1. Cryptography
Lecture 4
Arpita Patra
© Arpita Patra

2. Recall >> Perfect Security
Various Definitions and their equivalence (Shannon’s Theorem)
Inherent Drawbacks
Cannot afford perfect security
>> Relaxing Perfect Security
Make the adversary bounded/efficient/polynomial time
Allow the break with some small/negligible probability
Are they necessary?
Computational Security

3. Today’s Goal Both relaxations are necessary.
Computational/Cryptographic Security
Impossible to break
Infeasible to break with high prob.
Will make ‘polynomially bounded/efficient’ and ‘small/negligible prob.’ precise
Paradigm I- Semantic Security for SKE- computational analogue of Shannon’s perfect security
Paradigm II- Indistinguishability-based Security for SKE – computational analogue of game/experiment based security definition of perfect security
Look for assumptions needed for construction and construct a scheme

4. Necessity of relaxed Threat Model
- Assume a SKE that allows many messages to be encrypted using a single short key
- Allow the adversary to be unbounded powerful in contrast to bounded
- Assume the adversary knows many (m1, c1), (m2, c2), …,(mt, ct): ci = Enck(mi)
- Decrypt each ciphertext with all possible keys until it finds a matching key
---brute-force O(|K|) k1 k2 k3 k1
k ? k2
k ?
k ? k3
? Deck1(ci) = mi, for all i
(m1, c1), (m2, c2), …, (mt, ct): ci = Enck(mi)
? Deck2(ci) = mi, for all i
? Deck3(ci) = mi, for all i
Yes
Hurray : I got the key

5. Necessity of relaxed Break Model
- Assume a SKE that allows many messages to be encrypted using a single short key
- Break is allowed with only zero probability
- Let the adversary knows many (m1, c1), (m2, c2), …,(mt, ct): ci = Enck(mi)
- Make random guess about a k and decrypt each ciphertext that key to verify k1 k2 k3
- O(1) time
k ? k2
k ?
- Probability : 1 / |K|
k ?
? Deck2(ci) = mi, for all i
(m1, c1), (m2, c2), …, (mt, ct): ci = Enck(mi)
Yes
Hurray : my guess was correct

6. Making “Polynomially Bounded” Precise– Asymptotic Approach
>> “Feasible” /“Efficient” / “Probabilistic Poly time (PPT)” algorithm means:
- running time of the algorithm is polynomial in the input size
>> PPT adversary = PPT algorithm
- Input size: n
- Running time of adversary is polynomial in n.
- A function f: Z+  Z+ is polynomial in n if there exist finite number of {ci} such that f(n) < i ci ni for all n. Example: n3
>> Example of what PPT adversary cannot do:
Assume your key size is n bits
| K | = 2n
An efficient/PPT adversary CANNOT brute-force over K

7. Making “Very Small/Negligible” Precise– Asymptotic Approach
>> “ Very Small / negligible in n” means those f(n) :
- for every polynomial in n, p(n), there exists some positive integer N, such that f(n) < 1/p(n) , for all n > N
n: Security parameter. A tunable parameter that tunes how difficult it is to break a cryptosystem
- “grows slower than any inverse poly”
>> Example: 1/2n , 1/2n/2
>> How about 1/n10 ?
For 1/n20 there is NO N s.t. 1/n10 < 1/n20
>> An adversary running for n3 time breaks a scheme with probability at most 1/2n
- The more the value of n, the tougher the life of the adversary is.
>> Usually the key size is set to n

8. Closures properties of poly and negligible functions
Proposition: Let p1 and p2 be polynomials in n. Then,
(i) p1 + p2 is a poly.
(ii) p1 * p2 is a poly.
Proposition: Let negl1 and negl2 be negligible functions in n. Then,
(i) negl1 + negl2 is a negligible function.
(ii) p(n). negl1 is a negligible function for any poly p(n)

9. Asymptotic Approach: Summary
>> Security parameter n - publicly known (part of the scheme) ; inputs to all algorithms (including adversary) will be made of size polynomial in n.
Running time of the users
Running time of the attacker
Success probability of the attacker
Functions of a security parameter n
Polynomial in n
Negligible in n
>> Typically n is the size of secret-key (ex: n = 128, 256, etc)

10. Choosing n Carefully is Very Essential
A designer claims that an adversary running for n3 minutes can break his scheme with probability n
Physicists believe that the no. of seconds elapsed since the birth of Earth is on the order of 258
Something that occurs with probability 2-60/sec is expected to occur once every 100 billion years
n is negligible --- hence secure scheme
- But what value of n to select while implementing ?
- If n  40 then an adversary working for 403 minutes (6 weeks) can break the scheme with probability 1
- You will claim it’s a useless scheme, but you just made a foolish choice of n
- n = 50 ?
: adversary working for 503 minutes (3 months) succeed with probability 1/1000
- may be unacceptable
- n = 500: adversary working for 200 yrs can break the scheme with probability 2-460
- definitely acceptable

11. n = Knob User’s running time is also increasing 
Adv’s job becomes harder 
min
max n

12. Concrete Approach   >> Set the value of n
>> Run users and adversary on specific machines
No adversary running for 5 yrs on 4GHz Machine can break the scheme with probability better than 2-60
Concrete Statement 1
Concrete Statement 2
Asymptotic Statement ……
Concrete Statement n 
Asymptotic Approach 
Concrete Approach

13. Syntax of Secret Key Encryption (SKE) Revisited
Key-generation Algorithm: Gen(1n)
> Outputs a key k chosen according to some probability distribution.
> MUST be a Randomized algorithm
2. Encryption Algorithm: Enck(m); m in {0,1}l(n)
> c  Enck(m) when randomized and c:=Enck(m) when deterministic
> Deterministic/Randomized algorithm
3. Decryption Algorithm: Deck(c)
> Outputs m:= Deck(c)
> Deterministic

14. Semantic Security for SKE
S. Goldwasser and S. Micali. Probabilistic Encryption. Journal of Computer and System Sciences, 28(2): , 1984
Impossible to break
Infeasible to break with high prob.
Randomized
PPT
COA
Given prior information about message, the ciphertext leaks no additional information about the message
h(m): external info about m; history function
f(m): additional information about m that adv wants to compute

15. Semantic Security
Two worlds: In one adv gets ciphertext and in another it does not. If the difference between probabilities of guessing f(x) in the both worlds are negligibly apart, then semantic security is achieved. k
Enc m
Gen(1n)
c  Enck(m)
h(m)
guess about f(m)
h(m)
guess about f(m)
|m|
Computational Analogue of Shannon’s definition of perfect-security A’ A
 = (Gen, Enc, Dec) is semantically-secure if for every PPT A there exists a PPT A’ such that for any Samp and PPT functions f and h: | - |
Pr [ A(1n,c,h(m)) =f(m)]
Pr [ A’(1n,|m|,h(m)) =f(m)]
 negl(n)
Probability taken over
>> uniform k,
>> m output by Samp(1n),
>> the randomness of A and
>> the randomness of Enc
Probability taken over
>> m output by Samp(1n) and
>> the randomness of A’

16. Indistinguishability Security for SKE
S. Goldwasser and S. Micali. Probabilistic Encryption. Journal of Computer and System Sciences, 28(2): , 1984
Impossible to break
Infeasible to break with high prob.
Randomized
PPT
COA
Given the knowledge of two messages, it cannot be distinguished if the ciphertext corresponds to the first or second message.

17. Indistinguishability Based Definition
An Experiment / a game between a challenger and an adversary
Indistinguishability experiment
PrivK (n)
A, 
ind
 = (Gen, Enc, Dec), M
Attacker A
Challenger
b  {0, 1}
m0, m1 M ; |m0|=|m1|
(freedom to choose any pair)
c  Enck(mb)
I can break 
b’  {0, 1} k
Let me verify
Run time: Poly(n)
(Attacker’s guess about encrypted message)
Gen(1n)
PrivK (n)
A, 
ind
b = b’
b  b’
0 --- attacker lost
1 --- attacker won
Published in the same paper. It captures the fact that the adversary gets just a single ciphertext. Even though the adversary knows that the ciphertext corresponds to one two message m0 and m1, it cannot tell apart which message the ciphertext corresponds to. Very strong.
 has is ind-secure if for every PPT attacker A, there is a negligible function negl(n) such that
½ + negl(n) Pr
PrivK (n)
A, 
ind
= 1 
Probability is taken over the randomness used by A and the challenger

18. Semantic vs. Indistinguishability Security
SEM: Given prior information about message, the ciphertext leaks no additional information about the message
Randomized
PPT
COA
Given the knowledge of two messages, it cannot be distinguished if the ciphertext corresponds to the first or second message.
IND Security → SEM Security
IND Security ← SEM Security
IND Security is the de facto style followed in Crypto community
RA3: If a scheme is ind-secure then for all PPT A and any index i, there is a negligible function negl(n) s.t
Pr [ A(1n,c) =mi] ≤ ½ + negl(n)
For uniform distribution of k and m.
IND Security ↔ SEM Security

19. Indistinguishability Based Definition: Renaming
An Experiment / a game between a challenger and an adversary
Indistinguishability experiment
PrivK (n)
A, 
coa
 = (Gen, Enc, Dec), M
Attacker A
Challenger
b  {0, 1}
m0, m1 M ; |m0|=|m1|
(freedom to choose any pair)
c  Enck(mb)
I can break 
b’  {0, 1} k
Let me verify
Run time: Poly(n)
(Attacker’s guess about encrypted message)
Gen(1n)
PrivK (n)
A, 
coa
b = b’
b  b’
0 --- attacker lost
1 --- attacker won
Published in the same paper. It captures the fact that the adversary gets just a single ciphertext. Even though the adversary knows that the ciphertext corresponds to one two message m0 and m1, it cannot tell apart which message the ciphertext corresponds to. Very strong.
 has is coa-secure if for every PPT attacker A, there is a negligible function negl(n) such that
½ + negl(n) Pr
PrivK (n)
A, 
coa
= 1 
Probability is taken over the randomness used by A and the challenger

20. Equivalent Formulation of Ind Definition
Attacker A
 = (Gen, Enc, Dec), M , n
Challenger
b  {0, 1}
m0, m1 , |m0| = |m1|
(freedom to choose any pair)
c  Enck(mb)
I can break 
b’  {0, 1} k
Let me verify
Run time: Poly(n)
(Attacker’s guess about encrypted message)
Gen(1n)
b = b’
Game Output
b  b’
0 --- attacker lost
1 --- attacker won
½ + negl(n) Pr
PrivK (n)
A, 
coa
= 1 
Intuition behind the definition ?
>> Attacker should behave in the same way irrespective of m0 or m1
>> What does same behavior mean ? --- Attacker just outputs a bit
>> Same behavior means that attacker outputs 1 with almost the same probability in each case (irrespective of whether it sees an encryption of m0 or m1)

21. Equivalent Formulation
PrivK (n, b) : the experiment with mb selected by challenger
A, 
coa
Output(PrivK (n, b)) : output bit of the attacker during
A, 
coa
PrivK (n, b))
 = (Gen, Enc, Dec) is coa-secure if for every PPT adversary A, there is a negligible function negl, such that :
Pr[Output(PrivK (n, 0)) = 1]
A, 
coa
Pr[Output(PrivK (n, 1)) = 1]
A, 
coa | - |
 negl(n) A4
 = (Gen, Enc, Dec) is coa-secure if for every PPT adversary A, there is a negligible function negl, such that :
½ + negl(n) Pr
PrivK (n)
A, 
coa
= 1 

22. Assumptions for coa-Secure SKEs
A suggestion: Try to BE good rather than trying to pretend to be good.
Recall the promises of computational security?
- Shorter key for big message
- Key Reuse
Let’s go OTP style: key will be used to pad/mask the message
- The pad can’t be just the key
- Pad = f(key) and the function is length-expanding ??
- For perfect security the pad needed to be truly random
- For computational security, enough if the pad ‘looks’ random to a PPT adversary but actually not.
M = K = C = {0, 1}l k k
Dec
m:= ck
Enc
c:= mk
k R K
m  M c
c  C m
Gen

23. Assumptions for coa-Secure SKEs
M. Blum, S. Micali. How to Generate Cryptographically strong sequences of pseudo-random bits. SIAM Journal of Computing, 13(4), , 1984
A. C.-C. Yao. Theory and Applications of Trapdoor Functions. FOCS, 80-91, 1982.
Pseudorandom Generators (PRGs): Tool to cheat the PPT adversaries

24. Scribe?

25. Pseudorandomness { Set of all binary strings of length l }
- It’s a property of a probability distribution
{ Set of all binary strings of length l }
G: a prob. Dist. = { Set of probabilities }
U: Uniform probability Distribution
A string drawn according to U is called random
A string drawn according to G is called pseudorandom
Sampler for G and U
Give me a string w
G is pseudorandom if a string drawn according to G is indistinguishable from a string drawn according to U to a PPT distinguisher