Table 4 Concrete security estimation of LWE instances in the TU Darmstadt LWE challenge
Parameters | The BKW algorithms | The lattice-based algorithms | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
n | q | \(\alpha\) | Plain BKW | LMS-BKW | FFT-BKW | Coded-BKW | Sieve-Coded-BKW | BKW-FWHT-SR | Primal | Decoding | Dual |
40 | 1601 | 0.005 | 47.3 | 47.4 | 40.6 | 42.6 | 41.5 | 34.4 | 31.4 | 35.0 | 37.1 |
0.010 | 51.6 | 49.9 | 43.5 | 43.7 | 42.7 | 39.3 | 31.7 | 35.0 | 37.6 | ||
0.015 | 55.2 | 52.2 | 46.5 | 52.6 | 44.1 | 42.4 | 32.0 | 35.2 | 41.4 | ||
0.020 | 57.4 | 53.4 | 49.0 | 52.6 | 49.1 | 46.2 | 32.4 | 35.3 | 43.8 | ||
0.025 | 59.9 | 54.8 | 52.6 | 52.7 | 49.2 | 48.3 | 34.3 | 36.6 | 48.1 | ||
0.030 | 62.5 | 56.6 | 52.6 | 52.7 | 50.4 | 50.0 | 37.9 | 39.0 | 53.4 | ||
45 | 2027 | 0.005 | 51.7 | 51.6 | 44.1 | 55.2 | 45.0 | 37.7 | 31.6 | 35.1 | 37.4 |
0.010 | 56.6 | 54.4 | 48.8 | 55.2 | 45.3 | 43.5 | 32.0 | 35.3 | 39.9 | ||
0.015 | 60.8 | 57.0 | 51.8 | 55.2 | 54.7 | 48.3 | 32.5 | 35.5 | 43.9 | ||
0.020 | 63.3 | 58.5 | 55.4 | 55.2 | 54.8 | 51.2 | 35.7 | 37.3 | 50.7 | ||
0.025 | 66.1 | 60.2 | 55.7 | 55.3 | 54.8 | 54.1 | 39.9 | 41.0 | 57.1 | ||
0.030 | 72.8 | 65.2 | 59.7 | 64.1 | 63.3 | 56.3 | 44.2 | 45.1 | 64.1 | ||
50 | 2503 | 0.005 | 56.3 | 55.7 | 48.9 | 46.4 | 45.5 | 41.8 | 31.9 | 35.2 | 37.6 |
0.010 | 61.6 | 59.0 | 54.4 | 56.0 | 53.3 | 48.7 | 32.5 | 35.5 | 42.5 | ||
0.015 | 66.3 | 61.9 | 57.8 | 56.8 | 53.4 | 52.5 | 35.5 | 37.2 | 50.5 | ||
0.020 | 72.2 | 66.2 | 61.9 | 61.9 | 60.4 | 56.4 | 41.0 | 42.7 | 59.0 | ||
0.025 | 72.2 | 65.3 | 61.9 | 66.1 | 61.7 | 59.3 | 46.4 | 47.7 | 65.4 | ||
0.030 | 75.7 | 67.7 | 66.9 | 66.3 | 65.6 | 63.3 | 51.4 | 52.9 | 75.5 | ||
70 | 4903 | 0.005 | 75.9 | 74.0 | 66.2 | 62.3 | 62.2 | 62.3 | 34.6 | 37.5 | 49.5 |
0.010 | 82.8 | 79.5 | 72.9 | 67.1 | 70.6 | 73.7 | 47.1 | 49.6 | 66.3 | ||
0.015 | 88.2 | 81.1 | 77.5 | 73.3 | 72.7 | 75.6 | 57.4 | 60.9 | 81.2 | ||
120 | 14401 | 0.005 | 125.2 | 119.1 | 113.4 | 110.5 | 108.8 | 100.1 | 70.3 | 79.4 | 98.4 |
0.010 | 142.0 | 130.4 | 123.1 | 124.0 | 115.8 | 115.1 | 86.5 | 101.7 | 106.1 | ||
0.015 | 148.9 | 133.8 | 130.9 | 136.8 | 135.3 | 127.0 | 98.3 | 118.5 | 133.3 | ||