Transcript Foundations of Cryptography Lecture 1: Introduction, Identification, One-way functions Lecturer: Moni Naor Weizmann Institute of Science.

Foundations of Cryptography
Lecture 1: Introduction, 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
History of Cryptography
• Very ancient occupation
Biblical times ‫איך נלכדה ששך ותתפש תהלת כל הארץ‬
‫איך היתה לשמה בבל בגויים‬
• 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
– Exception: Shannon, Turing*
• 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.
Cryptography and Complexity
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!
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.
Our goal is to investigate this relationship
Administrivia
• Instructor: Moni Naor
• Grader: Gil Segev
• When: Wednesday 16:00--18:00
Where: Ziskind 1
Home page of the course:
www.wisdom.weizmann.ac.il/~naor/COURSE/foundations_of_crypto.html
• METHOD OF EVALUATION: around 8 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 (usually two weeks after they
are given)!
– 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 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
What you will learn in this course
• How to specify a cryptographic task
• How to specify a solution
• Relationship with complexity assumptions
Lectures Outline
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Identification, Authentication and encryption
One-way functions and their essential role in cryptography
Universal one-way functions, multiple identifications,
Amplification: from weak to strong one-way functions
Universal hashing and authentication.
One-way hashing
Signature Scheme: Existentially unforgeability
Pseudo-random generators
Hardcore predicates,
Pseudo-Random Functions and Permutations.
Semantic Security and Indistinguishability of Encryptions.
Zero-Knowledge Proofs and Arguments
Chosen ciphertext attacks and non-malleability
Oblivous Transfer and Secure Function Evaluation
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 her 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 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) = - ∑
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 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
• Relevant when they are many of 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 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?
– 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 yy
x’
y
The time and probability of success of breaking
the identification scheme by Eve
x’
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.
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