Quantum algorithm to find invariant linear structure of MD hash functions

• Published in: Quantum information processing Vol. 14; no. 3; pp. 813 - 829
• Main Authors: Wu, WanQing, Zhang, HuanGuo, Mao, ShaoWu, Wang, HouZhen
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.03.2015
• Subjects: Data Structures and Information Theory; Mathematical Physics; Physics; Physics and Astronomy; Quantum Computing; Quantum Information Technology; Quantum Physics; Spintronics; Hash functions; Quantum network; Quantum algorithm; Invariant linear structure
• ISSN: 1570-0755; 1573-1332
• DOI: 10.1007/s11128-014-0909-5

In this paper, we consider a special problem. “Given a function f : { 0 , 1 } n → { 0 , 1 } m . Suppose there exists a n -bit string α ∈ { 0 , 1 } n subject to f ( x ⊕ α ) = f ( x ) for ∀ x ∈ { 0 , 1 } n . We only know the Hamming weight W ( α ) = 1 , and find this α .” We present a quantum algorithm with “Oracle” to solve this problem. The successful probability of the quantum algorithm is ( 2 l - 1 2 l ) n - 1 , and the time complexity of the quantum algorithm is O ( log ( n - 1 ) ) for the given Hamming weight W ( α ) = 1 . As an application, we present a quantum algorithm to decide whether there exists such an invariant linear structure of the M D hash function family as a kind of collision. Then, we provide some consumptions of the quantum algorithms using the time–space trade-off.