I did this algorithm to find a primitive element of a multiplicative group
on a finite field. This is a basic algorithm
def random_primitive(p,h):
F.<x>=GF(p^h)
s=p^h-1
r=F.random_element()
j=0
if r!=0:
for t in prime_factors(s):
if r^(s/t)==1:
j=j+1
if j==0:
return r
else:
return random_primitive(p,h)
else:
return random_primitive(p,h)
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