An exact version of Shor's order finding algorithm and its applications

We present an efficient exact version of Shor's order finding algorithm when a multiple $m$ of the order $r$ is known. The algorithm consists of two main ingredients. The first ingredient is the exact quantum Fourier transform proposed by Mosca and Zalka in [MZ03]. The second ingredient is an amplitude amplification version of Brassard and Hoyer in [BH97] combined with some ideas from the exact discrete logarithm procedure by Mosca and Zalka in [MZ03]. As applications, we show how the algorithm derandomizes the quantum algorithm for primality testing proposed by Donis-Vela and Garcia-Escartin in [DVGE18], and serves as a subroutine of an efficient exact quantum algorithm for finding primitive elements in arbitrary finite fields in a slightly less general computational model proposed by Mosca in [Mos02].