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Almost Entirely Correct Mixing With Applications to Voting

Almost Entirely Correct Mixing With Applications to Voting. Philippe Golle Dan Boneh Stanford University. Inputs. Outputs. Proof. Mix Server. Mix Server. ?. A mix server is a cryptographic implementation of a hat. Proof. Proof. Proof. Inputs. Outputs. Mix Network. Mix network

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Almost Entirely Correct Mixing With Applications to Voting

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  1. Almost Entirely Correct MixingWith Applications to Voting Philippe Golle Dan Boneh Stanford University

  2. Inputs Outputs Proof Mix Server Mix Server ? A mix server is a cryptographic implementation of a hat.

  3. Proof Proof Proof Inputs Outputs Mix Network • Mix network • A group of mix servers that operate sequentially. Server 1 Server 2 Server 3 ? ? ? • If a single mix server is honest, global permutation is secret.

  4. Submission Tabulation Mix Applications • Anonymous voting • Other applications: • Anonymous payments • Anonymous channels All these applications require efficient schemes

  5. Properties • Privacy: outputs can’t be matched to inputs • Correctness: outputs match inputs • Robustness: an output is produced regardless of possible mix server failures or bad inputs • Verifiability: local or universal • Efficiency

  6. ? Inputs Outputs Zoology of Mix Networks • Decryption Mix Nets [Cha81,…]: • Inputs: ciphertexts • Outputs: decryption of the inputs. • Re-encryption Mix Nets[PIK93,…]: • Inputs: ciphertexts • Outputs: re-encryption of the inputs

  7. 1. Users encrypt their inputs: Input Input Pub-key 2. Encrypted inputs are mixed: Server 1 Server 2 Server 3 re-encrypt & mix re-encrypt & mix re-encrypt & mix Proof Proof Proof 3. A quorum of mix servers decrypts the outputs Priv-key Output Output Re-encryption Mixnet 0. Setup: mix servers generate a shared ElGamal key

  8. ElGamal Cryptosystem • ElGamal is a randomized public-key cryptosystem • Plaintexts in a group G of prime order q • Ciphertext are pairs (a,b) where a,b in G. • Malleable: Er(m)  Er+s(m) • ZK proof that two CT decrypt to the same PT (1 exp) • Multiplicative homomorphism: E(m) , E(m’)  E(mm’)

  9. Problem • Mix servers must prove correct re-encryption • Inputs: n ElGamal ciphertexts E(mi ) • Outputs: n ElGamal ciphertexts E(m’i) • Mix proves that there is a permutation π such that: without revealing π.

  10. Quick survey of proofs of re-encryption n = number of inputs k = number of servers

  11. Verifier: • Computes: E(i=1mi) and E(i=1m’i) • Ask Mix for ZK proof that these CT decrypt to same PT. n n Proving Correct Re-encryption • Mix server: • Receives: n ElGamal ciphertexts E(mi ) • Produces: n ElGamal ciphertexts E(m’i) • Observations: • Honest mix can always give this proof • Verification is necessary but not sufficient • Idea: use random subsets  the name PSP

  12. n n 2. Mix gives ZK proof that i=1mi = i=1m’i mod q Repeat α times 5. Mix gives ZK proof that Proof-of-Subproduct (PSP) Mix net Mix Server Inputs mi Outputs m’i S S’ • Mix the inputs 3. Verifiers choose random subset S 4. The mix server reveals image S’

  13. Theorem: cheating mix is detected with prob > Conjecture: cheating is detected with prob > where w is the number of wrong outputs Properties of PSP • PSP is sound • PSP is robust • Efficiency (per mix server, for n inputs): • Mixing: n exponentiations • Proof: α exponentiations (e.g. α = 5) • Constant in number of inputs! • Privacy:users only lose α bits of privacy on average

  14. Applications of PSP • Large elections: 160,000 ballots. • Suppose the mixnet corrupts 100 votes. • With α = 6: • Every ballot hidden among 2,500 others • Provable bound: prob > 94% cheating detected • Conjectured bound: prob > 99.9% cheating detected • PSP is compatible with other verification schemes that offer full correctness: • Use PSP to verify output • Announce the output • Run another slower scheme to verify the output

  15. Proof of Correctness • Theorem: cheating is detected with probability 1 – (5/8) • A cheating mix that fools the verifier with prob > 1 – (5/8)can compute discrete logarithm in G. • Reduction relies on the following theorem: Let S be a subset of {0,1}n such that |S| > (5/8)2n Let F : S  {0,1}n be a linear function such that: • F(S) spans all of Zqn • F preserves the L norm Then there exists a permutation matrix P such that F(v)=P.v for all v in S.

  16. Conclusion • The difficulty lies in giving efficient proofs of correctness. • We propose a new scheme: PSP • Exploit the multiplicative homomorphism of ElGamal • Exceptionally computationally efficient • PSP only guarantees near correctness • Full paper at: http://crypto.stanford.edu/~pgolle

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