<p>Quantum computing is a rapidly evolving scientific and technological field with the potential to revolutionize computational problem-solving by leveraging interdisciplinary research in computer science, physics, and mathematics based on quantum mechanics (Steane in Phy Rev Lett 77:793–797, 2003). One of the most significant implications of quantum computing is its ability to accelerate the solution of computationally complex problems that are challenging for classical von Neumann architectures. Among various quantum algorithms, Shor’s algorithm holds particular interest due to its ability to factorize integers in polynomial time, posing a significant threat to the security of traditional cryptographic systems such as RSA, DSA, EdDSA, GOST R 34.10-2012, and others that lack quantum resistance (Seifer in LNCS 2020:319–327, 2001, Parametric Selection of Cryptographic Primitives for Blockchain Platforms, Athena Publishing, 2022). However, the practical implementation of Shor’s algorithm faces several technical challenges, with the efficient construction of the Quantum Fourier Transform (QFT) being a critical hurdle. This article proposes a new hybrid approach to constructing an Approximate Quantum Fourier Transform (AQFT) (Coppersmith, An approximate Fourier transform useful in quantum factoring, 2002) with a limited number of quantum gates, aimed at optimizing Shor’s algorithm. The method involves the sequential application of various techniques to reduce quantum resources and enhance the efficiency of the QFT circuit. These optimizations are crucial for improving the performance of Shor’s algorithm while maintaining the desired level of precision.Keywords: Quantum Threat; Quantum Fourier Transform; Quantum Resistance; Quantum and Post-Quantum Cryptography; Cryptographic Primitive Security; Quantum-Resistant Cryptosystem</p>