Quantum Algorithm for Boolean Equation Solving and Quantum Algebraic Attack on Cryptosystems

Yu-Ao Chen, Xiao-Shan Gao

Abstract: This paper presents a quantum algorithm to decide whether a Boolean equation system $$\mathcal{F}$$ F has a solution and to compute one if $$\mathcal{F}$$ F does have solutions with any given success probability. The runtime complexity of the algorithm is polynomial in the size of $$\mathcal{F}$$ F and the condition number of certain Macaulay matrix associated with $$\mathcal{F}$$ F . As a consequence, the authors give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are polynomial in the size of $$\mathcal{F}$$ F . The authors apply the proposed quantum algorithm to the cryptanalysis of several important cryptosystems: The stream cipher Trivum, the block cipher AES, the hash function SHA-3/Keccak, the multivariate public key cryptosystems, and show that they are secure under quantum algebraic attack only if the corresponding condition numbers are large. This leads to a new criterion for designing such cryptosystems which are safe against the attack of quantum computers: The corresponding condition number.