2 Self-tallying Voting Scheme with Perfect Ballot Secrecy
2.1 Security Definitions
The requirements we want the voting scheme to satisfy are the following.
Perfect ballot secrecy: This is an extension of the usual privacy requirement. In a voting scheme with perfect ballot secrecy the partial tally of a group of voters is only accessible to a coalition consisting of all remaining voters. This is the best type of anonymity we can hope for in elections where we publish the result, since a coalition of voters may of course always subtract their own votes.
Self-tallying: After all votes have been cast, it is possible for anybody, both voters and third parties, to compute the result.
Fairness: Nobody has access to a partial tally before the deadline. We will interpret this demand in a relaxed way such that it is guaranteed by a hopefully honest authority.
Dispute-freeness: This notion extends universal verifiability. A scheme is dispute-free if everybody can check whether voters act according to the protocol or not. In particular, this means that the result is publicly verifiable.
2.2 The voting protocol
The basic idea. To quickly describe our idea let us use an analogue with the physical world. Assume a group of people want to vote yes or no to a proposal. To do this the voters take a box with a small slot and each voter puts a padlock on the box. Taking turns the voters one by one drop a white (yes) stone or a black (no) stone into the box and remove their padlock. When the last voter has removed his padlock, they may open the box and see the result of the election. The protocol has perfect ballot secrecy since the box cannot be opened before all honest voters have cast their vote, and thus any honest voter's vote is mixed in with the rest of the honest voters' votes.
Overview of the protocol. For simplicity, we first describe the protocol in the honest-but-curious setting, i.e., corrupted voters may leak information but follow the protocol. For simplicity, we also assume there are just two candidates that the voters can choose between.
Initialization: First, the voters agree on a group $G_q$ of order $q$ where the DDH problem is hard. Let $g$ be a generator for $G_q$. All voters now select at random an element in $\mathbb{Z}_q$. Each voter $j$ keeps his element $x_j$ secret but publishes $h_j = g^{x_j}$.
Casting votes: Voters may vote in any adaptively chosen order, however, for simplicity we assume in this example that they vote in the order $1, 2, \dots, n$. Let their choices be $v_1, v_2, \dots, v_n \in {0, 1}$. The election now proceeds like this:
- Voter 1 selects at random $r_1 \in \mathbb{Z}q$ and publishes $(g^{r_1}, (\prod{i=2}^n h_i)^{r_1} g^{r_1})$.