Selene: Voting with Transparent Verifiability and Coercion...
Transcript of Selene: Voting with Transparent Verifiability and Coercion...
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Selene: Voting with Transparent Verifiability and
Coercion Mitigation
Voting’16 26 Feb 2016 1
Peter Y A RyanVincenzo Iovino and Peter Roenne
Université du Luxembourg
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Outline
• End-To-End verifiability
• Outline of Selene I
• The sting in the tail-Selene II
• Discussion
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Why “Selene”?
• Greek god of the moon.
• A pale reflection of Helios.
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E2E Verifiability
• Goal: voters can confirm that their vote is accurately counted, without introducing coercion threats.
• Typically, voters get a “protected receipt”, i.e. an encrypted/encoded version of their vote.
• Cast receipts are posted to a secure web bulletin board (WBB). Voters can verify that their receipt is correctly posted.
• A (universally) verifiable, anonymising tabulation is performed on the posted receipts.
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Indirection
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Key Requirements
– Integrity/accuracy: the count accurately reflects (legitimate) votes cast.
– Individual verifiability: every voter can confirm that her vote is accurately recorded.
– Universal verifiability: anyone can verify that the recorded ballots are accurately tabulated.
– (Universal) Eligibility verifiability: anyone can verify than only valid votes are cast, and no voter casts more than one vote (needs PKI or similar).
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Secrecy Requirements
– Ballot secrecy: the way a voter casts her vote should be known only to the voter.
– Receipt-freeness: there must be no way for a voter to construct a proof of how she voted (post hoc).
– Coercion resistance: a voter can always cast a vote according to her intent while appearing to comply with a coercer’s instructions, (before, during and after the voting ceremony).
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And.....
–Usability–Understandability– Availability
– Accountability
– Accessibility
– Resilience/recoverability
– etc. etc....
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Prêt à Voter Ballot
BoufiltreAsterixObelixIdefix
PanoramixAbraroucourix
7490012
X
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Voter-friendly Verification
• But we often encounter hostility to the use of encrypted ballots and the expectation to be able to find a vote in the clear on the WBB.
• Voter verification steps can be burdensome and non-intuitive.
• Crypto-free verification: Randell-Ryan, Rivest’s ThreeBallot....
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Tracker Numbers
• A very simple approach: give each voter a private tracker number and post these along with the votes in the clear.
• Verification is simple and intuitive-no need to handle encrypted ballots etc.
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Tracker numbers
•
347563 Obelix947253 Asterix556884 Panoramix569331 Idefix586994 Idefix607855 Obelix374823 Obelix
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But….
• We have to guarantee that voters get unique trackers.
• The association of trackers to voters must be kept secret.
• Largely ignored by the crypto/security community, aside maybe for “boardroom” style contexts.
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Coercion Attack
• Coercer requires the voter to reveal her tracker number so that he can check how she voted.
• Note however: the coercer has to require the voter to reveal her tracker before the ballots are posted. Otherwise the voter just pulls something suitable off the WBB.
• So what if voters only learn their number after posting!?
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The goal
• To ensure that each voter is assigned a unique tracker number.
• To notify the voters of their trackers (after trackers/voters have been posted) in a way that provides high assurance but is deniable.
• And we want to do this in a way that ensures no single entity knows the assignment.
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Selene I
• Assume standard DH/ElGamal style setup:
• g generator of the large cyclic group of prime order q in which DDH believed to be “hard”.
• Tellers hold shares of a threshold election PK.
• Voters have secret signing and trapdoor keys.
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The Setup
• For each voter we want to post to the WBB:
• PKi, {n i}PK, TDCi{ni}
• TDCi =Trap Door Commitment for voter i.
• {ni}PK will be used in the tabulation.
• TDCi{ni} will be used in notifying the voter of the tracker.
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Selene Set-up
• Generate sufficient tracker numbers nj and compute g^nj and post to the WBB.
• Form (ElGamal) encryption under the Teller’s PK of the g^nj : {g^nj}PK.
• Put these through a sequence of verifiable re-encryption mixes and assign the resulting shuffled, re-encrypted numbers to the voters’ Ids (PKi).
• PKi:, {g^nπ(i)} PK_T
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Set-up
• The result is posted to the WBB rows of the form:
• PKi, {gn_π(i)}PK
• Note: since the nπ(i) have gone through multiple mixes, no single entity knows the assignment. But the verifiable shuffling preserves the uniqueness.
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Tracker Commitments
•
Now we want a distributed construction for the trap-door commitments to the tracker numbers.
• Assume voters have SK/PK pairs: (xi, hi:=g^xi)
• Pedersen like commitments:
• gn_i⋅hir_i
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Trapdoor commitments
• Suppose that we have t Trustees.
• Now, for each voter i, each Trustee Tj generates a fresh random r_i,j, computes gr_i,j and hir_i,j , where hi = PKi.
• and publishes the encryptions in a table:
• {gr_i,j}PK_T and {hir_i,j}PK
• Note that we can require ZK proofs of well-formedness of ({gr}PK, {hir}PK)
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Distributed construction
• For each voter i we now form the product over the t Tellers of these:
• {gr_i}PK = ∏ j=1t {gr_i,j}PK
• {hir_i}PK = ∏ j=1t {hir_i,j}PK
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Set-up
• Now take the product of the {hir_i} PK with the {gn_i}PK previously posted:
• {gn_i}PK ⊗ {hir_i}PK = {gn_i⋅hir_i}PK
• The Trustees now perform a (verified) threshold decryption to yield the (Pedersen) trapdoor commitments:
• gn_i⋅hir_i
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Set up
• On the WBB we now have rows of the form:
• PKi, {gn_i}PK, gn_i⋅h_ir_i
• Ready for the i-th voter ‘s ballot.
• The Tellers retain their gr_i,j terms in secret.
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Voting• To vote, the voter forms:
• SigVi({|Votei|}PK)
• Where {|X|} denotes plaintext aware encryption such as Cramer-Shoup, and sends this to the server:
• This is posted to the WBB:
• PKi, gn_i⋅h_ir_i,, {g^ ni} , SigVi({|Votei|}PK)
• Proofs and signatures are checked.
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Tabulation• We extract the last two terms of the tuple, and
strip off the signature and ZK proofs:
• ({gn_i}PK , {Votei}PK)
• These are now put through verifiable, parallel, re-encryption mixes and threshold decrypted:
• (gn_i, Votei)
• From which is derived:
• (ni , Votei)
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Revealing the trackers
• After the decrypted trackers and votes have been available on the WBB for a while, we notify the voters of their tracker.
• We treat the gn_i⋅h_ir_i as the “beta” components of an ElGamal encryption under the voter’s PKs.
• Trustee T_j reveals to V_i gr_i,j through a private (anonymous) channel.
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Revealing the trackers
• The voter can now form the product of these to give gr_i and can then form the ElGamal cryptogram:
• (gr_i, h_ir_i, ⋅ gn_i)
• which she can decrypt as usual with her secret key xi to reveal: gn_i, and hence n_i.
• Note: we could combine the gr_i,j and just send gr_i to the voter, but requires more trust.
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Coercion Resistance
• If V_i is coerced she can compute, with knowledge of the trapdoor, an alternative (gr_i)ʹ value which will open the encryption to whichever tracker number she needs to satisfy the coercer.
• On the other hand, without the knowledge of secret trapdoor, this is intractable, so revealing the wrong tracker to the voter should not be feasible for an attacker.
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(Preliminary)Analysis
• Uniqueness of trackers assigned to voters follows from the mix construction.
• Need to ensure that Vi gets the correct, assigned tracker. This follows from the fact that it will be intractable for anyone but the voter to compute alternative (gr_i)ʹ that will open the commitment to an alternative valid tracker.
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Discussion
• The fact that voters get to check their vote in the clear in the WBB seems to change the trust model, in particular for the voter device!?
• e.g., it may be possible to avoid the need for “Benaloh” challenges?!
• But dispute resolution becomes harder.
• Using voting codes could help.
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Increased Coercion Resistance
• The scheme doesn’t provide coercion resistance against a coercer who demands the voter reveal her SK.
• A further possibility is to re-encrypt the {Vote_i} before posting, see Belenios RF, using malleable signatures.
• Now the voter does not know the randomization of the posted, re-encrypted vote.
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Discussion
• So we seem to have a scheme which provides a very direct, intuitive way for voters to check that their vote is accurately included in the tally.
• It also provides a good degree of receipt-freeness.
• But there is a problem.....
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The sting in the tail!
• Suppose that the coercer C is himself a voter. A coerced voter V might by chance chose C’s tracker.
• Or, C might simply claim that this is his tracker number, even if in reality it isn’t!
• How should V respond?
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Discussion
• Not a problem if the coercer is not a voter.
• Maybe acceptable of the threat environment is mild.
• In many situations, the simplicity of the verification probably outweighs this threat.
• But it would be nice to address it.
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Alternative constructions
• Or generate an (equal) number of dummy trackers for each candidate. If coerced the voter can request a suitable dummy tracker instead of her real one.
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Selene as an add-on!?
• Seems possible to think of Selene (I and II) as possible add-ons to existing schemes.
• Could even potentially be added on dynamically:
• Add Selene I to a disputed election using encrypted ballots, e.g. Helios.
• Add Selene II if a high level of coercion is reporting prior to and during voting.
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Conclusions
• Selene seems to provide a high degree of transparency for the verification while providing a good level of coercion resistance.
• Where coercion threats are regarded as serious, Selene II could be used, but at a cost in terms of transparency of verifiability.
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Open questions
• Can we avoid TSitT while maintaining transparency and usability?
• Analysis, proofs.
• Dispute resolution
• Trust model for devices etc.
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Thank you!