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

Abstract

This paper presents a quantum algorithm to decide whether a Boolean equation system F has a solution and to compute one if F does have solutions with any given success probability. The runtime complexity of the algorithm is polynomial in the size of F and the condition number of certain Macaulay matrix associated with 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 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.