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We describe a straightforward method to generate a random prime q such that the multiplicative group GF(q)* also has a random large prime-order subgroup. The described algorithm also yields this order p as well as a p'th primitive root of unity. The methods here are efficient asymptotically, but due to large constants may not be very useful in practical settings.arXiv:2202.05955v2 fatcat:pa4saqzrlzgnjg7nqq6xzofad4