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Efficient Zero-Knowledge Argument for Correctness of a Shuffle Stephanie Bayer University College London Jens Groth University College London

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Motivation – e-voting Voting:- Voter casts secret vote - Authorities reveal votes in random permuted order E-voting:- voter casts secret votes on a computer -The votes are sent to a server who sends all votes to the central authorities -Authorities reveal votes in random permuted order

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Background - ElGamal encryption

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Shuffle...

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Mix-net: … Threshold decryption

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Problem: Corrupt mix-server … Threshold decryption

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Solution: Zero-knowledge argument … Threshold decryption ZK argument No message changed (soundness) ZK argument Permutation still secret (zero-knowledge)

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Zero-Knowledge Argument Requested Properties: –Soundness: The Verifier reject with overwhelming probability if the Prover tries to cheat –Zero-Knowledge: Nothing but the truth is revealed; permutation is secret –Efficient: Small computation and small communication complexity ProverVerifier The Shuffle was done correctly

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Public coin honest verifier zero-knowledge ProverVerifier Honest verifier zero-knowledge Nothing but truth revealed; permutation secret Can convert to standard zero-knowledge argument

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Our contribution 9-move public coin honest verifier zero-knowledge argument for correctness of shuffle in common reference string model

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RoundsProver in expos Verifier in expos Size in kbits Furukawa-Sako 013 FMMOS 025 Furukawa 05 (GL07)3 Terelius-Wikström 105 Neff 01,047 Groth 03,107 Groth-Ishai 087 Bayer-Groth 119

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Commitments – Length reducing – Computational binding – Perfectly hiding

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Techniques - Sublinear cost Length reducing commitments Batch verification Sublinear communication cost

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Shuffle argument

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3.The prover gives an argument that both commitments are constructed using the same permutation 4.The prover demonstrates that the input ciphertexts are permuted using the same permutation and knowledge of the randomizers used in the re-encryption.

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Shuffle argument Inexpensive See full paper Expensive Will sketch idea Both polynomials are equal, only the roots are permuted

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Notation

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Multi-exponentiation argument idea

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Multi-exponentiation argument

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Prover’s computation Computing this matrix costs m 2 n = mN ciphertext expos

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Reducing the prover’s computation Do not compute entire matrix Instead use techniques for multiplication of polynomials “in the exponent” of ciphertexts Fast Fourier Transform – O(N log m) exponentiations O (1) rounds Interaction – O (N) exponentiations O (log m) rounds

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Implementation Implementation in C++ using the NTL library and the GMP library Different levels of optimization –Multi-exponentiation techniques –Fast Fourier Transform –Extra Interaction and Toom-Cook

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Comparison Single argumentArgument Size Verificatum5 min37.7 MB 2 min0.7 MB

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Thank You

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Efficient Zero-Knowledge Proofs Jens Groth University College London.

Efficient Zero-Knowledge Proofs Jens Groth University College London.

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