How to find element g order q, given 2 large primes p and q where q|p-1

Question

i have calculated 2 large primes, $p$ (minimum 2048 bits) and $q$ (minimum 224 bits), where $p-1 \mod q = 0$ by using SageMath.

$p = $32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638215525193403303896028543209689578721838988682461578457274025662014413066681559$

$q = $26959946667150639794667015087019630673637144422540572481103610249951

Now, i need element $g$ order $q$. I try to find it, but it takes so long (still running). Anyone have idea how to find element $g$?

Answer 1

The multiplicative group $\mathbf{Z}_p^*$ of non-zero integers modulo $p$ is cyclic of order $p-1$, so it has exactly one subgroup of order $k$ for each divisor $k$ of $p-1$. In particular, it has exactly one subgroup of order $q$, which consists of those integers $a$ such that $a^q \bmod p = 1$. Let $G$ be this subgroup.

To obtain an element $g$ of $G$, take any element $a$ of $\mathbf{Z}_p^*$ and let $g = a^{(p-1)/q} \bmod p$. Then $g$ is an element of $G$ because $$g^q = \left(a^{(p-1)/q}\right)^q = a^{p-1} = 1 \pmod p.$$

The order of $g$ is a divisor of the order of $G$, so it is a divisor of $q$, and since $q$ is prime, it equals either $1$ or $q$. The only element of order $1$ is the identity $1$, so if $g \not\equiv 1 \pmod p$ you are done. Otherwise, try another $a$.

Answer 2

Let $G$ be the group with order $p-1$. Select random element from $G$ and call it $g'$. Then compute

$$g=\frac{p-1}{q}\cdot g'$$.