N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP G ROTH -S AHAI P ROOFS R EVISITED E. Ghadafi N.P. Smart B. Warinschi Department of Computer Science, University of Bristol 13th International Conference on Practice and Theory in Public Key Cryptography 2010 G ROTH -S AHAI P ROOFS R EVISITED
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP O UTLINE N ON -I NTERACTIVE P ROOF S YSTEMS 1 G ROTH -S AHAI P ROOFS 2 C ORRECTED G ROTH -S AHAI NIWI P ROOFS 3 G ROTH -S AHAI P ROOFS IN T YPE -2 P AIRINGS 4 R ESULTS AND C OMPARISON 5 S UMMARY 6 G ROTH -S AHAI P ROOFS R EVISITED
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP O UTLINE N ON -I NTERACTIVE P ROOF S YSTEMS 1 G ROTH -S AHAI P ROOFS 2 C ORRECTED G ROTH -S AHAI NIWI P ROOFS 3 G ROTH -S AHAI P ROOFS IN T YPE -2 P AIRINGS 4 R ESULTS AND C OMPARISON 5 S UMMARY 6 G ROTH -S AHAI P ROOFS R EVISITED
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP O UTLINE N ON -I NTERACTIVE P ROOF S YSTEMS 1 G ROTH -S AHAI P ROOFS 2 C ORRECTED G ROTH -S AHAI NIWI P ROOFS 3 G ROTH -S AHAI P ROOFS IN T YPE -2 P AIRINGS 4 R ESULTS AND C OMPARISON 5 S UMMARY 6 G ROTH -S AHAI P ROOFS R EVISITED
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP O UTLINE N ON -I NTERACTIVE P ROOF S YSTEMS 1 G ROTH -S AHAI P ROOFS 2 C ORRECTED G ROTH -S AHAI NIWI P ROOFS 3 G ROTH -S AHAI P ROOFS IN T YPE -2 P AIRINGS 4 R ESULTS AND C OMPARISON 5 S UMMARY 6 G ROTH -S AHAI P ROOFS R EVISITED
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP O UTLINE N ON -I NTERACTIVE P ROOF S YSTEMS 1 G ROTH -S AHAI P ROOFS 2 C ORRECTED G ROTH -S AHAI NIWI P ROOFS 3 G ROTH -S AHAI P ROOFS IN T YPE -2 P AIRINGS 4 R ESULTS AND C OMPARISON 5 S UMMARY 6 G ROTH -S AHAI P ROOFS R EVISITED
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP O UTLINE N ON -I NTERACTIVE P ROOF S YSTEMS 1 G ROTH -S AHAI P ROOFS 2 C ORRECTED G ROTH -S AHAI NIWI P ROOFS 3 G ROTH -S AHAI P ROOFS IN T YPE -2 P AIRINGS 4 R ESULTS AND C OMPARISON 5 S UMMARY 6 G ROTH -S AHAI P ROOFS R EVISITED
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP N ON -I NTERACTIVE P ROOFS "A proof is whatever convinces me.", Shimon Even. G ROTH -S AHAI P ROOFS R EVISITED 1 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP P ROPERTIES OF NIZK P ROOFS ◮ Completeness: Verifier always accepts a valid proof. ◮ Soundness: Prover only has a negligible probability in making the verifier accept a proof for a false statement. ◮ (Composable) Zero-Knowledge: Verifier cannot tell a real proof from a simulated one. G ROTH -S AHAI P ROOFS R EVISITED 2 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP A PPLICATIONS OF Z ERO -K NOWLEDGE P ROOFS Example applications: Anonymous Credentials: Client proves he possesses the required credentials without revealing them. Online Voting: Voter proves to the server that he has voted correctly without revealing his actual vote. E-Cash, Signature Schemes, Oblivious Transfer , CCA-2 Encryption Schemes, ... G ROTH -S AHAI P ROOFS R EVISITED 3 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP H ISTORY OF NIZK P ROOFS Blum-Feldman-Micali, 1988. Damgard, 1992. Killian-Petrank, 1998. Feige-Lapidot-Shamir, 1999. De Santis-Di Crescenzo-Persiano, 2002. Groth-Sahai, 2008. G ROTH -S AHAI P ROOFS R EVISITED 4 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP O UR C ONTRIBUTION ◮ We present a correction to a minor problem in GS NIWI proofs under the DLIN and XSDH assumptions. ◮ We extend GS proofs to work under Type-2 pairings; the previous formulation only worked under Type-1 and Type-3 pairings. G ROTH -S AHAI P ROOFS R EVISITED 5 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP B ILINEAR G ROUPS G 1 , G 2 , G T are finite cyclic groups of order n ( prime or composite number), where G 1 = < P 1 > and G 2 = < P 2 > . Pairing ( e : G 1 × G 2 − → G T ) : The function e must have the following properties: ◮ Bilinearity: ∀ Q 1 ∈ G 1 , Q 2 ∈ G 2 x , y ∈ Z n , we have e ([ x ] Q 1 , [ y ] Q 2 ) = e ( Q 1 , Q 2 ) xy . ◮ Non-Degeneracy: The value e ( P 1 , P 2 ) � = 1 generates G T . ◮ The function e is efficiently computable. G ROTH -S AHAI P ROOFS R EVISITED 6 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP P AIRINGS ’ T YPES ◮ Type-1 : This is the symmetric pairing setting in which G 1 = G 2 = G and e : G × G − → G T . ◮ Type-2 : e : G 1 × G 2 − → G T , where G 1 � = G 2 and there is an efficiently computable isomorphism ψ : G 2 − → G 1 where ψ ( P 2 ) = P 1 . ◮ Type-3 : e : G 1 × G 2 − → G T , where G 1 � = G 2 , but there is no known efficiently computable isomorphism. G ROTH -S AHAI P ROOFS R EVISITED 7 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP G ROTH -S AHAI P ROOFS f A 1 × A 2 → A T G ROTH -S AHAI P ROOFS R EVISITED 8 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP G ROTH -S AHAI P ROOFS f A 1 × A 2 → A T ι 1 ↓↑ ρ 1 ι 2 ↓↑ ρ 2 ι T ↓↑ ρ T F B 1 × B 2 − → B T G ROTH -S AHAI P ROOFS R EVISITED 8 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP G ROTH -S AHAI P ROOFS f A 1 × A 2 → A T ι 1 ↓↑ ρ 1 ι 2 ↓↑ ρ 2 ι T ↓↑ ρ T F B 1 × B 2 − → B T Properties: ∀ x ∈ A 1 , ∀ y ∈ A 2 : F ( ι 1 ( x ) , ι 2 ( y )) = ι T ( f ( x , y )) , ∀X ∈ B 1 , ∀Y ∈ B 2 : f ( p 1 ( X ) , p 2 ( Y )) = p T ( F ( X , Y )) . How does it work? Commit to the secrets(the witness), and just plug the commitments into the original equations you are proving! Binding Setting = ⇒ Perfect Soundness ( Allows witness extraction). Hiding Setting = ⇒ Perfect Witness Indistinguishability (Allows simulation). G ROTH -S AHAI P ROOFS R EVISITED 8 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP G ROTH -S AHAI P ROOFS Key Idea: Adversary cannot distinguish which setting we are working in. From NIWI to NIZK proofs ? In many cases (apart from a few Pairing Product Equations cases), it is easy to transform a NIWI proof into a NIZK proof. Just transform the equation into an equation with a trivial right-hand side and using the trapdoor information open a commitment to 1 to 0. What statements can be proven ? A variety of statements related to bilinear groups. G ROTH -S AHAI P ROOFS R EVISITED 9 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP T YPES OF EQUATIONS ◮ Pairing Product Equation n 1 n 2 n 1 n 2 e ( X i , Y j ) r i , j = T � � � � e ( A i , Y i ) · e ( X i , B i ) · i = 1 i = 1 i = 1 j = 1 here T ∈ G T ◮ Multi-scalar multiplication in G 1 n 1 n 2 n 1 n 2 � � � � y i A i + b i X i + r i , j y j X i = T 1 i = 1 i = 1 i = 1 i = j here T 1 ∈ G 1 ◮ Multi-scalar multiplication in G 2 n 1 n 2 n 1 n 2 � � � � a i Y i + x i B i + r i , j x i Y j = T 2 i = 1 i = 1 i = 1 i = j here T 2 ∈ G 2 ◮ Quadratic-equation in Z p n 1 n 2 n 1 n 2 � � � � a i y i + x i b i + r i , j x i y j = t i = 1 i = 1 i = 1 i = j here t ∈ Z p G ROTH -S AHAI P ROOFS R EVISITED 10 / 18
N ON -I NTERACTIVE P ROOF S YSTEMS G ROTH -S AHAI P ROOFS C ORRECTED G ROTH -S AHAI NIWI P ROOFS G ROTH -S AHAI P ROOFS IN T YP H ARD P ROBLEMS D EFINITION Symmetric External Diffie-Hellman (SXDH) Assumption: e : G 1 × G 2 − → G T (Type-3 Pairings) Setting : Assumption: DDH problem is hard in both G 1 and G 2 . D EFINITION Decisional Linear Problem(DLIN) Assumption: e : G × G − → G T (Type-1 Pairings) Setting : Input: ([ a ] P , [ b ] P , [ ra ] P , [ sb ] P , [ t ] P ) where a , b , r , s , t ∈ F q It is hard to tell whether t = r + s or t is random. Assumption: G ROTH -S AHAI P ROOFS R EVISITED 11 / 18
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