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

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 mat...

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Bibliographic Details
Published in:Journal of systems science and complexity Vol. 35; no. 1; pp. 373 - 412
Main Authors: Chen, Yu-Ao, Gao, Xiao-Shan
Format: Journal Article
Language:English
Published: Beijing Academy of Mathematics and Systems Science, Chinese Academy of Sciences 01.02.2022
Springer Nature B.V
Subjects:
Algorithms
Boolean
Boolean algebra
Complex Systems
Computer systems
Control
Cryptography
Encryption
Hash based algorithms
Mathematics
Mathematics and Statistics
Mathematics of Computing
Operations Research/Decision Theory
Polynomials
Quantum computers
Statistics
Systems Theory
Block cipher AES
hash function SHA-3/Keccak
Boolean equation solving
quantum algorithm
polynomial system solving
stream cipher Trivum
condition number
HHL algorithm
MPKC
ISSN:1009-6124, 1559-7067
Online Access:Get full text
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Summary: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.
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ISSN:1009-6124
1559-7067
DOI:10.1007/s11424-020-0028-6