Quantum reversible circuits for GF(2⁸) multiplication based on composite field arithmetic operations

In this paper, we mainly discuss the quantum reversible circuits of multiplication over GF(2(8)), which has many applications in modern cryptography. The quantum circuits of multiplication over GF(2(8)) implemented by using the existing methods need at least 64 Toffoli gates without auxiliary qubits. However, Toffoli gates need a lot of quantum resources in physical implementation. Therefore, we try to construct the quantum circuits with as few Toffoli gates as possible. We first convert multiplication over GF(2(8)) into multiplication over composite field GF((2(4))(2)), and then realize the quantum circuits of multiplication over GF(2(4)) by means of product matrix and converting the multiplication into composite field GF((2(2))(2)), respectively. In addition, we also discuss the case where the initial output qubits of the product are not |0)s, and give the quantum circuit of multiplication over GF(2(4)) in this case according to the principle of minimizing the number of Toffoli gates. Finally, according to the calculation formula of multiplication over composite field GF((2(4))(2)) and the isomorphic mappings between GF(28) and GF((2(4))(2)), the quantum circuits of multiplication over GF(2(8)) are realized. These quantum circuits without auxiliary qubits only needs 42 Toffoli gates, which are 22 less than the quantum circuits realized by the existing methods. Specifically, we give the specific quantum circuits with irreducible polynomials f(x) = x(8) + x(4) + x(3) + x + 1 and f(x) = x(8) + x(4 )+ x(3) + x(2) + 1, respectively.