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Receipt-Free Universally-Verifiable Voting With Everlasting Privacy

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  1. Tal Moran Joint work with Moni Naor Receipt-FreeUniversally-Verifiable Voting With Everlasting Privacy

  2. Outline of Talk • Motivation for Cryptographic Voting • Flavors of Privacy (and why we care) • Cryptographic Voting Scheme based on commitment with equivalence proof • We’ll use physical metaphors and a simplified model

  3. Voting: The Challenge • Requirements based on democratic principles: • Outcome should reflect the “people’s will” • Fairness • One person, one vote • Privacy • Not a principle in itself;required for fairness • Cast-as-intended • Counted-as-cast Additional requirements: Authorization, Availability

  4. A [Very] Brief History of Voting • Ancient Greece (5th century BCE) • Paper Ballots • Rome: 2nd century BCE(Papyrus) • USA: 17th century • Secret Ballots (19th century) • The Australian Ballot • Lever Machines • Optical Scan (20th century) • Direct Recording Electronic(DRE)

  5. The Case for Cryptographic Voting • Elections don’t just name the winnermust convince the loser they lost! • Elections need to be verifiable • Counting in public: • Completely verifiable • But no vote privacy • Using cryptography , we can get both!

  6. Voting with Mix-Nets • Idea due to David Chaum (1981) • Multiple “Election Authorities” • Assume at least one is honest • Each voter creates “Onion Ballot” • Authorities decrypt and shuffle • No Authority knows all permutations • Authorities can publish “proof of shuffle” No No Yes No Yes Yes No No Yes No No No No

  7. How Private is Private? • Intuition: No one can tell how you voted • This is not always possible • Best we can hope for: • As good as the “ideal” vote counter i1 i2 in … v1 v2 vn Tally

  8. Privacy and Coercion • Vote privacy is essential to prevent coercion • Computational privacy holds only as long as its underlying assumptions • Almost all universally verifiable voting schemes rely on public-key encryption • Belief in privacy violation isenough for coercion! Existing public-key schemes with current key lengths are likely to be broken in less than 30 years! [RSA conference ’06]

  9. Privacy is not Enough! • Voter can sell vote by disclosing randomness • Example: Italian Village Elections • System allows listing candidatesin any order • Bosses gave a different permutation of“approved” candidates to each voter • They could check which permutationsdidn’t appear • Need “Receipt-Freeness”[Benaloh&Tuinstra 1994]

  10. Who can you trust to encrypt? • Public-key encryption requires computers • Voting at home • Coercer can sit next to you • Voting in a polling booth • Can you trust the polling computer? • Verification should be possible for a human! • Receipt-freeness and privacy are also affected.

  11. Our Contributions First Universally Verifiable Voting SchemeBased on General Assumptions • First Universally Verifiable Scheme based onGeneral Assumption • Previous schemes required special properties(e.g. a homomorphic encryption scheme) • Our scheme can be based on any non-interactive commitment • First Receipt-Free Voting Scheme withEverlasting Privacy • Uses statistically hiding commitment instead of encryption • Formal definition of Receipt-Freeness • Proof of security (integrity) in UC model • Security against arbitrary coalitions “for free” First Receipt-Free Voting Scheme withEverlasting Privacy

  12. Alice and Bob for Class President • Cory “the Coercer” wants to rig the election • He can intimidate all the students • Only Mr. Drew is not afraid of Cory • Everybody trusts Mr. Drew to keep secrets • Unfortunately, Mr. Drew also wants to rig the election • Luckily, he doesn't stoop to blackmail • Sadly, all the students suffer severe RSI • They can't use their hands at all • Mr. Drew will have to cast their ballots for them

  13. Commitment with “Equivalence Proof” • We use a 20g weight for Alice... • ...and a 10g weight for Bob • Using a scale, we can tell if two votes are identical • Even if the weights are hidden in a box! • The only actions we allow are: • Open a box • Compare two boxes

  14. Additional Requirements • An “untappable channel” • Students can whisper in Mr. Drew's ear • Commitments are secret • Mr. Drew can put weights in the boxes privately • Everything else is public • Entire class can see all of Mr. Drew’s actions • They can hear anything that isn’t whispered • The whole show is recorded on video (external auditors) I’m whispering

  15. Ernie Casts a Ballot • Ernie whispers his choice to Mr. Drew I like Alice

  16. Ernie Casts a Ballot • Mr. Drew puts a box on the scale • Mr. Drew needs to prove to Ernie that the box contains 20g • If he opens the box, everyone else will see what Ernie voted for! • Mr. Drew uses a “Zero Knowledge Proof” Ernie

  17. Ernie Casts a Ballot Ernie Casts a Ballot • Mr. Drew puts k (=3) “proof” boxes on the table • Each box should contain a 20g weight • Once the boxes are on the table, Mr. Drew is committed to their contents Ernie

  18. Ernie Ernie Ernie Casts a Ballot Weigh 1Open 2Open 3 • Ernie “challenges” Mr. Drew; For each box, Ernie flips a coin and either: • Asks Mr. Drew to put the box on the scale (“prove equivalence”) • It should weigh the same as the “Ernie” box • Asks Mr. Drew to open the box • It should contain a 20g weight

  19. Ernie Casts a Ballot Open 1Weigh 2Open 3 • If the “Ernie” box doesn’tcontain a 20g weight, every proof box: • Either doesn’t contain a 20g weight • Or doesn’t weight the same as theErnie box • Mr. Drew can fool Ernie with probability at most 2-k Ernie

  20. Ernie Casts a Ballot • Why is this Zero Knowledge? • When Ernie whispers to Mr. Drew,he can tell Mr. Drew what hischallenge will be. • Mr. Drew can put 20g weights in the boxes he will open, and 10g weights in the boxes he weighs I like Bob Open 1Weigh 2Weigh 3

  21. Ernie Ernie Casts a Ballot: Full Protocol • Ernie whispers his choice and a fake challenge to Mr. Drew • Mr. Drew puts a box on the scale • it should contain a 20g weight • Mr. Drew puts k “Alice” proof boxesand k “Bob” proof boxes on the table • Bob boxes contain 10g or 20g weights according to the fake challenge I like Alice Open 1Weigh 2Weigh 3

  22. Ernie Ernie Ernie Casts a Ballot: Full Protocol Open 1Open 2Weigh 3 • Ernie shouts the “Alice” (real) challenge and the “Bob” (fake) challenge • Drew responds to the challenges • No matter who Ernie voted for,The protocol looks exactly the same! Open 1Weigh 2Weigh 3

  23. r Implementing “Boxes and Scales” • We can use Pedersen commitment • G: a cyclic (abelian) group of prime order p • g,h: generators of G • No one should know loggh • To commit to m2Zp: • Choose random r2Zp • Send x=gmhr • Statistically Hiding: • For any m, x is uniformly distributed in G • Computationally Binding: • If we can find m’m and r’ such that gm’hr’=x then: • gm-m’=hr-r’1, so we can compute loggh=(r-r’)/(m-m’)

  24. r s Implementing “Boxes and Scales” • To prove equivalence of x=gmhr and y=gmhs • Prover sends t=r-s • Verifier checks that yht=x h g h g t=r-s

  25. A “Real” System Hello Ernie, Welcome to VoteMaster Please choose your candidate: Alice Bob 1 Receipt for Ernie 2 o63ZJVxC91rN0uRv/DtgXxhl+UY= 3 - Challenges - 4 Alice: 5 Sn0w 619- ziggy p3 6 Bob: 7 l4st phone et spla 8 - Response - 9 9NKWoDpGQMWvUrJ5SKH8Q2CtwAQ= 0 === Certified ===

  26. A “Real” System Hello Ernie, You are voting for Alice Please enter a fake challenge for Bob Alice: l4st phone et spla Bob : Continue 1 Receipt for Ernie 2 o63ZJVxC91rN0uRv/DtgXxhl+UY= 3 - Challenges - 4 Alice: 5 Sn0w 619- ziggy p3 6 Bob: 7 l4st phone et spla 8 - Response - 9 9NKWoDpGQMWvUrJ5SKH8Q2CtwAQ= 0 === Certified ===

  27. A “Real” System Hello Ernie, You are voting for Alice Make sure the printer has output twolines (the second line will be covered)Now enter the real challenge for Alice Alice: Sn0w 619- ziggy p3 l4st phone et spla Bob : Continue 1 Receipt for Ernie 2 o63ZJVxC91rN0uRv/DtgXxhl+UY= 3 - Challenges - 4 Alice: 5 Sn0w 619- ziggy p3 6 Bob: 7 l4st phone et spla 8 - Response - 9 9NKWoDpGQMWvUrJ5SKH8Q2CtwAQ= 0 === Certified ===

  28. A “Real” System Hello Ernie, You are voting for Alice Please verify that the printed challengesmatch those you entered. Alice: Sn0w 619- ziggy p3 l4st phone et spla Bob : Finalize Vote 1 Receipt for Ernie 2 o63ZJVxC91rN0uRv/DtgXxhl+UY= 3 - Challenges - 4 Alice: 5 Sn0w 619- ziggy p3 6 Bob: 7 l4st phone et spla 8 - Response - 9 9NKWoDpGQMWvUrJ5SKH8Q2CtwAQ= 0 === Certified ===

  29. A “Real” System Hello Ernie, Thank you for voting Please take your receipt 1 Receipt for Ernie 2 o63ZJVxC91rN0uRv/DtgXxhl+UY= 3 - Challenges - 4 Alice: 5 Sn0w 619- ziggy p3 6 Bob: 7 l4st phone et spla 8 - Response - 9 9NKWoDpGQMWvUrJ5SKH8Q2CtwAQ= 0 === Certified ===12

  30. Ernie Fay Guy Heidi Counting the Votes • Mr. Drew announces the final tally • Mr. Drew must prove the tally correct • Without revealing who voted for what! • Recall: Mr. Drew is committed toeveryone’s votes Alice: 3Bob: 1

  31. Ernie Fay Guy Heidi Counting the Votes Weigh WeighOpen • Mr. Drew puts k rows ofnew boxes on the table • Each row should contain the same votes in a random order • A “random beacon” gives k challenges • Everyone trusts that Mr. Drewcannot anticipate thechallenges Alice: 3Bob: 1

  32. Ernie Fay Guy Heidi Ernie Fay Guy Heidi Counting the Votes Weigh WeighOpen • For each challenge: • Mr. Drew proves that the row contains a permutation of the real votes Alice: 3Bob: 1

  33. Ernie Fay Guy Heidi Counting the Votes Weigh WeighOpen • For each challenge: • Mr. Drew proves that the row contains a permutation of the real votes Or • Mr. Drew opens the boxes andshows they match the tally Alice: 3Bob: 1 Fay

  34. Ernie Fay Guy Heidi Counting the Votes Weigh WeighOpen • If Mr. Drew’s tally is bad • The new boxes don’t matchthe tally Or • They are not a permutationof the committed votes • Drew succeeds with prob.at most 2-k Alice: 3Bob: 1 Fay

  35. Ernie Fay Guy Heidi Counting the Votes Weigh WeighOpen • This prototocol does notreveal information aboutspecific votes: • No box is both opened andweighed • The opened boxes are ina random order Alice: 3Bob: 1 Fay

  36. Using “Standard” Commitment • Is the equivalence proof necessary? • Our new metaphor: Locks and Keys • Assumptions: • Every key fits a single lock • Every lock has only one key • No one can tell by just looking whether a key fits a lock

  37. Private Commitment with Locks and Keys • To commit to a message: • Privately lock the message using a key • Put the key (or lock) on the table • The key only fits one lock • To open the commitment, show the lock and open it

  38. Private Nested Commitments • We have an additional trick: • Commitment to a commitment • We can put a key on the lock instead of a message • The locked key is a commitment to the commitment to the message

  39. Private Nested Commitments • We can open the “external” commitment without giving any information about the “internal” • Or open the “internal” one without revealing the “external”

  40. Private Ernie Casts a Ballot • Ernie whispers his choice to Mr. Drew • Mr. Drew creates 2k doublecommitments to Ernie’s choice • Mr. Drew now proves to Ernie thatmost of the commitments are correct • He uses a Zero Knowledge proof I like Alice

  41. Private Ernie Casts a Ballot • Ernie chooses a random permutation • Drew rearranges keysand locks by this permutation 2314

  42. Private Ernie Casts a Ballot • Drew reveals k of the internalcommitments • Does not open external commitments! • Ernie makes k challenges Candidate 1Connection 2

  43. Private Ernie Casts a Ballot • Drew responds to challenges • Opens internal commitment Candidate 1Connection 2

  44. Private Ernie Casts a Ballot • Drew responds to challenges • Opens internal commitment Or • Opens external commitment Candidate 1Connection 2

  45. Ernie Casts a Ballot: Proof Intuition • If a large fraction of Drew’s commitments are bad • After shuffling, a large fraction of bad commitments will be in the first k • For each bad commitment: • Either Drew cannot open internal commitment Or • Drew cannot open external commitment • Drew cheats successfully with prob. exponentially small in k

  46. Ernie Casts a Ballot: Zero Knowledge • If Drew knows Ernie’s challengein advance • He creates “fake”internal commitments Candidate 1Connection 2 Private

  47. Ernie Casts a Ballot: Zero Knowledge • Drew can “prove” Ernievoted for Bob Candidate 1Connection 2 Private

  48. Ernie Casts a Ballot: Receipt Freeness • We use the same technique as previously • Ernie whispers his choiceand a fake challenge • Drew “proves” that Ernievoted for Bob using the fake challenge • And that Ernie voted for Alice usinga real challenge • The real and fake proofs are indistinguishable to everyone else I like Alice Candidate 1Candidate 2

  49. Private Counting the Votes Alice: 3Bob: 1 • Drew reveals the tally • Random beacon providesn permutations of 1,…,k • Drew permutes the columns Ernie: 12 Fay: 12Guy: 21Heidi: 21 Ernie Fay Guy Heidi Ernie Fay Guy Heidi

  50. Private Ernie Ernie Fay Fay Guy Heidi Heidi Ernie Fay Fay Guy Guy Heidi Heidi Counting the Votes • Drew chooses k randompermutations of 1,…,n • Drew permutes the rows(of internal commitments) Row1: 2431Row2: 1342