Jeremy Clark, Urs Hengartner, & Kate Larson University of Waterloo Not So Hidden Informa0on: Op0mal contracts for undue influence in E2E vo0ng systems
SeGng We consider vo0ng systems with end‐to‐end (E2E) verifiability. The correctness of these systems rely on mathema0cal assump0ons instead of chain‐ of‐custody, soQware, or hardware. (Custody independence and soQware independence) 2
Punchscan To illustrate the idea of contracts, we focus on one system: Punchscan Why just one? Why this one? 3
Punchscan 01-0031337 01-0031337 Alice Alice Y Bob Bob Z Z Y 4
01-0031337 01-0031337 Alice Alice Y Bob Bob Z Z Y 5
01-0031337 Alice Y Bob Z 6
01-0031337 Alice Y Bob Z Y Z 7
01-0031337 Alice Y Bob Z Z Y 8
01-0031337 01-0031337 Alice Alice Y Bob Bob Z Z Y 9
Contracts 10
Alice X Bob Y X Y 11
Alice Alice X X Bob Bob Y Y Y X X Y 12
Alice Alice X X Bob Bob Y Y Y X X Y Alice Y Bob X X Y 13
Alice Alice X X Bob Bob Y Y Y X X Y Alice Alice Y Y Bob Bob X X X Y Y X 14
A Simple Fix Order maders. If the voter choose top or bodom prior to seeing the ballot, the best possible contract is forced randomiza0on. This, however, does increase the role of poll worker procedure in the security of the system. We are aiming for custody‐independence. 15
Ques0ons about Contracts What tool is best for analysis? Of the exis0ng contracts, which are best? Can we define the best possible contract? What if we have more than 2 candidates? What if voters do not behave correctly and follow the contract? Is the contract financially sensible? 16
Game Theory 17
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Alice Alice Alice Alice X X Y Y Bob Bob Bob Bob Y Y X X X Y Y X X Y Y X Nature {XY,XY} {XY,YX} {YX,XY} {YX,YX} L,T u 0 , 1 u 0 , ‐1 u 2 , ‐1 u 2 , 1 L,B u 1 , 1 u 0 , ‐1 u 1 , ‐1 u 0 , 1 Voter R,T u 0 , ‐1 u 0 , 1 u 0 , 1 u 0 , ‐1 R,B u 0 , ‐1 u 1 , 1 u 0 , 1 u 1 , ‐1 20
Alice Alice Alice Alice X X Y Y Bob Bob Bob Bob Y Y X X X Y Y X X Y Y X Nature {XY,XY} {XY,YX} {YX,XY} {YX,YX} L,T u 0 , 1 u 0 , ‐1 u 2 , ‐1 u 2 , 1 L,B u 1 , 1 u 0 , ‐1 u 1 , ‐1 u 0 , 1 Voter R,T u 0 , ‐1 u 0 , 1 u 0 , 1 u 0 , ‐1 R,B u 0 , ‐1 u 1 , 1 u 0 , 1 u 1 , ‐1 21
Nature {XY,XY} {XY,YX} {YX,XY} {YX,YX} L,T u 0 , 1 u 0 , ‐1 u 2 , ‐1 u 2 , 1 L,B u 1 , 1 u 0 , ‐1 u 1 , ‐1 u 0 , 1 Voter R,T u 0 , ‐1 u 0 , 1 u 0 , 1 u 0 , ‐1 R,B u 0 , ‐1 u 1 , 1 u 0 , 1 u 1 , ‐1 22
Adversary Nature {XY,XY} {XY,YX} {YX,XY} {YX,YX} L,T u 0 , 1 u 0 , ‐1 u 2 , ‐1 u 2 , 1 L,B u 1 , 1 u 0 , ‐1 u 1 , ‐1 u 0 , 1 Voter R,T u 0 , ‐1 u 0 , 1 u 0 , 1 u 0 , ‐1 R,B u 0 , ‐1 u 1 , 1 u 0 , 1 u 1 , ‐1 23
Adversary Voter Nature {XY,XY} {XY,YX} {YX,XY} {YX,YX} L,T u 0 , 1 u 0 , ‐1 u 2 , ‐1 u 2 , 1 L,B u 1 , 1 u 0 , ‐1 u 1 , ‐1 u 0 , 1 Voter R,T u 0 , ‐1 u 0 , 1 u 0 , 1 u 0 , ‐1 R,B u 0 , ‐1 u 1 , 1 u 0 , 1 u 1 , ‐1 24
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What percent of the 0me, on average, will u0lity maximizing voters cast a vote for Alice? • Forced Randomiza0on: 50.0% • MN (Moran, Naor 07): 54.2% (or 62.5%) • BMR (Bohli, Muller‐Quade, Rohrich 07): 62.5% • KRMC (Kelsey, Regenscheid, Moran, Chaum 09): 75.0% 26
Op0mal Contract KRMC has the best proper0es but is it the best contract possible? Yes (for two candidates). What if there are more than 2 candidates? We provide an algorithm for genera0ng op0mal contracts of any number of candidates. KRMC improved on the previous contracts by adding a second level of u0lity. Could we not improve further by adding more levels of u0lity? No. 27
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✓ 30
✓ ✓ 31
✓ ✓ ✓ 32
✸ ✓ ✓ 33
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✓ ✓ ✓ ✓ 35
✓ ✓ ✓ ✓ ✓ 36
✓ ✓ ✓ ✓ ✓ n/n 1/n 1/n 37
Advantage vs. Number of Candidates 38
What if voters are not u0lity‐maximizing? We model voters of four types: ‐ Always vote for Alice ‐ Always vote for Bob ‐ Follow the contract to receive highest payoff (“u0lity maximizing”) ‐ Always vote contrary to the adversary (“vengeful”) 39
Coercion vs. Vote Buying? The language of u0li0es abstracts away the difference: u0li0es could be possible (vote buying) or nega0ve (coercion). However vote buying is voluntary while coercion is involuntary: we must this. For example, vengeful voters only mader for coercion. 40
Influence of vote type in coercion How many coopera0ve voters are needed to counter‐ act the influence of one vengeful voters in coercive contracts? • MN: at least 6.1 • BMR: at least 4 • KRMC: 0 (any cooperate voters add votes for Alice) 41
Influence of Voter Type in Buying With vote‐buying contracts, the voters who did not change their vote due to the contract may s0ll coincidentally meet the condi0ons of the contract and request payment. How much does the adversary actually pay and how does that relate to how many votes are actually being bought? 42
We provide some analysis and a general u0lity equa0on. Example numbers: For op0mal 2‐candidate contracts, assume Alice voters and Bob voters make up 45% of the electorate each. The remaining 10% will vote according to the contract for €10. The contract becomes profitable when a vote gained is worth at least €89 for the adversary. For 3 candidates and similar split, the number increases to €96. 43
Future Work • A general framework for elimina0ng contracts is leQ for future work • Elimina0ng, or moving forward, voter choices helps in this specific case • Screening techniques could improve contracts 44
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