# Quantum algorithm to find invariant linear structure of MD hash functions

Quantum algorithm to find invariant linear structure of MD hash functions In this paper, we consider a special problem. “Given a function $$f$$ f : $$\{0, 1\}^{n}\rightarrow \{0, 1\}^{m}$$ { 0 , 1 } n → { 0 , 1 } m . Suppose there exists a n-bit string $$\alpha \in \{0, 1\}^{n}$$ α ∈ { 0 , 1 } n subject to $$f(x\oplus \alpha )=f(x)$$ f ( x ⊕ α ) = f ( x ) for $$\forall x\in \{0, 1\}^{n}$$ ∀ x ∈ { 0 , 1 } n . We only know the Hamming weight $$W(\alpha )=1$$ W ( α ) = 1 , and find this $$\alpha$$ α .” We present a quantum algorithm with “Oracle” to solve this problem. The successful probability of the quantum algorithm is $$(\frac{2^{l}-1}{2^{l}})^{n-1}$$ ( 2 l - 1 2 l ) n - 1 , and the time complexity of the quantum algorithm is $$O(\log (n-1))$$ O ( log ( n - 1 ) ) for the given Hamming weight $$W(\alpha )=1$$ W ( α ) = 1 . As an application, we present a quantum algorithm to decide whether there exists such an invariant linear structure of the $$MD$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Quantum algorithm to find invariant linear structure of MD hash functions

, Volume 14 (3) – Jan 7, 2015
17 pages

/lp/springer_journal/quantum-algorithm-to-find-invariant-linear-structure-of-md-hash-SheSWCHveb
Publisher
Springer Journals
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-014-0909-5
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we consider a special problem. “Given a function $$f$$ f : $$\{0, 1\}^{n}\rightarrow \{0, 1\}^{m}$$ { 0 , 1 } n → { 0 , 1 } m . Suppose there exists a n-bit string $$\alpha \in \{0, 1\}^{n}$$ α ∈ { 0 , 1 } n subject to $$f(x\oplus \alpha )=f(x)$$ f ( x ⊕ α ) = f ( x ) for $$\forall x\in \{0, 1\}^{n}$$ ∀ x ∈ { 0 , 1 } n . We only know the Hamming weight $$W(\alpha )=1$$ W ( α ) = 1 , and find this $$\alpha$$ α .” We present a quantum algorithm with “Oracle” to solve this problem. The successful probability of the quantum algorithm is $$(\frac{2^{l}-1}{2^{l}})^{n-1}$$ ( 2 l - 1 2 l ) n - 1 , and the time complexity of the quantum algorithm is $$O(\log (n-1))$$ O ( log ( n - 1 ) ) for the given Hamming weight $$W(\alpha )=1$$ W ( α ) = 1 . As an application, we present a quantum algorithm to decide whether there exists such an invariant linear structure of the $$MD$$ 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.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Jan 7, 2015

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