Binary Field Montgomery Multiplication on Quantum Computers

Optimizing arithmetic operations into quantum circuits to utilize quantum algorithms, such as the Shor algorithm and Grover search algorithm for cryptanalysis, is an active research field in cryptography implementation. In particular, reducing quantum resources is important for efficient implementation. In this paper, binary field (GF (2)) Montgomery multiplication in quantum circuits is presented. We utilize the bit-level Montgomery algorithm to efficiently compute the Montgomery product C = A ·B · r−1 in the binary field GF (2). Additionally, we also present an efficient Montgomery multiplication quantum circuit in the case where the modulus of GF (2) is specified.

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