Travelling Ballot Based Protocol

This example protocol implements the task of [[Quantum Electronic Voting| Quantum E-voting. The protocol uses two entangled qudits, one as a blank ballot that travels from voter to voter and the second one for computing the election result. The first quantum scheme in this category was introduced by Vaccaro and later improved.

Outline

We consider N voters who wish to cast their vote secretly. The election authority prepares an entangled state, keeps one of the qudits, and passes the other one as the ballot qudit. Each voter receives the ballot qudit from the previous voter, casts her vote by applying a unitary and then forwards the qudit to the next voter. In the end, the authority obtains the election outcome by measuring the two qudits.

Notations

• 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}: i^{th}} voter
• c: number of possible candidates
• N: number of voters
• $v_{i}:$ vote of $i^{th}$ voter
• T: election authority
• m: number of yes votes

Requirements

• Quantum channel for qudit communication
• Qudit Measurement Device for election authority
• Quantum memory to store qudits

Properties

This type of protocol is subject to double voting and privacy attacks when several voters are colluding.

• Double voting: A corrupted voter can apply the “yes” unitary operation many times without being detected.
• Privacy attack: An adversary that corrupts 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_{k-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 V_{k+1}} can learn how voter 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} voted with probability 1.

Protocol Description

• Setup phase:

T prepares the state 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 |\phi_0\rangle =\dfrac{1}{\sqrt{N}}\sum_{j=0}^{N-1}|j\rangle_V |j\rangle_T} , keeps the second qudit and passes the first to voter 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_1} as the ballot qudit.

• Casting phase:

For k = 1, ... ,N, 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 } receives the ballot qudit and applies the unitary 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^{v_{k}}=\sum_{j=0}^{N-1}|j+1\rangle \langle j|} , 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 v_k = 1} signifies a yes vote 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 v_k = 0} a no vote.

Then, 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} forwards the ballot qudit to the next voter 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+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 V_N} to T.

• Tallying phase:

The global state held by T after all voters have voted is: 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 |\phi_N\rangle =\dfrac{1}{\sqrt{N}}\sum_{j=0}^{N-1}|j+m\rangle_V|j\rangle_T} T measures the two qudits in the computational basis, subtracts the two results, and obtains the outcome m.

Further Information

*contributed by Sara Sarfaraz