1. Lecturer: Moni Naor Weizmann Institute of Science
Foundations of Cryptography Lecture 1: Introduction, Identification, One-way functions
Lecturer: Moni Naor
Weizmann Institute of Science
Give the handouts (course web page)

2. What is Cryptography?
Traditionally: how to maintain secrecy in communication
Alice and Bob talk while Eve tries to listen
Alice
Bob
Eve

3. 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!
Kahn Book was important to the development of the science of crypto, since this is what people (like Diffie) knew
The two historical reference are an early one and a late one.

4. 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.

5. 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.
Key idea in cryptography: Use the computational infeasibility of problems in order to obtain security
The development of these two areas is tightly connected!

6. 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

7. Administrivia Instructor: Moni Naor Grader: Gil Segev
When:    Wednesday 16:00--18:00 Where:   Ziskind 1
Home page of the course:
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.

8. 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

9. What you will learn in this course
How to specify a cryptographic task
How to specify a solution
Relationship with complexity assumptions

10. 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

11. Three Basic Issues in Cryptography
Identification
Authentication
Encryption

12. 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

13. Rigorous Specification of Security
To define security of a system must specify:
What constitute a failure of the system
The power of the adversary
computational
access to the system
what it means to break the system.

14. 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

15. 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…

16. Identification X X
Alice
Bob ??
Eve

17. 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

18. 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?

19. 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

20. H(X) = - ∑ x Γ Px (x) log Px (x)
(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

21. 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) = - ∑ x  {0,1}n 1/2n log 1/2n = 2n/2n n = n

22. Properties of Entropy
Entropy is bounded H(X) ≤ log | Γ | with equality only if X is uniform over Γ

23. 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

24. 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?

25. 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!

26. Good source on Information Theory:
T. Cover and J. A. Thomas, Elements of Information Theory

27. One-time vs. many times
This was good for a single identification. What about many sessions of identification?
Later…

28. 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!

29. 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

30. 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!

31. 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

32. Computational Models Asymptotic: Turing Machines with random tape
For classical models: precise model does not matter up to polynomial factor
Random tape 1 1 1 1
Both algorithm for evaluating f and the adversary are modeled by PTM
Input tape

33. 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

34. Computational Models Concrete : Boolean circuits (example)
precise model makes a difference
Time = circuit size
Input
Output

35. 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)) ] ≤ 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)
circuit

36. 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

37. 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?
If f is a one way function is g(x) = f(f(x)) necessarily a one-way function?

38. 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

39. 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
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.
The time and probability of success of breaking the identification scheme by Eve
same as
The time and probability of inverting f by A’ A’
Eve y y y x’ x’

40. Reductions This is a simple example of a reduction
Simulate Eve’s algorithm in order to break the one-way function
Most reductions are much more involved
Do not preserve the parameters so well

41. 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

42. Examples of One-way functions
Examples of hard problems:
Subset sum
Discrete log
Factoring (numbers, polynomials) into prime components
How do we get a one-way function out of them?
Easy problem

43. Assumption  f is one way
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
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 )
Assumption  f is one way

44. Exercise Show a function f such that
if f is polynomial time invertible on all inputs, then P=NP
f is not one-way

45. x is called the discrete log of y to base g.
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

46. 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