SECURITY AND VERIFICATION

Lecture 3: What kind of attacks are there? - Chosen Ciphertexts AttacksTamara RezkINDES TEAM, INRIA January 17th, 2012

Plan Lecture 1 Chosen Plaintext Attacks (CPA assumption)

CPA schemes: ElGamal, Paillier

Lecture 2 Game-based proofs CPA proof: ElGamal

Today: CPA proof: Paillier Limits on provable cryptography Chosen Ciphertext Attacks (CCA assumption) CCA1 proof: using proof of knowledge-zero knowledge (PKZK) From interactive to non-interactive PKZK CCA2 an example of a CCA2 scheme

Observational EquivalenceP0 and P1 are observational equivalent with respect to

variable x, denoted P0 {x} P1 if

Pr[P0; x = v] = Pr[P1; x =v] for all v

P0 and P1 are observational equivalent with respect to variable x, denoted P0 {x1..xn} P1 if

Pr[P0; x1 = v1 ˄.. x2 = v2 ˄..] = Pr[P1; x1 = v1 ˄.. x2 = v2 ˄..]

for all v1…vn

Game-based proofsHow to prove cryptography?

G0 G1 G2 … Gn

For each arrow, we have that either :

Pr[Gi; g=b] ≤ Pr[Gi+1; g=b]

or

Gi {g} Gi+1

PAILLIER ENCRYPTIO

NPaillier encryption

Assume that generateN() is a probabilistic function that generates two primes with the property that gcd(p*q, (p*q) ) = 1 and g with g a generator for the multiplicative group {1 … n2-1}. Then Paillier encryption is defined by:

G() =

p,q,g:= generateN(); n := p * q;

ke := (n, g); kd:= (p,q)

Assume x is in {1…n-1}

E (x, (n,g)) = y := {1.. n-1}; c:= yn * g x mod n2

PROVABLE CRYPTO

GRAPH

Y

Decisional Reduosity Assumption

CR(x0, x1 ) = if (b = 0)

then {y:= {1..n-1}; c :=yn mod n2}

else {c:= {1.. n2 -1}}

DRA = b := {0,1};

p,q,q:= generateN(); n := p * q;

B[CR]

| Pr[DRA; g’ =b] - ½ | is negligible for ɳ (ɳ is called security parameter, order of the group , ie n2 -1 ) . Attacker B does not have p, or q.

PROVABLE CRYPTO

GRAPH

Y

Decisional Reduosity Assumption

CR(x0, x1 ) = if (b = 0)

then {y:= {1..n-1}; c :=yn mod n2}

else {c:= {1.. n2 -1}}

DRA = b := {0,1};

p,q,g:= generateN(); n := p * q;

B[CR]

| Pr[DRA; g’ =b] - ½ | is negligible for ɳ (ɳ is called security parameter, order of the group , ie n2 -1 )

nth residuo modulo n2

PROVABLE CRYPTO

GRAPH

Y

Chosen-plaintext attack (CPA)

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

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

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

CPA = b := {0,1};

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

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

THEO

REMTHEOREM

Theorem

Paillier encryption scheme is resistent to Chosen Plaintext Attacks

PROO

F OF CPA O

F PAILLIERGAME 0

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

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

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

CPApaillier = b := {0,1};

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

PROO

F OF CPA O

F PAILLIERstep 1: INLINE

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

then {y := {1.. n-1}; c:= yn * g x0 mod n2 }

else {y := {1.. n-1}; c:= yn * g x1 mod n2 }

CPApaillier1 = b := {0,1};

p,q,q:= generateN(); n := p * q;

ke := (n, g); kd:= (p,q);

A[E]

PROO

F OF CPA O

F PAILLIERstep 1: INLINE

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

then {y := {1.. n-1}; c:= yn * g x0 mod n2 }

else {y := {1.. n-1}; c:= yn * g x1 mod n2 }

CPApaillier1 = b := {0,1};

p,q,q:= generateN(); n := p * q;

ke := (n, g); kd:= (p,q);

A[E]

CPApaillier {g} CPApaillier1

PROO

F OF CPA O

F PAILLIERstep 2: DEADCODE

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

then {y := {1.. n-1}; c:= yn * g x0 mod n2 }

else {y := {1.. n-1}; c:= yn * g x1 mod n2 }

CPApaillier1 = b := {0,1};

p,q,q:= generateN(); n := p * q;

ke := (n, g); kd:= (p,q);

A[E]

PROO

F OF CPA O

F PAILLIERstep 2: DEADCODE

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

then {y := {1.. n-1}; c:= yn * g x0 mod n2 }

else {y := {1.. n-1}; c:= yn * g x1 mod n2 }

CPApaillier2 = b := {0,1};

p,q,q:= generateN(); n := p * q;

ke := (n, g); A[E]

CPApaillier1 {g} CPApaillier2

PROO

F OF CPA O

F PAILLIERstep 3 INLINE

CR(x0, x1 ) = if (b = 0)

then {y:= {1..n-1}; c :=yn mod n2}

else {c:= {1.. n2 -1}}

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

then {y := {1.. n-1}; c:= yn * g x0 mod n2 }

else {y := {1.. n-1}; c:= yn * g x1 mod n2 }

DRA = b := {0,1};

p,q,q:= generateN(); n := p * q; B[CR]

B = ke := (n, g);

A[CR; c:= c * g x0 mod n2 ]; g0:=g;

A[CR; c:= c * g x1 mod n2 ]; g1:=g;

if (g0 =0 OR g1 =1 ) then g’ = 0 else g’:= 1

PROO

F OF CPA O

F PAILLIERCalculating probabilities

CR(x0, x1 ) = if (b = 0)

then {y:= {1..n-1}; c :=xn mod n2}

else {c:= {1.. n2 -1}}

DRA = b := {0,1};

p,q,q:= generateN(); n := p * q; B[CR]

B = ke := (n, g);

A[CR; c:= c * g x0 mod n2 ]; g0:=g;

A[CR; c:= c * g x1 mod n2 ]; g1:=g;

if (g0 =0 OR g1 =1 ) then g’ = 0 else g’:= 1½ Pr[CPApaillier2;g=b] = Pr[DRA;g’=0 and b=0]

½ Pr[CPApaillier2;g=b] ≤ Pr[DRA;g’=b]

½ Pr[CPApaillier2;g=b] = Pr[DRA;g’=1 and b=1]

½ Pr[CPApaillier2;g=b] ≤ Pr[DRA;g’=b]

PROO

F OF CPA O

F PAILLIERstep 3 INLINE

CR(x0, x1 ) = if (b = 0)

then {y:= {1..n-1}; c :=xn mod n2}

else {c:= {1.. n2 -1}}

DRA = b := {0,1};

p,q,q:= generateN(); n := p * q; B[CR]

B = ke := (n, g);

A[CR; c:= c * g x0 mod n2 ]; g0:=g;

A[CR; c:= c * g x1 mod n2 ]; g1:=g;

if (g0 =0 OR g1 =1 ) then g’ = 1 else g’:= 0

negligible

NO

Assume that generateN() is a probabilistic function that generates two primes with the property that gcd(p*q, (p*q) ) = 1 and g with g a generator for the multiplicative group {1 … n2-1}. Then Paillier encryption is defined by:

G() = p,q,q:= generateN(); n := p * q; ke := (n, g); kd:= (p,q)Assume x is in {1…n-1}

E (x, (n,g)) = y := {1.. n-1}; c:= yn * g x mod n2

E (x0, (n,g)) * E (x1, (n,g)) =

y0n * g

x0 mod n2 * y1n * g

x1 mod n2 =

y0n *y1 n * g

x0 *g x1 mod n2 =

(y0 *y1 )n * g

x0 +x1 mod n2 =

E (x0+x1, (n,g))

A property of Paillier encryptions:

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

then {y := {1.. n-1}; c:= yn * g x0 mod n2 }

else {y := {1.. n-1}; c:= yn * g x1 mod n2 };

log := log + m

D(m) = if (m log)

then {x := 0}

else {x := D(m,kd)};

GamePaillier = b := {0,1}; p,q,q:= generateN();

n := p * q;ke := (n, g); kd:= (p,q);A[E, D]

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

then {m:=x0;y := {1.. n-1}; c:= yn * g x0 mod n2 }

else {m:=x1;y := {1.. n-1}; c:= yn * g x1 mod n2 };

log := log + c

D(m) = if (m log)

then {x := 0}

else {x := D(m,kd)};

GamePaillier = b := {0,1}; p,q,q:= generateN();

n := p * q;ke := (n, g); kd:= (p,q);A[E, D]

A[E, D] = x0 := 1; x1 := 2; E; m:=c * c; D;

if (x = 2) then g:=0 else g:=1

We have proved Paillier to be CPA.This is only one kind of attack. Paillier is secure for an adversary with the power of making chosen plaintext attacks (usually, the weaker kind of attack possible), but not for all possible attacks: for example, it is not secure for chosen ciphertext attacks.

Important: Provable cryptography only guarantees that no partial information is reveal for a given class of attack. It does not imply total security.

Another Look to Provable Cryptography

“the treatment of hashed ElGamal encryption in is in some sense a remarkable achievement … so successful in turning something that should be interesting and accessible to everyone into something lengthy, unreadable, and boring.”

Neal Koblitz

Another Look to ElGammal …

Another Look to Provable Cryptography• A security theorem is conditional in a strong sense — it assumes the intractability of some mathematical problem…

• Often the intractability assumption is made for a complicated and contrived problem that has never been carefully studied. In fact, in some cases the problem is trivially equivalent to the cryptanalysis problem for the protocol whose security is being "proved," and the "proof" is essentially circular.

• Certain attacks — especially side-channel attacks — are very hard to model, and the models that have been proposed are woefully inadequate. The problem is that the adversary is always coming up with ingenious new methods to compromise the security of a cryptographic system.

•AND MORE Neal Koblitz

Chosen Ciphertext Attacks (CCA)• CCA are strong forms of active attacks

• We will see two type of them a priori CCA and a posteriori CCA

• In both, the adversary has access to decryption requests

• CAVEAT: some use CCA to mean CCA2

Chosen-cyphertext attack 2 (CCA1)

E = if (b = 0)

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

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

CCA1 = b := {0,1};

ke, kd := Ge(); A[D]; E;A’

D = x := D(m,kd);

Example: A CCA1 scheme We will define a CCA1 scheme < G’, E’ , D’ >

It is based on a CPA scheme < G , E , D >

It is based on a non-interactive ZK scheme (P , V , R, S)

Proof of Knowledge Zero Knowledge

a prover gives a proof of some secret that he knows

Proof of Knowledge Zero Knowledge

a prover gives a proof of some secret that he knowsbut without revealing the secret!

Proof of Knowledge Zero Knowledge

a prover gives a proof of some secret that he knowsbut without revealing the secret!

Example: If x in Zq is the secret, the prover can exhibit witnesses based on gx

, showing that he knows x

(a concrete protocol later)

Proof of Knowledge Zero Knowledge: properties

ZK schemes have to satisfy: Soundness: the verification procedure cannot

“accept” valid false statements, except for negligible probability

Completeness: if a statement is true then the verifier “accepts” it, except for negligible probability

Zero-Knowledge: the adversary cannot guess the secret by using the scheme!

Proof Systems Schemes for ZKA proof of knowledge zero knowledge scheme is a tuple

(P , V , R, S) P (prover) is a probabilistic program that takes as inputs

a secret s, a witness w, and outputs a proof p in DV (verifier) is a probabilistic program that takes a witness

and a proof and outputs zero or oneR is a NP relation that depends on secret s S is a simulator, a probabilistic program that outputs a

“proof” in D without using secret s.

(we do not include here the algorithm for “extraction”)

Zero Knowledge (indistinguishability)O = if (b = 0)

then {p := P (s, w)}

else {p:= S(w)};

ZK = b := {0,1};

A[O]

Example: A CCA1 scheme (Naor-Yung) We will define a CCA1 scheme < G’, E’ , D’ >It is based on a CPA scheme < G , E , D >It is based on a ZK scheme (P , V , R, S)

G’‘ ( ) =

k0e, k0d:= G( ); k1e, k1d:= G( )

E ‘(x, (k0e , k1e)) =

e0, e1 := E (x, k0e ); E (x, k1e);

p:= P(e0, e1, x); c:= e0,e1, p0,p1,p

D ‘ ((e0,e1, p), (k0e , k1e)) =

if V(e0, e1,,p) = true then x: = D(e1, k1d)

Proof of CCA1 of Naor-Yung scheme Naor-Yung scheme is CCA1

Theorem

Naor-Yung encryption scheme is resistent to Chosen Ciphertext Attacks version 1 (CCA1)

E = if (b = 0)

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

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

CCA1 = b := {0,1};

ke, kd := Ge(); A[D]; E;A’

D = x := D(m,kd);

E = if (b = 0)

then {e0, e1 := E r0 (x0, k0e ); Er1 (x0, k1e);

p0,p1,p:= P (e0, e1, x0, r0,r1,); c:= e0,e1, p0,p1,p }

else {e0, e1 := E r0’ (x1, k0e ); Er1’ (x1, k1e);

p:= P(e0, e1, x1, r0’,r1’); c:= e0,e1, p0,p1,p

};

CCA1-1 = b := {0,1};

k0e, k0d:= G( );

k1e, k1d:= G( )

A[D]; E;A’

D = if V(e0, e1,p0,p1,p) = true then x: = D(e1, k1d)Inline

CCA1 {g} CCA1-1

E = if (b = 0)

then {e0, e1 := E (x0, k0e ); E (x0, k1e);

p0,p1,p:= S(e0, e1);c:= e0,e1, p0,p1,p }

else {e0, e1 := E (x1, k0e ); E (x1, k1e);

p0,p1,p:= S(e0, e1); c:= e0,e1, p0,p1,p

};

CCA1-2 = b := {0,1};

k0e, k0d:= G( );

k1e, k1d:= G( )

A[D]; E; A’

D = if V(e0, e1,p0,p1,p) = true then x: = D(e1, k1d)Zero knowledge

CCA1-1 {g} CCA1-2

E = if (b = 0)

then {e0, e1 := E (x0, k0e ); E (x0, k1e);

}

else {e0, e1 := E (x1, k0e ); E (x1, k1e);

};

CCA1-3 = b := {0,1};

k0e, k0d:= G( );

k1e, k1d:= G( )

A[D]; E; p0,p1,p:= S(e0, e1); c:= e0,e1, p0,p1,p ; A’

D = if V(e0, e1,p0,p1,p) = true then x: = D(e1, k1d)Code motion

CCA1-2 {g} CCA1-3

E = if (b = 0)

then {e0, e1 := E (x0, k0e ); E (x0, k1e);

}

else {

e0, e1 := E (x1, k0e ); E (x1, k1e); };

CCA1-4 = b := {0,1};

k0e, k0d:= G( );

BB = k1e, k1d:= G( ) ;

A[D]; E; p0,p1,p:= S(e0, e1); c:= e0,e1, p0,p1,p ; A’

D = if V(e0, e1,p0,p1,p) = true then x: = D(e1, k1d)Inline

CCA1-3 {g} CCA1-4

E = if (b = 0)

then {e0, e1 := E (x0, k0e ); E (x0, k1e); }

else {e0, e1 := E (x1, k0e ); E (x1, k1e); };

E’ = if (b = 0)

then {e0, := E (x0, k0e ) }

else {e0 := E (x1, k0e ) };

CPA = b := {0,1};

k0e, k0d:= G( );

BB = k1e, k1d:= G( ) ;A[D]; E’; e1 := E (x0, k1e );

p0,p1,p:= S(e0, e1); c:= e0,e1, p0,p1,p ; if V(e0, e1,p0,p1,p) = true then A’ else g:=1

D = if V(e0, e1,p0,p1,p) = true then x: = D(e1, k1d)A cpa attacker

E = if (b = 0)

then {e0, e1 := E (x0, k0e ); E (x0, k1e); }

else {e0, e1 := E (x1, k0e ); E (x1, k1e); };

E’ = if (b = 0)

then {e0, := E (x0, k0e ) }

else {e0 := E (x1, k0e ) };

CPA = b := {0,1};

0e, k0d:= G( );

BB = k1e, k1d:= G( ) ;A[D]; E’; e1 := E (x0, k1e );

p0,p1,p:= S(e0, e1); c:= e0,e1, p0,p1,p ; if V(e0, e1,p0,p1,p) = true then A’ else g:=1

D = if V(e0, e1,p0,p1,p) = true then x: = D(e1, k1d)A cpa attacker

Pr[CCA1-4;g=b]=

Pr[CCA1-4;g=0 and b=0] +

Pr[CCA1-4;g=1 and b=1] =

1/2 Pr[CPA;g=b] + 1/2

E = if (b = 0)

then {e0, e1 := E (x0, k0e ); E (x0, k1e); }

else {e0, e1 := E (x1, k0e ); E (x1, k1e); };

E’ = if (b = 0)

then {e0, := E (x0, k0e ) }

else {e0 := E (x1, k0e ) };

CPA = b := {0,1};

0e, k0d:= G( );

BB = k1e, k1d:= G( ) ;A[D]; E’; e1 := E (x0, k1e );

p0,p1,p:= S(e0, e1); c:= e0,e1, p0,p1,p ; if V(e0, e1,p0,p1,p) = true then A’ else g:=1

D = if V(e0, e1,p0,p1,p) = true then x: = D(e1, k1d)A cpa attacker

Pr[CCA1-4;g=b]=

Pr[CCA1-4;g=0 and b=0] +

Pr[CCA1-4;g=1 and b=1] =

1/2 Pr[CPA;g=b] + 1/2

negligeable

There is a secret x that the prover wants to prove that he knowsThe NP relation that depends on x is “logg z = x and logh z’ = x“ , where g and h are generators for the multiplicative group { 1…q-1}The protocol for generating a proof is P0;V0;P1 and to verify isV1 where:

P0(g,h) = w := {1…q-1} la, lb := gw, hw

V0 (la,lb) = lc := {1…q-1}; P1 (w,x ,lc) = p := w + x * lc mod q

V0 ( p, la,lb , gx, hx ) = if (gp = la * gx*lc and hp = lb * hx*lc ) then true else false

Exercise: Assume that lc := {1…q-1} and that lc is a parameter of P0. Show that in the protocol for generating a proof is P0; P1 and to verify V1 the prover can cheat (he can prove he knows x, without knowing it)

P0(g,h,lc) = w := {1…q-1} la, lb := gw, hw

P1 (w,x ,lc) = p := w + x * lc mod q

V0 ( p, la,lb , gx, hx ) = if (gp = la * gx*lc and hp = lb * hx*lc ) then true else false

From interactive to non-interactive

There is a secret x that the prover wants to prove that he knowsThe NP relation that depends on x is “logg z = x and logh z’ = x“ , where g and h are generators for the multiplicative group { 1…q-1}The protocol for generating a proof is P and to verify is V where:

P(g,h,x) = w := {1…q-1} a, b := gw, hw

lc := H( a + b); p := w + x * lc mod q

V ( p, lc , gx, hx ) = a, b := gx lc * gp, hx lc * hp if (H(a+b) = lc ) then true else false

Chosen-cyphertext attack 2 (CCA2)

E = if (b = 0)

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

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

log := log + m

CCA2 = b := {0,1};

log := nil;

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

D = if (m log)

then {x := 0}

else {x := D(m,kd)};

Let H : { 0,1}l {0,1}l

G : { 0,1}l {0,1}p-l

be two hash functions

RSA-OAEP –ENC (m,ke)=r := { 0,1}l ;s:= H( r ) + m; t := G(s) + rc:= rsa-enc(s++t,ke)

RSA-OAEP –DEC (c,kd)=(s,t) := rsa-dec(c,kd) ;r:= t + G(s) ;m: = s + H( r )

READIN

GSlides, Notes, Bibliography

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

• Public-key Cryptosystems Provably Secure againstChosen Ciphertext Attacks – Naor, Yung

• Non-Interactive Proof of Knowledge and ChosenCiphertext Attacks Rackoff, Simon

• Another Look to Provable Cryptography – Neal Koblitz http://anotherlook.ca/

• Code-based Game-Playing Proofs and the Security of Triple Encryption – Bellare, Rogaway