Editing Dual Basis Measurement Based Protocol (section)

==Protocol Description==
*'''Setup phase:'''
# <p>One of the voters prepares <math> N+N2^{\delta_0}</math> states of the form <math>|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 </math> and <math> 1 + N2^{\delta_0}</math> states of the form <math> |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 </math>. </p> Each voter <math>V_k</math> receives kth particle from each of the states.
# Voter <math>V_k</math> chooses at random <math>2^{\delta_0}</math> of the <math>|D_1\rangle</math> 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. <p>All voters simultaneously broadcast their results and if one of them notices a discrepancy, the protocol aborts.</P><p> The states <math>|D_2\rangle</math> are similarly checked.</p>
#<p> All voters measure their qudits in the computational basis.</p> Then each <math>V_k</math> holds a blank ballot of dimension N with the measurement outcomes corresponding to parts of <math>|D_1\rangle </math> states <math>B_k = [\xi_k^{1}...\xi_k^{sk_k}...\xi_k^{N}]^{T}</math> and a unique index, <math>sk_k \in \{1,...,N\}</math> from the measurement outcome of the qudit that belongs to <math>|D_2\rangle</math>.

*'''Casting phase''':
# <math>V_k</math> applies <math>\xi^{sk_k}_k\rightarrow \xi^{sk_k}_k+v_k</math>.
# All voters simultaneously broadcast their columns resulting in a public <math>N \times N</math> table.

*'''Tally phase''':
# Each <math>V_k</math> checks that the corresponding row of the matrix adds up to their vote. If this fails, the protocol aborts.
# final outcome of the election is the sum of the elements of each row of the public <math>N \times N</math> table