SECURITY AND VERIFICATION Lecture 1: Why to prove cryptography? The origins of provable cryptography...

SECURITY AND VERIFICATION

Lecture 1: Why to prove cryptography? The origins of provable cryptographyTamara RezkINDES TEAM, INRIA January 3rd, 2012

RSA INVEN

TORS G

OT BO

RED AN

D D

ECIDED

TO

PLAY POKER

Some history …

Mental Poker Adi Shamir, Ronald Rivest, Leonard Adleman, ’81

HOW TO PLAY MENTAL POKER?

MEN

TAL POKER PRO

TOCO

L Some history …

Mental Poker Shamir, Rivest, Adleman, ’81

how to write a protocol for mental poker without using a third trusted party?

in theory impossible

MEN

TAL POKER PRO

TOCO

LSome history …

in theory impossible: no such protocol exists

Information Theory: the ciphertext provides no information about the plaintext. Shannon’s entropy is a measure of this information.

Mental Poker Shamir, Rivest, Adleman, ’81

how to write a protocol for mental poker without using a third trusted party?

MEN

TAL POKER PRO

TOCO

LSome history …

in theory impossible

Mental Poker Shamir, Rivest, Adleman, ’81

how to write a protocol for mental poker without using a third trusted party?

MEN

TAL POKER PRO

TOCO

LSome history …

in theory impossible

solution based on SRA

Mental Poker Shamir, Rivest, Adleman, ’81

how to write a protocol for mental poker without using a third trusted party?

MEN

TAL POKER PRO

TOCO

LSome history …

SRA Protocol relies on commutative encryption

E (E (x, a), b) = E (E (x, b), a)

in theory impossible

solution based on SRA

Mental Poker Shamir, Rivest, Adleman, ’81

how to write a protocol for mental poker without using a third trusted party?

MEN

TAL POKER PRO

TOCO

LSome history …

Mental Poker Shamir, Rivest, Adleman, ’81

Encryption function E for SRA

* q is a large prime number* (q) = q-1* plaintext, ciphertext, key spaces all in Zq

*

* key a s.t. gcd(a, (q))= 1

E (x, a) = xa mod qD (c, a) = c-a mod q

MEN

TAL POKER PRO

TOCO

LSome history …

Mental Poker Shamir, Rivest, Adleman, ’81

Cast : Alice and Bob

MEN

TAL POKER PRO

TOCO

LHow SRA works

MEN

TAL POKER PRO

TOCO

LHow SRA works

MEN

TAL POKER PRO

TOCO

LHow SRA works

MEN

TAL POKER PRO

TOCO

LHow SRA works

E ( , b ) E ( , b ) E ( , b )

MEN

TAL POKER PRO

TOCO

LHow SRA works

E ( , b ) E ( , b ) E ( , b )

MEN

TAL POKER PRO

TOCO

LHow SRA works

E ( , b ) E ( , b) E ( , b )

E ( , b ) E ( , b ) E ( , b )

MEN

TAL POKER PRO

TOCO

LHow SRA works

E ( , b ) E ( , b) E ( , b )

E ( , b ) E ( , b ) E ( , b )

MEN

TAL POKER PRO

TOCO

LHow SRA works

E ( , b ) E ( , b ) E ( , b )

E ( , b ) E ( , b ) E ( , b )

MEN

TAL POKER PRO

TOCO

LHow SRA works

E ( , b ) E ( , b) E ( , b)

Bob Alice

E ( , b ) E ( , b ) E ( , b )

MEN

TAL POKER PRO

TOCO

LHow SRA works

E ( , b ) E ( , b ) E ( , b )

E ( , b ) E ( , b) E ( , b)

Bob Alice

E (E ( , b ) ,a )

MEN

TAL POKER PRO

TOCO

LHow SRA works

E ( , b ) E ( , b ) E ( , b )

E ( , b ) E ( , b) E ( , b)

Bob Alice

E ( , b ) E (E ( , b ) ,a )

MEN

TAL POKER PRO

TOCO

LHow SRA works

E ( , b ) E ( , b ) E ( , b )

E ( , b ) E ( , b) E ( , b)

Bob Alice

E ( , b ) E (E ( , b ) ,a )

D ( E (E ( , b ) ,a ), b)

A simple programming language

Semantics of expressions [e]

To define semantics of expressions, we need to define states .

A state is a function that maps each variable into its value.

We need to provide an interpretation for each operation op

var ::= x | y | z …op :: = + | - | * | < | = … expr :: = const| var | expr op

expr

Semantics of expressions [e]

Example: If (x) = 3 and (y) = 0 then

[x+y] = [x] + [y] =3 + 0 =

3

We say that the semantics of [x/y] is not defined.

var ::= x | y | z …op :: = + | - | * | < | = … expr :: = const| var | expr op expr

Operational semanticsSemantics precisely defines the meaning of programs:

We will define a “small-step operational semantics”Basic idea: execution of a program can be formalize as

a sequence of configurations: c0 c1 c2 ….

A configuration is a pair command and a state Example of configuration:

< x: = x+1; y: = y+x, [x:2 | y:0]>

The operational semantics is defined by a transition system (Configurations, ).

Configurations = { , , , , }

The relation can be

represented by a picture but

it should be formally defined

by a set of rules.

Operational semantics

In this example:

< x: = x+1; y: = y+x, [x:2 | y:0]>

<y: = y+x, [x:3 | y:0]>

< , [x:3 | y:0]>

Configurations =

{ < x: = x+1; y: = y+x, [x:2 | y:0]>,

<y: = y+x, [x:3 | y:0]>,

< , [x:3 | y:0]>

}

< x: = x+1; y: = y+x, [x:2 | y:0]>

<y: = y+x, [x:3 | y:0]>

< , [x:3 | y:0]>

Operational semanticsThe operational semantics is defined by a transition

system (Configurations, ).

The relation is defined by a set of semantic rules of the form:

[e] =v_________________________< x: = e, > < , [ | y:v]>

Operational semantics

Operational semantics

[e] =v_________________________< x: = e, > < , [ | x:v]>

Operational semantics

_________________________< skip, > < , >

Operational semantics[e] =0

_________________________< if (e) then {c1} else {c2} , > < c1 , >

Operational semantics[e] =0

_________________________< if (e) then {c1} else {c2} , > < c1 , >

[e] 0_________________________

< if (e) then {c1} else {c2} , > < c2 , >

Operational semantics[e] =0

_________________________< while (e) do {c} , > < c ; while (e) do {c} , >

[e] 0_________________________

< while (e) do {c} , > < , , >

Operational semantics

<c1, > < c’ , ’ > c’ _________________________

< c1;c2 , > < c’;c2 , ‘>

<c1, > < , ’ > _________________________< c1;c2 , > < c2 , ‘>

Operational semantics

________________< , , > < , , >

Probabilistic programming language

0.4 0.1 0.2 0.2 0.1

0.1 0.41

0.1

0.40.9 0.3 0.7 0.80.1 0.2

0.4 0.1 0.2 0.2 0.1

0.1 0.41

0.1

0.40.9 0.3 0.7 0.80.1 0.2

The sum is equal to 1This forms a distribution for

configurations reachable from

0.4 0.1 0.2 0.2 0.1

0.1 0.41

0.1

0.40.9 0.3 0.7 0.80.1 0.2

Distribution d induced by

d ( ) = 0.4d ( ) = 0.1 d ( ) = 0.2

…

0.4 0.1 0.2 0.2 0.1

0.1 0.41

0.1

0.40.9 0.3 0.7 0.80.1 0.2

What is the probability of reaching

from Pr[ ] ?

0.4 0.1 0.2 0.2 0.1

0.1 0.41

0.1

0.40.9 0.3 0.7 0.80.1 0.2

What is the probability of reaching

from Pr[ ] ?

0.4 0.1 0.2 0.2 0.1

0.1 0.41

0.1

0.40.9 0.3 0.7 0.80.1 0.2

The probability of reachingfrom

Pr[ ] = 0.2 * 0.7 = 0.14

0.4 0.1 0.2 0.2 0.1

0.1 0.41

0.1

0.40.9 0.3 0.7 0.80.1 0.2

The probability of reachingfrom

Pr[ ] = 0.2 * 0.7 = 0.14

Given by a sequence of probability distributions

And more formally, we need to provide a set of rules to define the probabilistictransition system

Now relation is probabilistic, annotated with a probability p p

Operational semantics[e] =v

_________________________< x: = e, > 1 < , [ | x:v]>

_________________________< skip, > 1 < , >

[e] =0_________________________

< if (e) then {c1} else {c2} , > 1

< c1 , >

[e] 0_________________________

< if (e) then {c1} else {c2} , > 1

< c2 , >

[e] =0_________________________

< while (e) do {c} , > 1 < c; while (e) do {c} , >

[e] 0_________________________

< while (e) do {c} , > 1 < , , >

<c1, > p < c’ , ’ > c’ _________________________

< c1;c2 , > p < c’;c2 , ‘>

<c1, > p < , ’ > _________________________< c1;c2 , > p< c2 , ‘>

________________< , , > 1 < , , >

Mental Poker in Whilep

shuffle ( ) =

c := {0,1,2,3,4,5} ;

if c=0 then

b0,b1,b2:= ;

else ….

MP =

cards:= shuffle( );

for c := 1 to 3 do ce[c]:= E (cards[c],b);

lce:= ce

ecards:= shuffle(E ( , b ) E ( , b ) E ( , b ) );

aliceCard := randomPick(lce, nil);

bobCard := randomPick(lce,aliceCard );

aliceCard:= E (aliceCard,a);

laliceCard:= aliceCard;

lbobCard:= bobCard

laliceCard:= D(laliceCard,b-1);

bobCard:= D(lbobCard,b-1);

MEN

TAL POKER

MP =

cards:= shuffle( );

for c := 1 to 3 do ce[c]:= E (cards[c],b);

lce:= ce

laliceCard:= D(laliceCard,b-1);

bobCard:= D(lbobCard,b-1);

HO

W TO

CHEAT IN

MEN

TAL POKER?

MP =

cards:= shuffle( );

for c := 1 to 3 do ce[c]:= E (cards[c],b);

lce:= ce

laliceCard:= D(laliceCard,b-1);

bobCard:= D(lbobCard,b-1);

HO

W TO

CHEAT IN

MEN

TAL POKER

A

HO

W TO

CHEAT IN

MEN

TAL POKER

How SRA works

Some facts to break the protocol:

HO

W TO

CHEAT IN

MEN

TAL POKER

How SRA works

Some facts to break the protocol:

• If “Alice” can decrypt without the key

HO

W TO

CHEAT IN

MEN

TAL POKER

How SRA works

Some facts to break the protocol:

• If “Alice” can decrypt without the key

An adversary that breaks the protocol by using brute force :

A =

for k = 1 to 2ɳ do

y:=D(lce[c],k);

if y = then aliceCard := lce[c]

HO

W TO

CHEAT IN

MEN

TAL POKER

How SRA works

Some facts to break the protocol:

• If “Alice” can decrypt without the key

A is polynomial, factorization is a hard problem

HO

W TO

CHEAT IN

MEN

TAL POKER

How SRA works

Some facts to break the protocol:

• If “Alice” can decrypt without the key

A is polynomial, factorization is a hard problem

HO

W TO

CHEAT IN

MEN

TAL POKER

How SRA works

Some facts to break the protocol:

• If “Alice” can decrypt without the key

• If Bob plays twice with the same key

HO

W TO

CHEAT IN

MEN

TAL POKER

How SRA works

Some facts to break the protocol:

• If “Alice” can decrypt without the key

• If Bob plays twice with the same key

An adversary that breaks the protocol if Bob plays twice with the same key

HO

W TO

CHEAT IN

MEN

TAL POKER

How SRA works

Some facts to break the protocol:

• If “Alice” can decrypt without the key

• If Bob plays twice with the same key A =

if oldEncACE = lce[c] then

aliceCard := lce[c];

HO

W TO

CHEAT IN

MEN

TAL POKER

How SRA works

Some facts to break the protocol:

• If “Alice” can decrypt without the key

• If Bob plays twice with the same key

• Observing quadratic residues!! (R.J. Lipton)

x Qq b x2 b (mod q)

x Qq xk (mod q) Qq

HO

W TO

CHEAT IN

MEN

TAL POKER

How SRA works

Observing quadratic residues!! (R.J. Lipton)

x Qq b x2 b (mod q)

x Qq xk (mod q) Qq

How to cheat in Mental PokerLipton’81

PROVABLE CRYPTO

GRAPH

Yhow to prove security of encryption algorithms?

PROVABLE CRYPTOGRAPHY

PROVABLE CRYPTO

GRAPH

Yhow to prove security of encryption algorithms?

Probabilistic Encryption and How to Play Mental Poker Keeping Secret All Partial Information Goldwasser and Micali ’82

PROVABLE CRYPTOGRAPHY

Probabilistic Encryption and How to Play Mental Poker …Goldwasser and Micali ’82

The fact that f is a trapdoor function does not rule out: 1.the possibility of computing x from f(x) when x is of a special form.2.the possibility of computing some partial information about x (even every other bit of x) from f(x).

TRAPDOOR FUNCTION

x f(x)

easy

hard

PROVABLE CRYPTO

GRAPH

Yhow to prove security of encryption algorithms?

Probabilistic Encryption and How to Play Mental Poker …Goldwasser and Micali ’82 - probabilistic setting- notion of semantic security

PROVABLE CRYPTOGRAPHY

PROVABLE CRYPTO

GRAPH

YSemantic security or Chosen Plaintext Attack (a.k.a. CPA)

E ( message1 , b)

message1, message 2

E ( message2 , b)

“I guess that the coin was tail”

| Pr[CPA; g = c] - ½ | is negligible for |b| (|b| is called security parameter)

NEGLIGIBLE FUNCTION

A function f (x) is negligible for x when for all c>0, there is a constant nc such

that nc≤ x implies f(x) ≤ 1/xc

NEGLIGIBLE FUNCTION

A function f (x) is negligible for x when for all c>0, there is a constant nc such that nc≤ x implies f(x) ≤ 1/xc

Are the following functions negligible?

f(x) = x2 f(x) =1/xf(x) =1/x2

f(x)= 1/3x

encryption scheme

Definition of encryption scheme

An encryption scheme is a triple (G, E, D) of probabilistic polynomial-time algorithms such that:

- On input ɳ , algorithm G outputs a pair e, d of bitstrings

- D(E(x,e),d) = x

PROVABLE CRYPTO

GRAPH

Y

Chosen-plaintext attack (CPA)

E(x0, x1 ) = if (c = 0)

then {m := E (x0, ke)}

else {m := E(x1,ke)};

CPA = c := {0,1};

ke, kd := Ge(); A[E]

| Pr[CPA; g =c] - ½ | is negligible for ɳ (ɳ is called security parameter)

READIN

GSlides, Notes, Bibliography

• Slides and lecture notes: www-sop.inria.fr/members/Tamara.Rezk/teaching

• Mental Poker – Shamir, Rivest, Adleman

• Probabilistic Encryption & How to Play Mental Poker Keeping Secret all Partial Information – Goldwasser , Micali