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Once you have a generator $g$ of a cyclic group $G$ of order $n$, all the other generators are given by $g^a$ where $\gcd(a,n)=1$. So you pick a random element, test it: $$\forall p|n,\qquad g^{n/p}\neq e,$$ and as soon as you find a generator, you find them all. Since the order of $(\mathbb{Z}/251\mathbb{Z})^*$ is $250=2\cdot 5^3$, any element $a\in ...


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Yes, there are better approaches. A better approach is to verify that $g^{n-1} \equiv 1 \mod n$ (which is always true) and $g^{\frac{n-1}{d}} \not \equiv 1 \mod n$ for the factors $d$ of $n-1$. This can be optimized a little bit. See this answer by André Nicolas for a bit more there. Once you've found one, you can get all the others "for free." In ...


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if i understand you right, that is not possible. if you take a=3, b=5, c=7, it is the same thing as if you take a=4,b=4, c=7 that means, if you are able to get c if you know the real $a_1$ and $b_1$, you can get the same c if you use "wrong" a,b, lets say $a_2,b_2$ such that $a_1+b_1 = a_2 b_2$



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