Dual Basis Measurement Based Protocol

This example protocol implements the task 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.

Assumptions

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

Outline

We 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 state. After voters verify that they received correct states by cut and choose technique, 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.

Notations

• ${\displaystyle V_{i}:i^{th}}$ voter
• c: number of possible candidates
• m: dimension of qudits
• ${\displaystyle \delta _{0}}$: security parameter
• N: number of voters
• ${\displaystyle v_{i}}$: vote of ${\displaystyle i^{th}}$ voter
• ${\displaystyle P_{N}}$: set of all possible permutations with N elements
• ${\displaystyle B_{k}:k^{th}}$ voter’s blank ballot

Properties

• 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 ${\displaystyle \delta _{0}}$.

Requirements

• Quantum memory for each party to store qubits
• Measurement Devices for each party
• Quantum channel capable of sending qubits
• Classical channel to send multiple bits

Protocol Description

• Setup phase:
1. One of the voters prepares ${\displaystyle N+N2^{\delta _{0}}}$ states of the form ${\displaystyle |D_{1}\rangle ={\dfrac {1}{\sqrt {m^{N-1}}}}\sum _{\sum _{k=1}^{N}i_{k}=0{\text{ }}mod{\text{ }}c}|i_{1}\rangle |i_{2}\rangle ...|i_{N}\rangle }$ and ${\displaystyle 1+N2^{\delta _{0}}}$ states of the form ${\displaystyle |D_{2}\rangle ={\dfrac {1}{\sqrt {N!}}}\sum _{(i_{1},i_{2},...,i_{N})\in P_{N}}|i_{1}\rangle |i_{2}\rangle ...|i_{N}\rangle }$.

Each voter ${\displaystyle V_{k}}$ receives kth particle from each of the states.
2. Voter ${\displaystyle V_{k}}$ chooses at random ${\displaystyle 2^{\delta _{0}}}$ of the ${\displaystyle |D_{1}\rangle }$ states. The other voters measure half of their particles in the computational and half in the Fourier basis.
• if the chosen basis is computational then the measurement results need to add up to 0,
• else: the measurement results are all the same.

All voters simultaneously broadcast their results and if one of them notices a discrepancy, the protocol aborts.

The states ${\displaystyle |D_{2}\rangle }$ are similarly checked.

3. All voters measure their qudits in the computational basis.

Then each ${\displaystyle V_{k}}$ holds a blank ballot of dimension N with the measurement outcomes corresponding to parts of ${\displaystyle |D_{1}\rangle }$ states ${\displaystyle B_{k}=[\xi _{k}^{1}...\xi _{k}^{sk_{k}}...\xi _{k}^{N}]^{T}}$ and a unique index, ${\displaystyle sk_{k}\in \{1,...,N\}}$ from the measurement outcome of the qudit that belongs to ${\displaystyle |D_{2}\rangle }$.
• Casting phase:
1. ${\displaystyle V_{k}}$ applies ${\displaystyle \xi _{k}^{sk_{k}}\rightarrow \xi _{k}^{sk_{k}}+v_{k}}$.
2. All voters simultaneously broadcast their columns resulting in a public ${\displaystyle N\times N}$ table.
• Tally phase:
1. Each ${\displaystyle V_{k}}$ checks that the corresponding row of the matrix adds up to their vote. If this fails, the protocol aborts.
2. final outcome of the election is the sum of the elements of each row of the public ${\displaystyle N\times N}$ table

Further Information

*contributed by Sara Sarfaraz