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

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

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, not necessarily trusted, prepares $N+N2^{\delta _{0}}$ states of the form $|D_{1}\rangle ={\dfrac {1}{\sqrt {m^{N-1}}}}\sum _{\sum _{k=1}^{N}i_{k}=0{\text{ }}mod{\text{ }}c}|i_{1}\rangle |i_{2}\rangle ...|i_{N}\rangle$ and $1+N2^{\delta _{0}}$ states of the form $|D_{2}\rangle ={\dfrac {1}{\sqrt {N!}}}\sum _{(i_{1},i_{2},...,i_{N})\in P_{N}}|i_{1}\rangle |i_{2}\rangle ...|i_{N}\rangle$.

   Each voter 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 kth particle from each of the states.
2. Voter $V_{k}$ chooses at random $2^{\delta _{0}}$ of the $|D_{1}\rangle$ states The other voters measure half of their particles in the computational and half in the Fourier basis. Whenever the chosen basis is computational, the measurement results need to add up to 0, while when the basis is the Fourier, 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 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 |D_2\rangle} are similarly checked.

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

   Then each $V_{k}$ holds a blank ballot of dimension N with the measurement outcomes corresponding to parts of 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 |D_1\rangle } states 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 B_k = [\xi_k^{1}...\xi_k^{sk_k}...\xi_k^{N}]^{T}} and a unique index, $sk_{k}\in \{1,...,N\}$ from the measurement outcome of the qudit that belongs to $|D_{2}\rangle$.

• Casting phase:

1. $V_{k}$ applies 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 \xi^{sk_k}_k\rightarrow \xi^{sk_k}_k+v_k} .
2. All voters simultaneously broadcast their columns resulting in a public $N\times N$ table.

• Tally 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} 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 $N\times N$ table

Further Information

*contributed by Sara Sarfaraz