A multiplicative group modulo a prime p contains the nonzero residue classes 1 through p−1 under multiplication modulo p. In Z₁₃*, exponentiation cycles through these elements; a generator is a base whose powers visit every 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 |
The exponentiation table for modulo 13 makes generators visible: bases 2, 6, 7, and 11 each produce every nonzero residue exactly once across their powers. section 8.3.1
Exponentiating a known generator is straightforward, but reversing y = gⁿ mod p to recover n is the discrete logarithm problem. This asymmetry supports Diffie–Hellman key exchange and ElGamal encryption. section 8.3.1
