B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS E FFICIENT T WO -M OVE B LIND S IGNATURES IN THE C OMMON R EFERENCE S TRING M ODEL E. Ghadafi N.P. Smart Department of Computer Science, University of Bristol Information Security Conference – ISC 2012 E FFICIENT T WO -M OVE B LIND S IGNATURES . . .
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS O UTLINE B LIND S IGNATURES 1 S ECURITY M ODEL 2 R ELATED W ORK 3 O UR C ONSTRUCTION 4 E FFICIENCY C OMPARISON 5 O PEN P ROBLEMS 6 E FFICIENT T WO -M OVE B LIND S IGNATURES . . .
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS O UTLINE B LIND S IGNATURES 1 S ECURITY M ODEL 2 R ELATED W ORK 3 O UR C ONSTRUCTION 4 E FFICIENCY C OMPARISON 5 O PEN P ROBLEMS 6 E FFICIENT T WO -M OVE B LIND S IGNATURES . . .
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS O UTLINE B LIND S IGNATURES 1 S ECURITY M ODEL 2 R ELATED W ORK 3 O UR C ONSTRUCTION 4 E FFICIENCY C OMPARISON 5 O PEN P ROBLEMS 6 E FFICIENT T WO -M OVE B LIND S IGNATURES . . .
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS O UTLINE B LIND S IGNATURES 1 S ECURITY M ODEL 2 R ELATED W ORK 3 O UR C ONSTRUCTION 4 E FFICIENCY C OMPARISON 5 O PEN P ROBLEMS 6 E FFICIENT T WO -M OVE B LIND S IGNATURES . . .
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS O UTLINE B LIND S IGNATURES 1 S ECURITY M ODEL 2 R ELATED W ORK 3 O UR C ONSTRUCTION 4 E FFICIENCY C OMPARISON 5 O PEN P ROBLEMS 6 E FFICIENT T WO -M OVE B LIND S IGNATURES . . .
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS O UTLINE B LIND S IGNATURES 1 S ECURITY M ODEL 2 R ELATED W ORK 3 O UR C ONSTRUCTION 4 E FFICIENCY C OMPARISON 5 O PEN P ROBLEMS 6 E FFICIENT T WO -M OVE B LIND S IGNATURES . . .
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS (T WO -M OVE ) B LIND S IGNATURES pk sk USER SIGNER E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 1 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS (T WO -M OVE ) B LIND S IGNATURES pk sk USER SIGNER E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 1 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS (T WO -M OVE ) B LIND S IGNATURES pk sk Sig USER SIGNER E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 1 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS (T WO -M OVE ) B LIND S IGNATURES pk sk Sig USER SIGNER E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 1 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS A PPLICATIONS OF B LIND S IGNATURES Example applications: ◮ E-Cash: A bank signs a coin without learning its serial number (provides unlinkability between withdrawal and spend transactions). ◮ E-Voting: Authority certifies a ballot without learning its content. The client cannot vote for more than one candidate. ◮ Many other applications where anonymity/privacy or unlinkability are required (Anonymous Access Control, ... etc. ). E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 2 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS A LGORITHMS OF A B LIND S IGNATURE ◮ Setup − Setup BS ( 1 λ ) crs BS ← ◮ Key Generation ( sk BS , pk BS ) ← − KeyGen BS ( crs BS ) ◮ Signing ( ⊥ , σ ) ← − � Request BS ( pk BS , m ) , Issue BS ( sk BS ) � ◮ Verification 1 / 0 ← − Verify BS ( pk BS , m , σ ) E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 3 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS S ECURITY OF B LIND S IGNATURES ◮ Blindness [JLO97,PS00]: The Signer does not learn what message he is signing nor can he link a signature to its sign request. m 0 ,m 1 pk BS ,sk BS b {0,1} σ b Request BS ( pk BS ,m b ) Request BS ( pk BS ,m b ) σ 1-b Request BS ( pk BS ,m 1-b ) Request BS ( pk BS ,m 1-b ) (σ 0 ,σ 1 ) or ( , ) ⟂ ⟂ b * The adversary wins if b ∗ = b . • Malicious Keys [Oka06]: The adversary generates the keys. E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 4 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS S ECURITY OF B LIND S IGNATURES ◮ Blindness [JLO97,PS00]: The Signer does not learn what message he is signing nor can he link a signature to its sign request. m 0 ,m 1 pk BS ,sk BS b {0,1} σ b Request BS ( pk BS ,m b ) Request BS ( pk BS ,m b ) σ 1-b Request BS ( pk BS ,m 1-b ) Request BS ( pk BS ,m 1-b ) (σ 0 ,σ 1 ) or ( , ) ⟂ ⟂ b * The adversary wins if b ∗ = b . • Malicious Keys [Oka06]: The adversary generates the keys. E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 4 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS S ECURITY OF B LIND S IGNATURES ◮ (Weak) Unforgeability [JLO97,PS00]: The User cannot output more signatures than the number of interactions with the signer. pk BS Issue BS (sk BS ) Issue BS (sk BS ) (n times) (m 1 ,σ 1 ),…,(m n+1 ,σ n+1 ) The adversary wins if all σ i verify and the messages are distinct. E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 5 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS R ELATED W ORK Some previous two-move constructions: ◮ Chaum 1983: using RSA signatures (ROM). ◮ Boldyreva 2003: using BLS signatures (ROM). ◮ Fischlin 2006: generic construction (CRS). ◮ Fuchsbauer 2009: special case instantiation of Fischlin 2006 (CRS). ◮ AHO 2010: efficient instantiation of Fischlin 2006 (CRS). ◮ MSF 2010: using Waters signatures in composite-order groups (CRS). ◮ Garg et al. 2011: generic construction (Standard Model). E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 6 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS O UR A PPROACH We follow the Blind-Unblind paradigm ... pk sk m m' m'←Blind(m,r) σ'← Sign(sk,m') USER SIGNER σ←Unblind(σ',r) However, we dispense with the need for random oracles by requiring a common reference string. E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 7 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS (P RIME -O RDER ) B ILINEAR G ROUPS G 1 , G 2 , G T are finite cyclic groups of prime order q , 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 , 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. Type-3 [GPS08]: G 1 � = G 2 and no efficiently computable isomorphism between G 1 and G 2 . E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 8 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS I NTRACTABILITY A SSUMPTIONS D EFINITION (LRSW A SSUMPTION [LRSW99]) Given ( X ← [ x ] P 2 , Y ← [ y ] P 2 ) and access to an oracle O X , Y ( · ) that, on input f i ∈ Z q outputs ( A i , B i , C i ) ← ( A i , [ y ] A i , [ x + f i · x · y ] A i ) , for some random A i ∈ G 1 , it is hard to output ( f ∗ , A ∗ , B ∗ , C ∗ ) where f ∗ / ∈ { f i } ∪ { 0 } . D EFINITION (B-LRSW A SSUMPTION [CMS09]) Given ( X ← [ x ] P 2 , Y ← [ y ] P 2 ) and access to an oracle O B X , Y ( · ) that, on input F i = [ f i ] P 1 ∈ G 1 outputs ( A i , B i , C i ) ← ( A i , [ y ] A i , [ x + f i · x · y ] A i ) , for some random A i ∈ G 1 , it is hard to output ( f ∗ , A ∗ , B ∗ , C ∗ ) where [ f ∗ ] P 1 / ∈ { F i } ∪ { 0 G 1 } . E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 9 / 18
B LIND S IGNATURES S ECURITY M ODEL R ELATED W ORK O UR C ONSTRUCTION E FFICIENCY C OMPARISON O PEN P ROBLEMS I NTRACTABILITY A SSUMPTIONS D EFINITION (LRSW A SSUMPTION [LRSW99]) Given ( X ← [ x ] P 2 , Y ← [ y ] P 2 ) and access to an oracle O X , Y ( · ) that, on input f i ∈ Z q outputs ( A i , B i , C i ) ← ( A i , [ y ] A i , [ x + f i · x · y ] A i ) , for some random A i ∈ G 1 , it is hard to output ( f ∗ , A ∗ , B ∗ , C ∗ ) where f ∗ / ∈ { f i } ∪ { 0 } . D EFINITION (B-LRSW A SSUMPTION [CMS09]) Given ( X ← [ x ] P 2 , Y ← [ y ] P 2 ) and access to an oracle O B X , Y ( · ) that, on input F i = [ f i ] P 1 ∈ G 1 outputs ( A i , B i , C i ) ← ( A i , [ y ] A i , [ x + f i · x · y ] A i ) , for some random A i ∈ G 1 , it is hard to output ( f ∗ , A ∗ , B ∗ , C ∗ ) where [ f ∗ ] P 1 / ∈ { F i } ∪ { 0 G 1 } . E FFICIENT T WO -M OVE B LIND S IGNATURES . . . 9 / 18
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