Distributed Ballot Based Protocol

This example protocol implements the task of Quantum E-voting. In this protocol, the election authority prepares and distributes to each voter a blank ballot, and gathers it back after all voters have cast their vote in order to compute the final outcome.

Assumptions

• The tallier is assumed to be trusted to correctly prepare specific states.

Outline

In the beginning, the election authority prepares an N-qudit ballot state where the kth qudit of the state corresponds to $V_{k}$’s blank ballot and sends the corresponding blank ballot to $V_{k}$ together with two option qudits, one for the “yes” and one for the “no” vote. then each voter decides on “yes” or “no” by appending the corresponding option qudit to the blank ballot and performing a 2-qudit measurement, then based on its result she performs a unitary correction and sends the 2-qudits ballot along with the measurement result back to the election authority. At the end of the election, the election authority applies a unitary operation on one of the qudits in the global state and another unitary operation on one of the qudits to find the number of yes votes.

Notations

• $V_{i}:i^{th}$ voter
• c: number of possible candidates
• N: number of voters
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{i}} : vote of $i^{th}$ voter
• T: election authority
• m: number of yes votes

Requirements

• Quantum channel capable of sending qubits -> (qudit) between the election authority and voters
• Qudit Measurement Device for election authority and voters

Properties

This protocol is vulnerable to double voting. Specifically, an adversary can mount a “d-transfer attack”, and transfer d votes for one option of the referendum election to the other.

Protocol Description

• Setup phase:

1. T prepares an N-qudit ballot state $|\Phi \rangle ={\dfrac {1}{\sqrt {D}}}\sum _{j=0}^{D-1}|j\rangle ^{\otimes N}$.

   The states $|j\rangle ,j=0,...,D-1,$ form an orthonormal basis for the D-dimensional Hilbert space, and D > N. The k-th qudit of $\Phi$ is $V_{k}$'s blank ballot.

2. T sends to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_k} the corresponding blank ballot and two option qudits,for the "yes" and "no" option:

   $yes:|\psi (\theta _{y})\rangle ={\dfrac {1}{\sqrt {D}}}\sum _{j=0}^{D-1}e^{ij\theta _{y}}|j\rangle$, no:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi(\theta_n)\rangle=\dfrac{1}{\sqrt{D}}\sum_{j=0}^{D-1}e^{ij\theta_n}|j\rangle} .

   For $v\in \{y,n\}$ we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_v = (2\pi l_v/D) + \delta} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_v \in \{0,...,D- 1\}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta \in [0, 2\pi/D)} . Values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_y} and $\delta$ are chosen uniformly at random from their domain and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_n} is chosen such that $N(l_{y}-l_{n}{\text{ }}mod{\text{ }}D)$ < D.

• Casting phase:

1. Each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_k} appends the corresponding option qudit to the blank ballot and performs a 2-qudit measurement $R=\sum _{r=0}^{D-1}rP_{r}$ where $P_{r}=\sum _{j=0}^{D-1}|j+r\rangle \langle j+r|\otimes |j\rangle \langle j|.$

   According to the result $r_{k},V_{k}$ performs a unitary correction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{r_k} = I \otimes \sum_{j=0}^{D-1}|j+r_k\rangle \langle j |} and sends the 2-qudits ballot and $r_{k}$ back to T

• Tally phase:

1. The global state of the system is: ${\dfrac {1}{\sqrt {D}}}\sum _{j=0}^{D-1}\Pi _{k=1}^{N}\alpha _{j,r_{k}}|j\rangle ^{\otimes 2N}$ where , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_{j,r_k}= \begin{cases} e^{i(D+j-r_k)\theta^{k}_{v}},\text{ }0 \leq j \leq r_k -1,\\ e^{i(j-r_k)\theta^{k}_{v}}\text{ }r_k \leq j \leq D -1 \end{cases} }

   For every k, T applies $W_{k}=\sum _{j=0}^{r_{k}-1}e^{-iD\delta }|j\rangle |\langle j|+\sum _{j=r_{k}}^{D-1}|j\rangle |\langle j|$ on one of the qudits in the global state.

2. By applying the unitary operator $\sum _{j=0}^{D-1}e^{-ijN\theta _{n}}|j\rangle \langle j|$on one of the qudits we have $|\phi _{q}\rangle ={\dfrac {1}{\sqrt {D}}}\sum _{j=0}^{D-1}e^{2\pi ijq/D}|j\rangle ^{\otimes 2N}$ where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q=m(l_y-l_n)} . with the corresponding measurement, T retrieves q and uses values $l_{y},l_{n}$ to compute m.

Further Information

*contributed by Sara Sarfaraz