Discrete logarithm with 2 solutions? A clarification request I need some clarification on the discrete logarithm problem...
When a friend and I were solving for the discrete logarithm problem of 9 = 2 ^ x mod 11, we got two different answers. My initial answer was 6 for x.
But his answer was 12 for x. So we decided to cross check our answers and indeed we were very correct.
So what does this mean? Does it mean that there can be more than one answer for a single discrete logarithm problem?
And does this also relate to the circumstance in a real discrete logarithm problem?
The powers of $2$ modulo 11;
\begin{array}{c|rrrrrrrrrrrrrrrr}
x&amp;  1&amp; 2&amp; 3&amp; 4&amp; 5&amp; 6&amp; 7&amp; 8&amp; 9&amp; 10\\ \hline
2 ^x \bmod 11&amp;  2&amp; 4&amp; 8&amp; 5&amp; 10&amp; 9&amp; 7&amp; 3&amp; 6&amp; 1
 \end{array}
We don't need the above 10 due to the Little Fermat Theorem; for a prime $p$ and $p\not| a$ then
$$a^{p-1} \equiv 1 \pmod p$$ We can use this theorem as
$$a^x \equiv  a^{x \bmod{p-1}} \pmod p$$
Therefore your friend's power $12$ is equal to $2$; $12 \equiv 2 \pmod{10}$. In power;
$$ 2^{12} = 2^{12 \bmod{10}} = 2^2 = 4 \bmod 11.$$

A little theory;
The non-zero elements of modulus $11$ form a cyclic multiplicative group. The order of the group is given by $\varphi(p) = p-1$ where $\varphi$ is the Euler's totient function, so it has order $10$.
By the Lagrange theorem, the order of the subgroups must divide the order of the group. The converse, in general, is not true, that is for a divisor of the order of the group, there may not be a subgroup.
$2$ is a multiplicative generator, $\langle 2 \rangle = \mathbb{Z}_{11}^*$ and we can see this in the above table. Besides, $3$ is not a generator, since $\langle 3 \rangle = \{1,3,4,5,9\}$, with $1=3^5,3=3^1,4 = 3^4,5 = 3^3,9 =3^3$. As we can see, the order of $3$ is $ord(3)=5 | 10$.
What about order 2? It is generated by $\langle 10 \rangle = \{1,10\}$. The reason is $10 = -1 \bmod 11$
Note that for a prime $p$ forms a field $\mathbb{F}_{p}$ (also written as $GF(P)$) with integer addition and multiplication.