1 Lossy Trapdoor Functions and Their Applications Brent Waters SRI International Chris Peikert SRI International

2 Public Key Cryptography M M PK SK Factoring (RSA) Cyclic Groups (DH) Linear equations (lattices)

3 Trapdoor Functions (TDF) [DH76] f(x) x PK: f( * ) TD Receiver recovers all input Input = x

4 Uses of TDFs  Public Key Encryption (PKE)  PKE against active attackers CCA-security [NY90,DDN91]  NIZKs [BFM88]

5 PKE  TDF E(M,r) M PK: E(*,*) SK Message: M Randomness: r r Input not recovered. Not a TDF!

6 Building TDFs from PKE (a failure) E(x,x) x PK: E(*,*) SK Input: x Insecure! BB-Impossible [GMR05] Factoring (RSA) Cyclic Groups (DH) Linear equations (lattices)

7 This Talk First “non-native” TDF constructions New CCA-secure cryptosystems DDH TDF CCA-Enc Lattices Factoring [CS98] [NY90, DDN91][RSA78] [PW07]

8 Key Idea: Lossy TDFs Concepts Realizations

9 Lossy TDFs: A Tale of Two Keys x PK: f( * ) TD Injective Keys x’ f inj ( ) x TD Lossy Keys x’ f lossy ( ) PK: f( * ) 

10 Properties 1)Injective: 8 x,x’ f inj ( x )  f inj ( x’ ) f -1 (TD, f inj ( x )) = x 2) Lossy: n input size r < n residual leakage (range < 2 r ) k = n-r lossiness

11 Key-Type Indist. Attacker cannot tell key-type Injective Lossy Prob. < ½ + negl. ?

12 Building A Trapdoor Function Use Lossy-TDF with Injective Keys PK: f inj ( * ) TD Correctness: Direct Security ??

13 Sequence of Game Proofs Define Games: Game-1, …, Game-N Game-1 is actual security game Properties 1)Game-i  c Game-i+1 2)Advantage(Game-N)  0 (info theoretic)

14 Proving Non-Invertability f lossy ( ) f inj ( ) f inj ( x ) x’ Game-1 Game-2 Key Indist. Game-2: 9 ¼ 2 k z s.t. f losssy (x) = f lossy (z) ) negl. advantage Big Idea: Challenge over Public Key Type! x

15 Public Key Enc. (Chosen-plaintext) KeyGen PubKey: SK: f inj ( * ) TD, d (extractor seed) Enc(M,PK) x CT = (C 1,C 2 ) = f inj (x), M © Ext(d, x) Dec(CT,SK) x’ = f -1 (C 1 ) M= C 1 © Ext(x’,d)

16 CPA Security f lossy ( ) f inj ( ) M 0, M 1 Enc(PK,M b ) Game-1 Game-2 Key Indist. Wins if b’=b Game-2: Ext(x,d) ¼ Uniform | f lossy (x) ) negl. advantage b b’

17 CCA Security[RS91] M PK SK M’

18 Preventing CCA Attacks Non-Interactive Zero Knowledge (NIZK) [DY90,RS91,DDN91, CS98,S99, CS02, ES02] CT = Enc(M,r) + NIZK Decrypt: 1) Check NIZK 2) Decrypt Factoring (RSA) Cyclic Groups (DH) Linear equations (lattices) Theme: Decryptor not recover r

19 “Witness Recovering” Encryption E(M,r) M PK: E(*,*) SK Message: M Randomness: r r “Re-encrypt” to test

20 All-but-One (ABO) Encryption g b* ( *,* ) TD b* Generate “lossy branch” b* x x’ g b* (b=b*,x ) x x’ g b* (b  b*,x ) Correctness: g -1 (TD, b, g b* (b  b*, x)) = x Security: Lossy Branch indist.

21 CCA Enc KeyGen PubKey: SK: f inj ( * ) TD f, d (extractor seed) Enc(M,PK) x, ( VK, SigSK ) CT = VK, C 1 = f inj (x), C 2 =g b* (VK,x), C 3 = M © Ext(d, x),  = Sig(SK Sig, (C 1 …C 3 )) Dec(CT,SK) 2) x’ = f -1 (C 1 ) g b* (*,*) TD g 1) Check  4) M= C 3 © Ext(x’,d) 3) Re-encrypt with x’

22 Chosen Ciphertext Security f lossy ( ) f inj ( ) M 0, M 1 Enc(PK,M b )=CT* Game-1 Game-2 Signature Wins if b’=b Game-5: Ext(x,d) ¼ Uniform | g(b*,x), f lossy (x) ) negl. advantage b b’ CT i  CT*=(VK*…) Dec(CT_i) Game-3 Hidden Branch Game-4 Equivalent Game-5 Key Indist. g b* (*,*)g VK* (*,*) Game-2: Reject sigs from VK*Game-3: Lossy Branch = VK*Game-4: Decrypt with ABO keyGame-5: Make key Lossy

23 …,but Where do they Come From?

24 Homomorphic Encryption E(a) © E(b) = E(a+b) c ¢ E(a) = E(c ¢ a)

25 Creating Lossy TDFs E(1) E(0) x1x1 xnxn = E(x 1 ) E(x n ) Injective: Encrypt Identity Matrix Evaluate: Matrix Multiplication E(0)

26 Creating Lossy TDFs E(0) x1x1 xnxn = Lossy: Encrypt Zero Matrix E(0) Msg. output independent of input, but …

27 DDH-Construction Group G order q Input size: n > 3 lg(q) Pick: g, h 1 = g a 1, …, h n =g a n 2 G r 1, …, r n 2 Z q

28 Creating Lossy TDFs (injective) h 1 r 1 g hnrn ghnrn g h1r2h1r2 h1rnh1rn hnr1hnr1 x1x1 xnxn = h2r1h2r1 gr1gr1 if i =j A i,,j = h j r i g 1 else A i,,j = h j r i grngrn Use a i ’s to recover x i ’s,g a 1  x i r i g x 1 g  x i r i,g a n  x i r i g x n

29 Creating Lossy TDFs (lossy) h1r1h1r1 hnrnhnrn h1r2h1r2 h1rnh1rn hnr1hnr1 x1x1 xnxn = h2r1h2r1 gr1gr1 A i,,j = h j r i grngrn,g a 1  x i r i g  x i r i g a n  x i r i Only lg(q) bits of information ) n- lg(q) bits lost! DDH ) Key Indist.

30 Lattice Realization Similar Structure Gaussian Noise Issue Reduce to Learning w/ Error Lattices [R05]

31 Conclusions First TDFs w/o factoring First CCA from lattices Main Ideas: Loose Information Simulator changes parameters Future: CCA-secure PKE