A generator of the finite multiplicative group modulo a prime p is an element g whose successive powers produce every member of {1, 2, ..., p−1} modulo p. For Z₁₃*, the generators are 2, 6, 7, and 11; each generator's powers contain every group element exactly once. section 8.3.1
Table 8.1 Exponentiation table for multiplicative group modulo 13
| Base \ Exponent (n) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| \(1^n \bmod 13\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| \(2^n \bmod 13\) | 1 | 2 | 4 | 8 | 3 | 6 | 12 | 11 | 9 | 5 | 10 | 7 | 1 |
| \(3^n \bmod 13\) | 1 | 3 | 9 | 1 | 3 | 9 | 1 | 3 | 9 | 1 | 3 | 9 | 1 |
| \(4^n \bmod 13\) | 1 | 4 | 3 | 12 | 9 | 10 | 1 | 4 | 3 | 12 | 9 | 10 | 1 |
| \(5^n \bmod 13\) | 1 | 5 | 12 | 8 | 1 | 5 | 12 | 8 | 1 | 5 | 12 | 8 | 1 |
| \(6^n \bmod 13\) | 1 | 6 | 10 | 8 | 9 | 2 | 12 | 7 | 3 | 5 | 4 | 11 | 1 |
| \(7^n \bmod 13\) | 1 | 7 | 10 | 5 | 9 | 11 | 12 | 6 | 3 | 8 | 4 | 2 | 1 |
| \(8^n \bmod 13\) | 1 | 8 | 12 | 5 | 1 | 8 | 12 | 5 | 1 | 8 | 12 | 5 | 1 |
| \(9^n \bmod 13\) | 1 | 9 | 3 | 1 | 9 | 3 | 1 | 9 | 3 | 1 | 9 | 3 | 1 |
| \(10^n \bmod 13\) | 1 | 10 | 9 | 12 | 3 | 4 | 1 | 10 | 9 | 12 | 3 | 4 | 1 |
| \(11^n \bmod 13\) | 1 | 11 | 4 | 5 | 3 | 7 | 12 | 2 | 9 | 8 | 6 | 10 | 1 |
| \(12^n \bmod 13\) | 1 | 12 | 1 | 12 | 1 | 12 | 1 | 12 | 1 | 12 | 1 | 12 | 1 |
Use the exponentiation table modulo 13 to compare bases: generator rows visit all twelve nonzero residues, while other rows repeat a smaller cycle. section 8.3.1
Generator 2 modulo 13
The powers of 2 modulo 13 are 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, and 7 before returning to 1. They therefore cover every element of Z₁₃*. In contrast, powers of 3 repeat the shorter cycle 1, 3, 9. section 8.3.1
