$F_q$ is finite field, $g$ - generating element of multiplicative group. Assume that for some element $f$ from multiplicative group we have $\log_g (f^3) = 3x$. Is it true, that $\log_g (f) = x$?
2026-02-22 23:30:00.1771803000
For generator $g$ of multiplicative group: if $\log_g (f^3) = 3x$, then $\log_g (f) = x$?
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The question may be rephrased as follows:
Let $\log_g (f) = y$. Then $g^{3x}=f^3=g^{3y}$ iff $3x \equiv 3y \bmod q-1$, because $g$ has order $q-1$.
Therefore, it is true that $x=y$ if $\gcd(3,q-1)=1$.
It may be false otherwise. For instance, take $q=16$.