Quantum reversible circuits for mathrm{GF}(2^{8}) multiplicative inverse

Abstract

The synthesis of quantum circuits for multiplicative inverse over operatorname{GF}(2^{8}) are discussed in this paper. We first convert the multiplicative inverse operation in operatorname{GF}(2^{8}) to arithmetic operations in the composite field operatorname{GF}((2^{4})^{2}), and then discuss the expressions of the square calculation, the inversion calculation and the multiplication calculation separately in the finite field operatorname{GF}(2^{4}), where the expressions of multiplication calculation in operatorname{GF}(2^{4}) are given directly in operatorname{GF}(2^{4}) and given through being transformed into the composite field operatorname{GF}((2^{2})^{2}). Then the quantum circuits of these calculations are realized one by one. Finally, two quantum circuits for multiplicative inverse over operatorname{GF}(2^{8}) are synthesized. They both use 21 qubits, the first quantum circuit uses 55 Toffoli gates and 107 CNOT gates and the second one uses 37 Toffoli gates and 209 CNOT gates. As an example of the application of multiplication inverse, we apply these quantum circuits to the implementations of the S-box quantum circuit of the AES cryptographic algorithm. Two quantum circuits for implementing the S-box of the AES cryptographic algorithm are presented. The first quantum circuit uses 21 qubits, 55 Toffoli gates, 131 CNOT gates and 4 NOT gates and the second one uses 21 qubits, 37 Toffoli gates, 233 CNOT gates and 4 NOT gates. Through the evaluation of quantum cost, the two quantum circuits of the S-box of AES cryptographic algorithm use less quantum resources than the existing schemes.