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Conditional Encrypted Mapping and Comparing Encrypted Numbers

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Conditional Encrypted Mapping and Comparing Encrypted Numbers

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  1. Conditional Encrypted Mapping andComparing Encrypted Numbers Vladimir Kolesnikov Joint work with Ian F. Blake University of Toronto

  2. Privacy in Auctions Note to self: spam Austin with $999 tickets offers I am selling a ticket to Anguilla. $1000 One hundred million dollars! Sorry Deal!

  3. Comparing Encrypted Numbers I have no idea what the bids were Enc($100000000) Enc($1000) Enc(0) Enc(1) I lost I won What if bidders lie about the result?

  4. Conditional Encrypted Mapping (CEM) Prepare two secrets: s1 – signed contract s0– loser notification Enc($100000000) Enc($1000) Enc(s0) Enc(s1) s0 s1

  5. Q-CEM s0, s1 Q(x,y) m=Rmap(s0, s1, e0, e1, pk) e0 = Enc(x) e1 = Enc(y) m ? ? Rec(m, sk) = sQ(x,y) Pair (Rmap, Rec) for Q is a Q-CEM

  6. Definitional Choices CEM: Rmap(s0, s1, e0, e1, pk), Rec(m, sk) • Strong notion of privacy • Output of Rmap contains no statistical information other than the value sQ(x,y) • Strong composability • Holds for all generated key pairs, valid inputs and randomness used in encryption • E.g. Adv does not benefit from maliciously • choosing randomness when encrypting inputs

  7. Definitional Choices CEM: Rmap(s0, s1, e0, e1, pk), Rec(m, sk) • Do not specify security requirements of the • encryption scheme • One definition is useable in most settings • Delay discussion of easy but tedious details (e.g. what if inputs contain decryption keys) • Q-CEM with semantically secure encryption gives a protocol in the semi-honest model • can be modified to withstand malicious players (ZK or the light-weight CDS)

  8. Some of Related Work • Auctions and GT • Naor, Pinkas, Sumner 1999 • Di Crescenzo 2000 • Fischlin 2001 • Laur, Lipmaa 2005 • Many others • CEM • Conditional Oblivious Transfer and variants • Di Crescenzo, Ostrovsky, Rajagopalan 1999 • Gertner, Ishai, Kushilevitz, Malkin 1998 • Aiello, Ishai, Reingold 2001 • Di Crescenzo 2000 • Laur Lipmaa 2005

  9. Tools – Homomorphic Encryption Encryption scheme, such that: Given E(m1), E(m2) and public key, allows to compute E(m1­ m2) We will need: • Additively homomorphic (­ = +) schemes • Large plaintext group The Paillier scheme satisfies our requirements Can compute E(cm1 + m2) from c, E(m1), E(m2)

  10. The GT-CEM Construction y1, …, yn x1-y1, …, xn-yn :0 = 0, i = rii-1+di 1 -1 1 -1 1 -1 d = 0 0 0 0 0… 1 -1 = 0 0 0 t1 t2 t3 t4 t5… s0, s1 x y x1, …, xn d = Linear Map 0  R -1  s01  s1 0  R -1  ES01  ES1 • ESi is a randomized encoding of si • contains no other information

  11. Randomized Mapping Given s0, s1 f(-1) = b-a = ES0 (1) f(1) = a+b = ES1 (2) f(0) = b = ½ (ES0 + ES1) ES0, ES1, f(x) = ax + b Assume s0, s1 contain redundancy Choose R 2R ZN. View R as blocks r0, r1: R = r0 2k + r1 r0 r0 s0 r1 s0 _ _ _ _ _ _. _ _ _ _ _ _ _ _ _ _ _. _ _ _ _ _ ES0 = ES1 = s1 r1 r1 r0 s1 _ _ _ _ _ _. _ _ _ _ _ _ _ _ _ _ _. _ _ _ _ _ c=0 c=1 c 2R {0,1} • Set f = ax+b to satisfy (1),(2) • f(-1), f(1) contain s0, s1 and no extra information* • f(0) = ½ (ES0 + ES1) = ½ (s0 2k + r1 + r0 2k + s1) = • ½ (R + … ) = R’

  12. Resource Comparison Orders of magnitude improvement over GM-based schemes Performance similar to previous Paillier-based COT schemes • c-bit secrets are transferred based on comparison of n-bit numbers. • and  are the correctness and security parameter

  13. Conclusions • General and convenient definition of CEM • CEM for any NC1 predicate • GT-CEM Constructions • Simple and composable • Especially efficient for transferring larger secrets ( e.g. ¼500-1000 bits ) • Applications to auctions, etc