New answers tagged safe-prime
You are probably getting confused with the meaning of "generator" here. You are correct that a generator for the entire multiplicative group modulo $p$ cannot satisfy $g^q \equiv 1$ by definition. However - assuming $k = 2$ here - what you probably read is that $g$ is the generator of the quadratic residues modulo $p$, and this group has order $(p - 1)/2$. ...
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