Comments on search procedures for primitive roots

Abstract

Let p p be an odd prime. Assuming the Extended Riemann Hypothesis, we show how to construct O ( ( log ⁡ p ) 4 ( log ⁡ log ⁡ p ) − 3 ) O( {(\log p)^{4} (\log \log p)^{-3} } ) residues modulo p p , one of which must be a primitive root, in deterministic polynomial time. Granting some well-known character sum bounds, the proof is elementary, leading to an explicit algorithm.