Binary field multiplication is widely used in quantum information processing, such as quantum algorithms, cryptanalysis and mathematical arithmetic. The core quantum resources of binary field multiplication are the qubit count and Toffoli depth of its quantum circuit, both of which are largely dependent on the Toffoli gate count. In this paper, we analyze the multiplicative complexity of binary field and present quantum circuits for \(\mathbb {F}_{2^8}\) multiplication from the perspective of time and space. We find that the Toffoli gate count of quantum circuit corresponds to the bilinear complexity in \(\mathbb {F}_{2^n}\) multiplication. The Toffoli gate count obtained by the algebraic curve method increases linearly with \({\varvec{n}}\) , which is slower than the sub-quadratic complexity of Karatsuba algorithm and the iterated logarithm complexity of Chinese remainder theorem (CRT). To demonstrate the advantages of the algebraic curve method, we use elliptic curve bilinear algorithm in \(\mathbb {F}_{(2^2)^4}\) and composite field arithmetic (CFA) to present two types quantum circuits for \(\mathbb {F}_{2^8}\) multiplication, both of which have 24 Toffoli gates and are the lowest at present. The Toffoli depth of the time-efficient quantum circuit is only 1, and the product \({\varvec{D\cdot W}}\) of the depth and width of the circuit is 72, which is lower than before. The space-efficient quantum circuits require 24 qubits and maintain the Toffoli depth of 4, their \({\varvec{D\cdot W}}\) and Toffoli depth are reduced by at least 77.8 \(\%\) compared with the most advanced research.
