This protocol implements the functionality of Quantum E-voting. The protocol uses an entangled state with a special property as a blank ballot and is self-tallying, i.e. the voters, without the presence of any trusted authority or tallier, need to verify that they share specific quantum states.

Self-tallying quantum anonymous voting

Consider $N$ voters who wish to cast their vote secretly.

• One of the voters prepares some states in two forms, and each voter receives a specific particle of each global state.
• After voters verify that they received the correct states by the cut-and-choose technique, such as described in [2], they perform certain measurements on their qudits and cast their vote based on the measurement outcome.
• In the end, all voters simultaneously broadcast their votes in encoded form, and everyone can compute the election result by a simple summation.

All classical communication in the protocol takes place using pairwise authenticated channels.

The protocol assumed the ability to prepare a multipartite entangled quantum state, as well as a quantum memory that would last throughout the protocol before the final measurement. 

Network Stage: Quantum Memory 

Other requirement:

• Quantum memory
• Entangled states with the dimension of the number of voters.
• Pairwise authenticated classical channel

• $V_i$: $i^{\text{th}}$ voter
• $c$: number of possible candidates
• $m$: dimension of qudits
• $\delta_0$: security parameter
• $N$: number of voters
• $v_i$: vote of the $i^{\text{th}}$ voter
• $P_N$: set of all possible permutations with $N$ elements
• $B_k$: $k^{\text{th}}$ voter’s blank ballot

NOTE: According to [3], this protocol is not secure. (doesn’t satisfy quantum privacy property.)
We can construct an adversary that violates privacy by an attack on the cut-and-choose technique of the protocol with a non-negligible advantage in $\delta _{0}$.

• Setup phase:
  1. One of the voters prepares $N + N2^{\delta_0}$ states of the form:
     $$
     |D_1\rangle = \frac{1}{\sqrt{m^{N-1}}} \sum_{\sum_{k=1}^N i_k = 0 \mod c} |i_1\rangle |i_2\rangle \ldots |i_N\rangle
     $$
     and $1 + N2^{\delta_0}$ states of the form:
     $$
     |D_2\rangle = \frac{1}{\sqrt{N!}} \sum_{(i_1, i_2, \ldots, i_N) \in P_N} |i_1\rangle |i_2\rangle \ldots |i_N\rangle
     $$
     Each voter $V_k$ receives the $k^{\text{th}}$ particle from each of the states.
  2. Voter $V_k$ randomly chooses $2^{\delta_0}$ of the $|D_1\rangle$ states.
     The other voters measure half of their particles in the computational basis and half in the Fourier basis:
     • If the chosen basis is computational, the measurement results should sum to $0$
     • If the chosen basis is Fourier, the measurement results should be equal
  3. All voters simultaneously broadcast their results. If any discrepancy is found, the protocol aborts.
  4. The $|D_2\rangle$ states are similarly checked.
  5. All voters measure their qudits in the computational basis.
     Each $V_k$ now holds a blank ballot of dimension $N$, represented as:
     $$ B_k = [\xi_k^1, \ldots, \xi_k^{s_k^k}, \ldots, \xi_k^N]^T $$
     and a unique index $s_k^k \in \{1, \ldots, N\}$ from the measurement outcome of the qudit belonging to $|D_2\rangle$.
• Casting phase:
  1. $V_k$ applies the transformation: $\xi_k^{s_k^k} \rightarrow \xi_k^{s_k^k} + v_k $.
  2. All voters simultaneously broadcast their columns, forming a public $N \times N$ table.
• Tally phase:
  1. Each $V_k$ checks that its corresponding row sum equals its own vote $v_k$. If not, the protocol aborts.
  2. The final outcome of the election is the sum of the elements in each row of the public $N \times N$ table.

1. Wang, Qingle, Chaohua Yu, Fei Gao, Haoyu Qi, and Qiaoyan Wen. “Self-tallying quantum anonymous voting.” Physical Review A 94, no. 2 (2016): 022333.
2. Kashefi, Elham, Luka Music, and Petros Wallden. “The quantum cut-and-choose technique and quantum two-party computation.” arXiv preprint arXiv:1703.03754 (2017).
3. Arapinis, Myrto, Nikolaos Lamprou, Elham Kashefi, and Anna Pappa. “Definitions and security of quantum electronic voting.” ACM Transactions on Quantum Computing 2, no. 1 (2021): 1-33.