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A Verifiable Secret Shuffle of Homomorphic Encryptions Jens Groth UCLA On ePrint archive: http://eprint.iacr.org/2005/246

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Agenda Motivation – anonymous communication Motivation – anonymous communication What is What is A shuffle? Homomorphic encryption? Zero- knowledge proofs? A shuffle? Homomorphic encryption? Zero- knowledge proofs? ZK proof for shuffle of known contents ZK proof for shuffle of known contents Tool: Homomorphic commitments Tool: Homomorphic commitments ZK proof for shuffle of homomorphic encryptions ZK proof for shuffle of homomorphic encryptions Comparison with other ZK proofs Comparison with other ZK proofs Efficiency improvements Efficiency improvements

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Anonymous communication Mixer π m1m1 mnmn … … m π(1) m π(n) Sender 1 Sender n mix- servers

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Encryption Rerandomization property E(m) E´(m) Threshold decryption property t mix-servers can decrypt t-1 mix-servers do not learn anything

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Mix-net Mix-net π m1m1 mnmn … … E´(m π(1) )E´(m π(n) ) E(m 1 )E(m n ) Threshold-decryption … m π(1) m π(n) senders mix-servers at least t mix-servers

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Mix-net Mix-server 1 π 1 … E´(m π 1 (1) )E´(m π 1 (n) ) E(m 1 )E(m n ) Mix-server N π N E´´´(m π(1) )E´´´(m π(n) ) π = π N... π 1

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A shuffle π E´(m π(1) )E´(m π(n) ) E(m 1 )E(m n )

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Agenda Motivation – anonymous communication Motivation – anonymous communication Mix-nets Mix-nets What is What is A shuffle? Homomorphic encryption? Zero- knowledge proofs? A shuffle? Homomorphic encryption? Zero- knowledge proofs? ZK proof for shuffle of known contents ZK proof for shuffle of known contents Tool: Homomorphic commitments Tool: Homomorphic commitments ZK proof for shuffle of homomorphic encryptions ZK proof for shuffle of homomorphic encryptions Comparison with other ZK proofs Comparison with other ZK proofs Efficiency improvements Efficiency improvements

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Homomorphic encryption Homomorphic property E(m 1 m 2 ; R 1 +R 2 ) = E(m 1 ; R 1 ) E(m 2 ; R 2 ) Rerandomization E(m; R 1 +R 2 ) = E(m; R 1 ) E(1; R 2 ) Message space order Q no small prime factors Root extraction property see paper

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ElGamal variant Keys Primes Q, P so P = 2Q +1 Random elements G, Y of order Q PK = (Q, P, G, Y) SK = (PK, x) so Y = G x Encryption E(m; (±1, ±1, R)) = (±G R mod P, ±Y R m mod P) Ciphertext verification (U, V) valid ciphertext if 0 < U < P and 0 < V < P

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A shuffle of homomorphic encryptions π, R 1,...,R n e π(1) E(1;R 1 )e π(n) E(1;R n ) e1e1 enen

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Verifiability? π, R 1,...,R n ? E1E1 E n e1e1 enen

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Zero-knowledge proof Complete prover with π, R 1,...,R n can convince anybody of correctness of shuffle Complete prover with π, R 1,...,R n can convince anybody of correctness of shuffle Sound if not a valid shuffle impossible to convince others of correctness of shuffle Sound if not a valid shuffle impossible to convince others of correctness of shuffle Zero-knowledge prover does not reveal anything beyond correctness of shuffle Zero-knowledge prover does not reveal anything beyond correctness of shuffle

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Statement: PK, e 1,..., e n, E 1,..., E n (and a little more) Real proof (π, R 1,...) Simulated proof (c 1,...) a 1 a 1 c 1 c 1 a 2 a 2...... (a 1, c 1, a 2,... ) indistinguishable from (a 1, c 1, a 2,...) Special honest verifier zero- knowledge (SHVZK)

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Computational/statistical Soundness Soundness Unconditional: No adversary can make a valid proof for a false statement Unconditional: No adversary can make a valid proof for a false statement Computational: A polynomial time adversary cannot make a valid proof for a false statement Computational: A polynomial time adversary cannot make a valid proof for a false statement Special honest verifier zero-knowledge Special honest verifier zero-knowledge Statistical: No adversary can distinguish real proofs from simulated proofs Statistical: No adversary can distinguish real proofs from simulated proofs Computational: A polynomial time adversary cannot distinguish real proofs from simulated proofs Computational: A polynomial time adversary cannot distinguish real proofs from simulated proofs

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Main result A 7-round public coin SHVZK proof for correctness of a shuffle of homomorphic encryptions Optional - unconditional soundness or statistical SHVZK - key length vs efficiency

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Agenda Motivation – anonymous communication Motivation – anonymous communication Mix-nets Mix-nets What is What is A shuffle? Homomorphic encryption? Zero- knowledge proofs? A shuffle? Homomorphic encryption? Zero- knowledge proofs? ZK proof for shuffle of known contents ZK proof for shuffle of known contents Tool: Homomorphic commitments Tool: Homomorphic commitments ZK proof for shuffle of homomorphic encryptions ZK proof for shuffle of homomorphic encryptions Comparison with other ZK proofs Comparison with other ZK proofs Efficiency improvements Efficiency improvements

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Non-interactive commitment Public key Commitment c = commit(m; r) Opening given c, m, r check that c = commit(m; r)

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Commitment Binding Binding Unconditional: There is at most one way the comitter can open a commitment c Unconditional: There is at most one way the comitter can open a commitment c Computational: A polynomial time adversary cannot find c, m 1, r 1, m 2, r 2 so c = commit(m 1 ; r 1 ) = commit(m 2 ; r 2 ) and m 1 m 2 Computational: A polynomial time adversary cannot find c, m 1, r 1, m 2, r 2 so c = commit(m 1 ; r 1 ) = commit(m 2 ; r 2 ) and m 1 m 2 Hiding Hiding Statistical: Commitments to m and 0 have the same distribution Statistical: Commitments to m and 0 have the same distribution Computational: A polynomial time adversary cannot distinguish a random commitment to m 0 from a random commitment to 0 Computational: A polynomial time adversary cannot distinguish a random commitment to m 0 from a random commitment to 0

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Homomorphic commitment Homomorphic property com(m 1 +m 1 ´,..., m n +m n ´; r 1 +r 2 ) = com(m 1,..., m n ; r 1 ) com(m 1 ´,..., m n ´; r 2 ) Message space Z q n with q prime Root extraction property given c, m 1,...,m n, r, e so gcd(e,q) = 1 and c e = com(m 1,...,m n ; r) we can efficiently compute r´ so c = com(m 1 /e,...,m n /e; r´)

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Pedersen commitment variant Public key Primes q, p so p = kq+1 Random elements g 1,..., g n, h of order q pk = (q, p, g 1,..., g n, h) Commitment com(m 1,..., m n ; (u,r)) = ug 1 m 1 …g n m n h r mod p, where 1 = u k mod p Commitment verification Valid if 0 < c < p

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Shuffle of known content π, r com(m π(1),..., m π(n) ; r) m 1 m n...

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SHVZK proof for shuffle of known content A 4-round public coin SHVZK proof of knowledge for a commitment to a permutation of publicly known messages m 1,...,m n Optional - unconditional soundness or statistical SHVZK - key length vs efficiency

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Knowledge of contents Common: pk, c, m 1,..., m n Prover: π, r so c = com(m π(1),..., m π(n) ; r) c d = com(d 1,...,d n ; r d ) e {0,1} f i = em π(1) + d i, z = er+r d Check c e c d = com(f 1,...,f n ; z)

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Special HVZK Common: pk, c, m 1,..., m n Simulator: e {0,1} c d = com(f 1,...,f n ; z) c -e e f i Z q, z Z q Check c e c d = com(f 1,...,f n ; z)

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Knowledge Common: pk, c, m 1,..., m n c d = com(d 1,...,d n ; r d ) e, e´ {0,1} f i, z, f i ´, z´ c e c d = com(f 1,...,f n ; z) c e´ c d = com(f 1 ´,...,f n ´; z´) c e-e´ = com(f 1 -f 1 ´,...,f n -f n ´; z-z´) Root extraction: c = com(μ 1,...,μ n ; r)

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Idea (Neff 2001) Consider the polynomials (m i -X)and (μ i -X)in Z q [X] Are identical exactly when there exists π so μ i = m π(i) Pick x at random and demonstrate (m i -x) = (μ i -x) mod q With overwhelming probability not the case unless π exists

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Identical polynomials Common: pk, c, m 1,..., m n x {0,1} c d, c a, c Δ e {0,1} f i, z, f Δi, z Δ c e c d = com(f 1,...,f n ; z) c a e c Δ = com(f Δ1,...,f Δn-1 ; z Δ ) f i = eμ i + d i, f Δi = eα i + δ i

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Checking the polynomials f i = eμ i + d i, f Δi = eα i + δ i Let F 1 = f 1 -ex = e(μ 1 -x)+ d 1 Let eF i+1 = F i (f i+1 -ex) + f Δi e i F i+1 = e i-1 F i (f i+1 -ex) + f Δi = e i ( i (μ j -x) + poly i-1 (e)) (e(μ i+1 -x)+ d i+1 ) + e i-1 (eα i + δ i ) = e i+1 i+1 (μ j -x) + poly i (e) Check F n = e (m i -x) meaning e n (μ j -x) + poly n-1 (e) = e n (m i -x)

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Completeness F i = e i (μ j -x) + Δ i F 1 = f 1 -ex = e(m π(1) -x) + d 1 Δ 1 = d 1 eF i+1 = F i (f i+1 -ex) + f Δi eα i + δ i = e 2 i+1 (m π(j) -x) + eΔ i+1 - e( i (m π(j) -x) + Δ i )(e(m π(i+1) -x) + d i+1 ) = e(Δ i+1 - i (m π(j) -x) d i+1 - Δ i (m π(i+1) -x)) - Δ i d i+1 F n = e (m i -x) Δ n = 0

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SHVZK proof for known content 4-round public coin protocol 4-round public coin protocol Soundness – computational/unconditional Soundness – computational/unconditional SHVZK – statistical/computational SHVZK – statistical/computational With Pedersen commitment variant Prover3n expos2|q|n bits Verifier2n expos

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Agenda Motivation – anonymous communication Motivation – anonymous communication Mix-nets Mix-nets What is What is A shuffle? Homomorphic encryption? Zero- knowledge proofs? A shuffle? Homomorphic encryption? Zero- knowledge proofs? ZK proof for shuffle of known contents ZK proof for shuffle of known contents Tool: Homomorphic commitments Tool: Homomorphic commitments ZK proof for shuffle of homomorphic encryptions ZK proof for shuffle of homomorphic encryptions Comparison with other ZK proofs Comparison with other ZK proofs Efficiency improvements Efficiency improvements

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A shuffle of homomorphic encryptions π, R 1,...,R n e π(1) E(1;R 1 )e π(n) E(1;R n ) e1e1 enen

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Idea Want to show that e 1,..., e n and E 1,..., E n have the same plaintexts 1. Reveal π 2. Receive random challenges t 1,...,t n {0,1} 3. Release Z so E(1;Z) e i t i = E i t π(i) m i t i = M i t π(i) 1 = (M i /m π(i) ) t π(i) Since Q has no small prime factors M i = m π(i)

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Idea 1.Commit to π, commit to d 1,...,d n {0,1} +80 Form E d = E(1;R d ) E i -d i 2. Receive challenges t 1,...,t n {0,1} 3. Release f 1,...,f n, Z so f i = t π(i) + d i and E(1;Z) e i t i = E d E i f i m i t i = (M d M i d i ) M i t π(i) Z = R d + t π(i) R i

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Idea 1. Commit to π and d 1,...,d n c = com(π(1),...,π(n); r) c d = com(-d 1,...,-d n ; r d ) 2. Receive challenges t 1,...,t n 3. Send f 1,...,f n |q|> + 80 4. Receive challenge λ 5. Make SHVZK proof of known content for c λ c d com(f 1,...,f n ; 0) containing a permutation of λ + t 1,..., λn + t n π so π(i) + t π(i) With overwhelming probability over we have π(i) Exists π so λμ i + f i - d i = λ π(i) + t π(i) With overwhelming probability over λ we have μ i = π(i) and f i = t π(i) + d i

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Full protocol Common:pk, PK, e 1,...,e n and E 1,...,E n Prover: π, R 1,...,R n c, c d, E d t 1,...,t n {0,1} f 1,...,f n, Z λ {0,1} SHVZK proof Verify SHVZK proof Check E(1;Z) e i t i = E d E i f i

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Properties of shuffle proof 7-round public coin protocol 7-round public coin protocol Soundness – computational/unconditional Soundness – computational/unconditional SHVZK – statistical/computational SHVZK – statistical/computational With Pedersen commitment and ElGamal variants Prover4n p-expos, 2n P-expos 3|q|n bits Verifier2n p-expos, 4n P-expos

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Implementation (Stamer 2005) Pedersen commitment |p| = 1024, |q| = 160 ElGamal encryption|P| = 1024, |Q| =160 SHVZK proof of correct shuffle of 1024 ElGamal ciphertexts on AMD Duron 1.3 GHz Prover 14 seconds Verifier 5 seconds

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Agenda Motivation – anonymous communication Motivation – anonymous communication Mix-nets Mix-nets What is What is A shuffle? Homomorphic encryption? Zero- knowledge proofs? A shuffle? Homomorphic encryption? Zero- knowledge proofs? ZK proof for shuffle of known contents ZK proof for shuffle of known contents Tool: Homomorphic commitments Tool: Homomorphic commitments ZK proof for shuffle of homomorphic encryptions ZK proof for shuffle of homomorphic encryptions Comparison with other ZK proofs Comparison with other ZK proofs Efficiency improvements Efficiency improvements

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Other shuffle proofs Invariance of roots of polynomials Neff CCS01, Groth PKC03, Neff 03, Groth 05 Permutation matrices Furukawa & Sako Crypto01, Furukawa IEICE05 Integer commitments Wikström Asiacrypt05 Linear ignorance assumption Peng et al. Crypto05

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Comparison of approaches Pedersen, ElGamal |p|= 1024, |q| = 160 Roots of polyPermutation matrix Rounds7 3 Soundnessuncond./comp. computational SHVZKcomp./statistical statistical Prover expos6n7n Prover sends 480n bits 1344n bits Verifier expos6n8n Key lengthflexible (e.g. O(n)) 1024n bits

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Agenda Motivation – anonymous communication Motivation – anonymous communication Mix-nets Mix-nets What is What is A shuffle? Homomorphic encryption? Zero- knowledge proofs? A shuffle? Homomorphic encryption? Zero- knowledge proofs? ZK proof for shuffle of known contents ZK proof for shuffle of known contents Tool: Homomorphic commitments Tool: Homomorphic commitments ZK proof for shuffle of homomorphic encryptions ZK proof for shuffle of homomorphic encryptions Comparison with other ZK proofs Comparison with other ZK proofs Efficiency improvements Efficiency improvements

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Adjusting the key length Suggested Pedersen commitment variant had public key (q, p, g 1,..., g n, h) Assume wlog n = kl then we can instead use public key (q, p, g 1,..., g k, h) and commit as c = (c 1,...,c l ) (com(m 1,...,m k ), com(m k+1,...,m 2k ),...)

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Randomization c e c d = com(f 1,...,f n ; z) c a e c Δ = com(f Δ1,...,f Δn-1,0; z Δ ) Pick α {0,1} at random and check (c e c d ) α c a e c Δ = com(αf 1 +f Δ1,..., αf n +0; αz+z Δ ) Many other randomization/batch verification possibilities

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On-line/off-line computation Prover can precompute most values off-line (and in a mix-net also precompute the rerandomization of the ciphertexts) Prover can precompute most values off-line (and in a mix-net also precompute the rerandomization of the ciphertexts) Only needs to compute E d and c a on-line Only needs to compute E d and c a on-line

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Picking the challenges Verifier picks seed for pseudorandom number generator and sends it to prover Verifier picks seed for pseudorandom number generator and sends it to prover Prover generates t 1,...,t n from this seed If Q = q verifier can simply send challenge t and let prover use t 1 = t 1 mod q,..., t n = t n mod q If Q = q verifier can simply send challenge t and let prover use t 1 = t 1 mod q,..., t n = t n mod q

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Multi-exponentiation (Lim 00) Computing a product g i e i can be done in |e|n/(log n – log log n) multiplications Prover, Verifier 0.5n naïve single expos each for shuffling 100,000 ElGamal ciphertexts

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