Table 2 A comparison of different distinguishers in the hypothesis testing stage
Distinguisher | Time complexity |
Sample complexity | |
Optimal distinguisher | \(\mathcal {O}\left( m\cdot {{\left( 2d+1\right) }^{k}}\right)\) |
\(8\cdot \ln \left( \frac{{{q}^{k}}}{\varepsilon }\right) \cdot {{\left( 1-{2{{\pi }^{2}}{{\sigma }^{2}}/{{q}^{2}}}\right) }^{-{{2}^{t+1}}}}\) | |
FFT distinguisher | \(\mathcal {O}\left( m+k\cdot {{q}^{k}}\cdot \log q\right)\) |
\(8\cdot \ln \left( \frac{{{q}^{k}}}{\varepsilon }\right) \cdot {{\left( 1-{2{{\pi }^{2}}{{\sigma }^{2}}/{{q}^{2}}}\right) }^{-{{2}^{t+1}}}}\) | |
Pruned FFT distinguisher | \(\mathcal {O}\left( m+k\cdot {{q}^{k}}\cdot \log \left( 2d+1\right) \right)\) |
\(8\cdot \ln \left( \frac{{{\left( 2d+1\right) }^{\ \ \ k}}}{\varepsilon }\right) \cdot {{\left( 1-{2{{\pi }^{2}}{{\sigma }^{2}}/{{q}^{2}}}\right) }^{-{{2}^{t+1}}}}\) | |
Subspace hypothesis testing | \(\mathcal {O}\left( M\cdot {{n}_{test}}+{{\left( 2d+1\right) }^{{{n}_{top}}}}\cdot \left( {{q}^{l+1}}\cdot {{\log }_{2}}{{q}^{l+1}}+{{q}^{l+1}}\right) \right)\) |
\(\frac{4\ln \left( {{\left( 2d+1\right) }^{{{\ \ \ n\ \ }_{top}}}}\ \ {{q}^{l}}\right) }{\Delta \left( {{\chi }_{{\sigma }_{final}}}\ \ \ \|\ U\right) }\) | |
FWHT distinguisher | \(\mathcal {O}\left( {{2}^{k}}\log {{2}^{k}}\right)\) |
\(\frac{4\ln \left( {{2}^{{{n}_{\ t}}}}\right) }{\Delta \left( {{\chi }_{{{\sigma }_{final}}\ \ \ ,\ 2q}} \ \ \ \| {{U}_{\ 2q}}\right) }\) |