Linearly-Homomorphic Signatures and Scalable Mix-Nets Chlo Hbant, - PowerPoint PPT Presentation

Linearly-Homomorphic Signatures and Scalable Mix-Nets Chloé Hébant, Duong Hieu Phan and David Pointcheval

Outline 1. Mix-Nets in drawings 2. Building blocks 3. Spirit of our scheme 4. Difficulties 2

Mix-Net 3

Mix-Net 3

Mix-Net 3

Electronic Voting 4

Electronic Voting Ballot 4

Electronic Voting Ballot 4

Electronic Voting Ballot 3 1 2 4

Some Guarantees Authorized voters Ballot Unforgeability 5

Some Guarantees Authorized voters Indistinguishable Ballot Unforgeability 5

Some Guarantees No added ballot No deleted ballot Sealed No modified ballot Authorized voters Indistinguishable Ballot Unforgeability 5

Some Guarantees No added ballot No deleted ballot Sealed No modified ballot Authorized voters Indistinguishable 3 Ballot Unforgeability 1 Publicly verifiable 2 5

Mix-Net: Security Notions 1. Soundness 2. Unlinkability 6

Mix-Net: Soundness … Permutation 7

Mix-Net: Unlinkability Unlinkable 8

Building Blocks 9

Key Ingredients • Ciphertext randomization • “Signature randomization” 10

Key Ingredients • Ciphertext randomization • e.g. El Gamal • “Signature randomization” 10

Key Ingredients • Ciphertext randomization • e.g. El Gamal • “Signature randomization” • Linearly Homomorphic Signature 10

Signature 11

One-Time Linearly Homomorphic Signature (OT-LH-Sig) 12

Linearly Homomorphic Signature (LH-Sig) 13

Linearly Homomorphic Signature (LH-Sig) 13

Linearly Homomorphic Signature (LH-Sig) 13

Linearly Homomorphic Signature (LH-Sig) 13

Linearly Homomorphic Signature (LH-Sig) 13

Linearly Homomorphic Signature (LH-Sig) 13

Linearly Homomorphic Signature (LH-Sig) 14

Summary • Randomizable Ciphertexts: ElGamal • Linearly Homomorphic Signatures • 3 properties: • Message Homomorphism • Key Homomorphism • Tag Randomizability 15

Summary • Randomizable Ciphertexts: ElGamal • Linearly Homomorphic Signatures • 3 properties: • Message Homomorphism • Key Homomorphism • Tag Randomizability • 2 schemes: • One-Time Linearly Homomorphic Signature (Keygen,Sign,Verif,…) • Linearly Homomorphic Signature (Keygen*,Sign*,Verif*,…) 15

Spirit of our Mix-Net 16

Mix-Nets 17

Mix-Nets Mi Mix 18

Mix-Nets 19

Problems 20

Problems • Expanded vectors 21

Problems • Expanded vectors • Non-trivial transformation 21

Problems • Expanded vectors • Non-trivial transformation • Legitimate ballots 21

Problems • Expanded vectors • Non-trivial transformation • Legitimate ballots • Multiple servers 21

Aggregation 22

Aggregation • Groth-Sahai proofs 2008 22

Aggregation • Groth-Sahai proofs 2008 • Multi-signatures of Boneh-Drijvers-Neven 2018 22

Conclusion We saw: • (One-Time) Linearly Homomorphic Signatures schemes Message Homomorphism • Their properties Key Homomorphism Tag Randomizability • A new method to construct scalable Mix-Nets 23