# Strong Accumulators from Collision-Resistant Hashing

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1 Strong Accumulators from Collision-Resistant Hashing ISC 2008 Taipei - Taiwan Philippe Camacho (University of Chile Alejandro Hevia (University of Chile Marcos Kiwi (University of Chile Roberto Opazo (CEO Acepta.com

2 Outline Notion of accumulator Motivation e-invoice Factoring Our construction Conclusion

3 Notion of accumulator Problem A set X. Given an element x we wish to prove that this element belongs or not to X. Let X={x 1,x 2,,x n }: X will be represented by a short value Acc. Belongs(Acc,x,w = True x belongs to X. Witness

4 Notion of accumulator Accumulator Manager Computes setup values. Computes the accumulated value Acc. Computes the witness w x for a given x. Accumulator Users Check that an element belongs or not to the set, using Acc, w x and x.

5 Applications Time-stamping [BeMa94] Certificate Revocation List [LLX07] Anonymous credentials [CamLys02] E-Cash [AWSM07] Broadcast Encryption [GeRa04]

6 Nothing to see with Number Theory! Factoring Industry in Chile Factoring Entity Provider (Milk seller Client (Supermarket

7 Nothing to see with Number Theory! Factoring Industry in Chile Factoring Entity Provider (Milk seller 1 I want (a lot of milk now *. Client (Supermarket (* but I do not want to pay yet.

8 Nothing to see with Number Theory! Factoring Industry in Chile Factoring Entity Provider (Milk seller 1 I want (a lot of milk now *. 2 Here is your milk. Client (Supermarket (* but I do not want to pay yet.

9 Nothing to see with Number Theory! Factoring Industry in Chile Factoring Entity 3 Please pay the invoice. Provider (Milk seller 1 I want (a lot of milk now *. 2 Here is your milk. Client (Supermarket (* but I do not want to pay yet.

10 Nothing to see with Number Theory! Factoring Industry in Chile Factoring Entity 3 Please pay the invoice. 4 Here is your money (**. Provider (Milk seller 1 I want (a lot of milk now *. 2 Here is your milk. Client (Supermarket (* but I do not want to pay yet. (** minus a fee.

11 Nothing to see with Number Theory! Factoring Industry in Chile Factoring Entity 3 Please pay the invoice. 4 Here is your money (**. 5 It s time to pay. Provider (Milk seller 1 I want (a lot of milk now *. 2 Here is your milk. Client (Supermarket (* but I do not want to pay yet. (** minus a fee.

12 Nothing to see with Number Theory! Factoring Industry in Chile Factoring Entity 3 Please pay the invoice. 4 Here is your money (**. 5 It s time to pay. 6 Here is the money. Provider (Milk seller 1 I want (a lot of milk now *. 2 Here is your milk. Client (Supermarket (* but I do not want to pay yet. (** minus a fee.

13 The Problem A malicious provider could send the same invoice to various Factoring Entities. Then he leaves to a far away country with all the money. Later, several Factoring Entities will try to charge the invoice to the same client. Losts must be shared

14 Solution with Factoring Authority Factoring Authority (4 Is there the invoice? (5 YES / NO FE 1 FE 2 FE i FE n (3 Invoice, Ack Provider (1 Invoice (2 Ack Client

15 Caveat This solution is quite simple. However Trusted Factoring Authority is needed. Can we remove this requirement?

16 Properties Dynamic Allows insertion/deletion of elements. Universal Allows proofs of membership and nonmembership. Strong No need to trust in the Accumulator Manager.

17 Prior work Dynamic Strong Universal Security Efficiency (witness size Note [BeMa94] RSA + RO O(1 First definition [BarPfi97] Strong RSA O(1 - [CamLys02] Strong RSA O(1 First dynamic accumulator [LLX07] Strong RSA O(1 First universal accumultor [AWSM07] Pairings O(1 E-cash [WWP08] estrong RSA Paillier O(1 Batch Update

18 Prior work Dynamic Strong Universal Security Efficiency (witness size Note [BeMa94] RSA + RO O(1 First definition [BarPfi97] Strong RSA O(1 - [CamLys02] Strong RSA O(1 First dynamic accumulator [LLX07] Strong RSA O(1 First universal accumultor [AWSM07] Pairings O(1 E-cash [WWP08] estrong RSA Paillier O(1 Batch Update [CHKO08] Collision-Resistant Hashing O(ln(n Our work

19 Notation H: {0,1}* {0,1} k randomly chosen function from a family of collision-resistant hash functions. x 1,x 2,x 3, є {0,1} k x 1 < x 2 < x 3 < where < is the lexicographic order on binary strings. -, Special values such that For all x є {0,1} k : - < x < denotes the concatenation operator.

20 Ideas Merkle-trees P=H(Z 1 Z 2 Root value: Z 1 =H(Y 1 Y 2 Z 2 =H(Y 3 Y 4 Represents the set {x 1,,x 8 } Y 1 =H(x 4 x 1 Y 2 =H(x 5 x 6 Y 3 =H(x 2 x 8 Y 4 =H(x 7 x 3 x 4 x 1 x 5 x 6 x 2 x 8 x 7 x 3

21 Ideas Merkle-trees P=H(Z 1 Z 2 Root value: Represents the set {x 1,,x 8 } Z 1 =H(Y 1 Y 2 Z 2 =H(Y 3 Y 4 Y 1 =H(x 4 x 1 Y 2 =H(x 5 x 6 Y 3 =H(x 2 x 8 Y 4 =H(x 7 x 3 O(ln(n x 4 x 1 x 5 x 6 x 2 x 8 x 7 x 3

22 Ideas How to prove non-membership? Kocher s trick [Koch98]: store pair of consecutive values X={1,3,5,6,11} X ={(-,1,(1,3,(3,5,(5,6,(6,11,(11, } y=3 belongs to X (1,3 or (-,1 belongs to X. y=2 does not belong to X (1,3 belongs to X.

23 Public Data Structure Called Memory. Compute efficiently the accumulated value and the witnesses. In our construction the Memory will be a binary tree.

24 How to insert elements? (-, X=Ø, next: x 1

25 How to insert elements? (-,x 1 (x 1, X={x 1 }, next: x 2

26 How to insert elements? (-,x 1 (x 1, x 2 (x 2, X={x 1,x 2 }, next: x 5

27 How to insert elements? (-,x 1 (x 1, x 2 (x 2, x 5 (x 5, X={x 1,x 2,x 5 }, next: x 3

28 How to insert elements? (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, (x 3, x 5 X={x 1,x 2,x 3,x 5 }, next: x 4

29 How to insert elements? (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, (x 3, x 4 (x 4, x 5 X={x 1,x 2,x 3,x 4,x 5 }, next: x 6

30 How to insert elements? (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, X={x 1,x 2,x 3,x 4,x 5,x 6 }

31 value? (-,x 1 How to compute the accumulated (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, x 7 (x 9, (x 7, x 9 Proof N =H(Proof left Proof right value Proof Nil = Acc = Proof Root A pair (x i,x j

32 How to update the accumulated value? (Insertion (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, x 7 (x 9, (x 7, x 9 Next element to be inserted: x 8 We will need to recompute proof node values.

33 How to update the accumulated value? (Insertion (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, x 7 (x 9, (x 7, x 8 (x 8, x 9 New element: x 8. Proof N stored in each node. Dark nodes do not require recomputing Proof N. Only a logarithmic number of values needs recomputation.

34 Security Consistency Difficult to find witnesses that allow to prove inconsistent statements. X={1,2} Update Hard to compute a membership witness for 3. Hard to compute a nonmembership witness for 2. Guarantees that the accumulated value represents the set after insertion/deletion of x.

35 Security Lemma: Given a tree T with accumulated value Acc T, finding a tree T, T T such that Acc T = Acc T is difficult. Proof (Sketch: Proof N = H(Proof left Proof right value (-,x 1 (x 1, x 2 (x 2, x 3 (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, x 7 (x 5, x 6 (x 3, x 4 (x 4, x 7 (x 6, x 7

36 Security Lemma: Given a tree T with accumulated value Acc T, finding a tree T, T T such that Acc T = Acc T is difficult. Proof (Sketch: Proof N = H(Proof left Proof right value Collision for H (-,x 1 (x 1, x 2 (x 2, x 3 (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, x 7 (x 5, x 6 (x 3, x 4 (x 4, x 7 (x 6, x 7

37 Security Lemma: Given a tree T with accumulated value Acc T, finding a tree T, T T such that Acc T = Acc T is difficult. Proof (Sketch: Proof N = H(Proof left Proof right value Collision for H (-,x 1 (x 1, x 2 (x 2, x 3 (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, x 7 (x 5, x 6 (x 3, x 4 (x 4, x 7 (x 6, x 7

38 Security Lemma: Given a tree T with accumulated value Acc T, finding a tree T, T T such that Acc T = Acc T is difficult. Proof (Sketch: Proof N = H(Proof left Proof right value Collision for H (-,x 1 (x 1, x 2 (x 2, x 3 (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, x 7 (x 5, x 6 (x 3, x 4 (x 4, x 7 (x 6, x 7

39 Security (Consistency (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, x 7 (x 9, (x 7, x 9 Witness: blue nodes and the (x 3,x 4 pair, size in O(ln(n Checking that x belongs (or not to X: 1 compute recursively the proof P and verify that P=Acc 2 check that: x=x 3 or x=x 4 (membership x 3 < x < x 4 (nonmembership

40 Security (Update Before After (-,x 1 (-,x 1 Acc before Acc after (x 1, x 2 (x 2, x 3 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, x 7 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, x 7 (x 9, (x 7, x 9 (x 9, (x 7, x 8 (x 8, x 9 Insertion of x 8

41 Conclusion & Open Problem First dynamic, universal, strong accumulator. Simple. Security Existence of collision-resistant hash functions. Solves the e-invoice Factoring Problem. Less efficient than other constructions Size of witness in O(ln(n. Open Problem Is it possible to build a strong,dynamic and universal accumulator with witness size lower than O(ln(n?

42 Thank you!

43 Invoice Factoring using accumulator We need a secure broadcast channel If a message m is published, every participant sees the same m. Depending on the security level required Trusted http of ftp server Bulletin Board [CGS97]

44 Invoice Factoring using accumulator Factoring Authority (4 Is there the invoice? (5 YES / NO We need to see in detail this step FE 1 FE 2 FE i FE n (3 Invoice, Ack Provider (1 Invoice (2 Ack Client

45 Invoice Factoring using accumulator Step 5 (Details FE Factoring Authority Have you got invoice x? YES/NO, w x w x = Witness(m before,x Check(Acc before,w x,x If NO, insert x Acc new,w up,id FE Acc new,w up = Update Add (m before,x CheckUpdate(Acc before,acc after,w x All tests pass => I can buy x.

46 Distributed solutions? Complex to implement Hard to make them robust High bandwith communication Need to be online synchronization problems That s why we focus on a centralized solution.

47 Checking for (non-membership User Accumulator Manager Does x belong to X? Memory w x = Witness(m,x w x Belongs(Acc,w x = True x є X If w x is not valid Belongs returns.

48 Update of the accumulated value User Accumulator Manager Insert or Delete x m after, Acc after, w up = Update Add/Del (m before,x Acc after, w up CheckUpdate(Acc before,acc after,w up

49 How to delete elements? (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6, X={x 1,x 2,x 3,x 4,x 5,x 6 } element to be deleted: x 2

50 How to delete elements? (x 1,x 3 (-,x 1 (x 1, x 2 (x 2, x 3 (x 5, x 6 (x 3, x 4 (x 4, x 5 (x 6,

51 How to delete elements? (-,x 1 (x 1, x 3 (x 6, (x 5, x 6 (x 3, x 4 (x 4, x 5

52 Bibliography [BeMa92] Efficient Broadcast Time-Stamping Josh Benaloh and Michael de Mare 1992 [BeMa94] One-way Accumulators: A decentralized Alternative to Digital Signatures Josh Benaloh and Michael de Mare, 1994 [BarPfi97] Collision-Free Accumulators and Fail-Stop Signature Schemes Without Trees Niko Barić and Birgit Pfitzmann 1997 [CGS97] A secure and optimally efficient multi-authority election scheme R. Cramer, R. Gennaro, and B. Schoenmakers 1997 [Koch98] On certificate revocation and validation P.C. Kocher 1998 [CGH98] The random oracle methodoly revisited R. Canetti, O. Goldreich and S. Halevi 1998 [Sand99] Efficient Accumulators Without Trapdoor Tomas Sanders 1999 [GoTa01] An efficient and Distributed Cryptographic Accumulator Michael T. Goodrich and Roberto Tamassia 2001 [CamLys02] Dynamic Accumulators And Application to Efficient Revocation of Anonymous Credentials Jan Camenisch Anna Lysyanskaya 2002 [GeRa04] RSA Accumulator Based Broadcast Encryption Craig Gentry and Zulfikar Ramzan 2004 [LLX07] Universal Accumulators with Efficient Nonmembership Proofs Jiangtao Li, Ninghui Li and Rui Xue 2007 [AWSM07] Compact E-Cash from Bounded Accumulator Man Ho Au, Qianhong Wu, Willy Susilo and Yi Mu 2007 [WWP08] A new Dynamic Accumulator for Batch Updates Peishun Wang, Huaxiong Wang and Josef Pieprzyk 2008 [CKHO08] Strong Accumulators from Collision-Resistant Hashing Philippe Camacho, Alejandro Hevia, Marcos Kiwi, and Roberto Opazo 2008

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