Optimized reversible quantum circuits for ${\mathbb {F}}_{2^8}$ multiplication

Abstract

Quantum computers represent a serious threat to the safety of modern encryption standards. Within symmetric cryptography, Advanced Encryption Standard (AES) is believed to be quantum resistant if the key sizes are large enough. Arithmetic operations in AES are performed over the binary field ${\mathbb {F}}_{2^m}$ generated by an irreducible pentanomial of degree $m = 8$ using polynomial basis (PB) representation. Multiplication over ${\mathbb {F}}_{2^m}$ is the most complex and important arithmetic operation, so efficient implementations are highly desired. A number of quantum circuits realizing ${\mathbb {F}}_{2^m}$ multiplication have been proposed, where the number of qubits, the number of quantum gates and the depth of the circuit are mainly considered as optimization objectives. In this work, optimized reversible quantum circuits for ${\mathbb {F}}_{2^8}$ multiplication using PB generated by two irreducible pentanomials are presented. The proposed reversible multipliers require the minimum number of qubits and CNOT gates, and the minimum depth among similar ${\mathbb {F}}_{2^8}$ multipliers found in the literature.