago.
– Quantum computers – can factor “efficiently”
• One of the most useful problems
for
cryptography
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Key Idea of Cryptography
Use the intractability of some problems for the
advantage of constructing secure system
Almost any cryptographic task requires using this idea.
Large research effort devoted to studying the relationship
between cryptography and complexity
Our goal is to investigate this relationship
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Administrivia
• Instructor: Moni Naor
• When: Tuesday 14:00--16:00
Where: Ziskind 1 ?
Home page of the course:
www.wisdom.weizmann.ac.il/~naor/COURSE/theoretical_crypto.html
• METHOD OF EVALUATION: several homework assignments and a
final (in class) exam. Also must prepare notes for (at least) one lecture.
– Homework assignments should be turned in on time!
– Try and do as many problems from each set.
– You may discuss the problems with other students, but the write-up should be
individual.
– There will also be reading assignments.
Official Description
• Cryptography deals with methods for protecting the privacy,
integrity and functionality of computer and communication
systems.
• The goal of the Theoretical Cryptography course is to
address the foundations of cryptography and in particular
the relationship with computational complexity theory.
Topics Covered
• The standard model of cryptography,
• Notions of security of a cryptosystem
– signatures and encryption schemes,
• Proof techniques for demonstrating security
• Cryptographic primitives:
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–
–
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One-way functions and
Trapdoor permutations,
Zero-knowledge proofs,
Fully homomorphic encryption and secure function
evaluation.
Relationship with “Practical Cryptography”
• A sequence of two courses in cryptography will be
offered this year (at the same time slot):
• "Theoretical Cryptography" taught by Moni Naor in
the first semester and
• "Practical Cryptography", taught by Adi shamir in
the second semester
• These are two independent courses but
complimentary.
• Attending both is highly recommended
What you will learn in this course
• How to specify a cryptographic task
• How to specify a solution
• Relationship with complexity assumptions
Lectures Outline
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Identification, Authentication and encryption
One-way functions and their essential role in cryptography
Amplification: from weak to strong one-way functions
Universal hashing and authentication.
One-way hashing
Signature Scheme: Existentially unforgeability
Pseudo-randomness:
– Pseudo-random generators
– Hardcore predicates,
– Pseudo-Random Functions and Permutations.
•
•
•
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Semantic Security and Indistinguishability of Encryptions.
Zero-Knowledge Proofs and Arguments
Chosen ciphertext attacks and non-malleability
Fully Homomorphic Encryption
Oblivious Transfer and Secure Function Evaluation
Typical Scenario in Cryptography
Want to maintain secrecy in communication
Alice and bob talk while Eve tries to listen
Alice
Bob
Eve
Modeling an Attack
Foundations of Cryptography: Rigorous
specification of security of protocols


The power of the adversary
 Access to the system
 Computational power
What it means to break the system
Ek(m)
“Standard model”
Adversarial Models
STANDARD MODEL:




Abstract models of computation
Interactive Turing machines
Private memory, randomness
 ...
Well-defined adversarial access
 Can model powerful attacks
REAL LIFE:


Physical implementations leak information
Adversarial access not always captured by
abstract models
Ek(m)
Adversarial Models
Attacks - standard model:
 Chosen-plaintext attacks
 Chosen-ciphertext attacks
 Composition
 Self-referential encryption
 Circular encryption

....
Attacks outside standard model:
 Timing attacks [Kocher 96]
 Fault detection [BDL 97, BS 97]
 Power analysis [KJJ 99]
 Cache attacks [OST 05]
 Memory attacks [HSHCPCFAF 08]

Ek(m)
...
Adversarial Models
Attacks - standard model:
 Chosen-plaintext attacks
 Chosen-ciphertext attacks
 Composition
 Self-referential encryption
 Circular encryption

....
Attacks outside standard model:
 Timing attacks [Kocher 96]
 Fault detection [BDL 97, BS 97]
 Power analysis [KJJ 99]
 Cache attacks [OST 05]
 Memory attacks [HSHCPCFAF 08]

...
Side channel:
Any information not captured by the
abstract “standard” model
Adversarial Models
http://xkcd.com/538/
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 channel
Bob has two external states {N,Y}
Eve completely controls the channel
Requirements:
• If Alice wants to approve and Eve does not interfere –
Bob moves to state Y
• If Alice does not approve, then for any behavior from
Eve, Bob stays in 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 only with low probability.
– How low? Related to the string length that Alice sends…
Identification
X
X
Alice
Bob
Eve
??
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 Y
• If not equal moves permanently to N
Eve’s probability of success
• If Alice did not send X and Eve put some string X’
on the channel, then
– Bob moves to Y 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 over some set of symbols Γ
• What is Eve’s best strategy?
• What is Eve’s probability of success
(Shannon) Entropy
Let X be random variable over alphabet Γ with distribution Px
The (Shannon) entropy of X is
H(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 is when x is constant
– All other cases H(x) >0
• If Γ = {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 Γ = {0,1}n and X is uniformly distributed, then
H(X) = - ∑
n
x  {0,1}n 1/2
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 1{0,1} n-1 Prob[X=x ] = 1/2n
Then H(X) = n/2+1/2
But 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 is
Hmin(X) = - log max x Γ Px (x)
The min entropy represents the most likely value of X
Property: 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!
Good source on Information Theory:
T. Cover and J. A. Thomas, Elements of Information Theory
One-time vs. many times
• This was good for a single identification. What
about many sessions of identification?
• Later…
A different scenario – now Charlie is
involved
• Bob has no proof that Alice indeed identiferd herself
(`approved’).
• 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
• Relevant when they are many possible verifiers!
The new requirement
• If Alice wants to approve and Eve does not
interfere – Bob moves to state Y
• If Alice does not approve, then for any behavior
from Eve and Charlie, Bob stays in N
• Similarly if Bob and Charlie are switched
Charlie
Alice
Bob
Eve
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 - complexity
Alice 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
Computational Models
• Asymptotic: Turing Machines with random tape
– For classical models: precise model does not matter up to
polynomial factor
Random tape
1 0
Both algorithm for
evaluating f and the
adversary are
modeled by PTM
1 1 0
1 0
Input tape
One-way functions - asymptotic
A function f: {0,1}* → {0,1}* is called a one-way function, if
• f is a polynomial-time computable function
– Also polynomial relationship between input and output length
• 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
Computational Models
• Concrete : Boolean circuits (example)
– precise model makes a difference
– Time = circuit size
Input
Output
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,
circuit
Prob[A(f(x))  f-1(f(x)) ] ≤ ε
Where x is chosen uniformly in {0,1}n and the probability is also
over the internal coin flips of A
Can 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 Lf :
– Consisting of strings of the form {y, b1, b2,…,bk}
– There is an x  {0,1}n such that y=f(x) and
– The first k bits of x are b1, b2…bk
Lf is NP – guess x and check
If Lf is P then f is invertible in polynomial time:
Self reducibility
A few properties and questions concerning
one-way functions
• Major open problem: connect the existence of one-way functions and
the 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?
– good exercise!
• If f’ is a one-way function, is f’ where f’(x) is f(x) with the last bit
chopped necessarily 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) necessarily a one-way function?
– good exercise!
• If f is a one way function, is g(x) = f(f(x)) necessarily a one-way
function?
– good exercise!
Solution to the password problem
• Assume that
– f: {0,1}n → {0,1}n is a (t,ε) one-way function
– Adversary’s run times is bounded by t
• Setup phase:
– Alice chooses xR {0,1}n
– computes y=f(x)
– Gives y to Bob and Charlie
• 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 Y
– If not equal moves permanently to state N
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 Y 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
A’
Eve y
x’
y
x’
The time and probability of success of
breaking the identification scheme by Eve
same as
The time and probability of inverting f by A’
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 (password=key and encryption of ‘0’)
Reductions
• This is a simple example of a reduction
• Simulate Eve’s algorithm in order to break the oneway function
• Most reductions are much more involved
– Do not preserve the parameters so well
Cryptographic Reductions
Show how to use an adversary for breaking primitive 1
in order to break primitive 2
Important
• Run time: how does T1 relate to T2
• Probability of success: how does 1 relate to 2
• Access to the system 1 vs. 2
Examples of One-way functions
Examples of hard problems:
• Subset sum
• Discrete log
• Factoring (numbers, polynomials) into prime
components
Easy problem
How do we get a one-way function out of them?
Subset Sum
• Subset sum problem: given
– n numbers 0 ≤ a1, a2 ,…, an ≤ 2m
– Target sum T
– Find subset S⊆ {1,...,n} ∑ i S ai,=T
• (n,m)-subset sum assumption: for uniformly chosen
– a1, a2 ,…, an R{0,…2m -1} and S⊆ {1,...,n}
– For any probabilistic polynomial time algorithm, the probability of finding S’⊆
{1,...,n} such that
∑ i S ai= ∑ i S’ ai
is negligible, where the probability is over the random choice of the ai‘s, S and the
inner coin flips of the algorithm
– Not true for very small or very large m
Assumption  f is one way
• Subset sum one-way function f:{0,1}mn+n → {0,1}mn+m
f(a1, a2 ,…, an , b1, b2 ,…, bn ) =
(a1, a2 ,…, an , ∑ i=1n bi ai mod 2m )
Exercise
• Show a function f such that
– if f is polynomial time invertible on all inputs, then
P=NP
– f is not one-way
Discrete Log Problem
• Let G be a group and g an element in G.
• Let y=gz and x the minimal non negative
integer satisfying the equation.
x is called the discrete log of y to base g.
• Example: y=gx mod p in the multiplicative group of Zp
• In general: easy to exponentiate via repeated squaring
– Consider binary representation
• What about discrete log?
– If difficult, f(g,x) = (g, gx ) is a one-way function
Integer Factoring
• Consider f(x,y) = x • y
• Easy to compute
• Is it one-way?
– No: if f(x,y) is even can set inverse as (f(x,y)/2,2)
• If factoring a number into prime factors is hard:
– Specifically given N= P • Q , the product of two random large (n-bit) primes, it
is hard to factor
– Then somewhat hard – there are a non-negligible fraction of such numbers ~
1/n2 from the density of primes
– Hence a weak one-way function
• Alternatively:
– let g(r) be a function mapping random bits into random primes.
– The function f(r1,r2) = g(r1) • g(r2) is one-way