On Finding Quantum Multi-collisions

The current paper improves the number of queries of the previous quantum multi-collision finding algorithms presented by Hosoyamada et al. at Asiacrypt 2017. Let an l-collision be a tuple of l distinct inputs that result in the same output of a target function. In cryptology, it is important to study how many queries are required to find l-collisions for random functions of which domains are larger than ranges. The previous algorithm finds an l-collision for a random function by recursively calling the algorithm for finding (l − 1)-collisions, and it achieves the average quantum query complexity of O(N (3 l−1 −1)/(2·3 l−1 ) ), where N is the range size of target functions. The new algorithm removes the redundancy of the previous recursive algorithm so that different recursive calls can share a part of computations. The new algorithm finds an l-collision for random functions with the average quantum query complexity of O(N (2 l−1 −1)/(2 l −1) ), which improves the previous bound for all l ≥ 3 (the new and previous algorithms achieve the optimal bound for l = 2). More generally, the new algorithm achieves the average quantum query complexity of O cfor a random function f : X → Y such that |X| ≥ l · |Y |/cN for any 1 ≤ cN ∈ o(N 1 2 l −1 ). With the same query complexity, it also finds a multiclaw for random functions, which is harder to find than a multicollision.

show abstract

“…[HSX17] and matches with the lower bound proved by Liu and Zhandry [LZ18]. The complexities for small l's are listed in Table 1.…”

Section: Introductionsupporting

confidence: 72%

“…Concurrent Work. Very recently, Liu and Zhandry [LZ18] showed that for every integer constant l ≥ 2, Θ N 1 2 (1− 1 2 l −1 ) quantum queries are both necessary and sufficient to find a l-collision with constant probability, for a random Table 1. Query complexities of l-collision finding quantum algorithms.…”

Section: Introductionmentioning

confidence: 99%

Improved Quantum Multicollision-Finding Algorithm

Hosoyamada

Sasaki

Tani

et al. 2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…In [Zha19], Zhandry showed how we can overcome this obstacle with his proof technique named "compressed oracle technique," which enables us to record transcripts of queries made to quantum random oracles to some extent. His technique is so powerful that it can be used to show the indifferentiability of Merkle-Damgård construction, post-quantum security of Fujisaki-Okamato transformation [Zha19], the quantum query lower bound for the multicollision-finding problem on random functions [LZ19], and that the 4-round Luby-Rackoff construction is a qPRP [HI19,HI20]. The technique not only enables us to record quantum queries but also efficiently simulate random functions.…”

Section: Proof Techniques In the Quantum Settingmentioning

confidence: 99%

Provably Quantum-Secure Tweakable Block Ciphers

Hosoyamada

Iwata

2021

ToSC

View full text Add to dashboard Cite

Recent results on quantum cryptanalysis show that some symmetric key schemes can be broken in polynomial time even if they are proven to be secure in the classical setting. Liskov, Rivest, and Wagner showed that secure tweakable block ciphers can be constructed from secure block ciphers in the classical setting. However, Kaplan et al. showed that their scheme can be broken by polynomial time quantum superposition attacks, even if underlying block ciphers are quantum-secure. Since then, it remains open if there exists a mode of block ciphers to build quantum-secure tweakable block ciphers. This paper settles the problem in the reduction-based provable security paradigm. We show the first design of quantum-secure tweakable block ciphers based on quantum-secure block ciphers, and present a provable security bound. Our construction is simple, and when instantiated with a quantum-secure n-bit block cipher, it is secure against attacks that query arbitrary quantum superpositions of plaintexts and tweaks up to O(2n/6) quantum queries. Our security proofs use the compressed oracle technique introduced by Zhandry. More precisely, we use an alternative formalization of the technique introduced by Hosoyamada and Iwata.

show abstract

“…Liu and Zhandry in [18] proposed an improved k-collision search algorithm. The attack process of their k-collision search algorithm can be explained in Algorithm 1 as follows.…”

Section: The Improved Multi-collision Search Algorithm Proposed By Liu and Zhandry (Lz18)mentioning

confidence: 99%

“…) quantum queries and O(2 n/3 ) quantum memory. In 2018, Liu and Zhandry [18] proposed an improved quantum k-collision search algorithm (call LZ18) with Θ(2…”

Section: Introductionmentioning

confidence: 99%

An Efficient Quantum Multi-Collision Search Algorithm

2020

View full text Add to dashboard Cite

In this article, we propose an efficient quantum k-collision search algorithm with low quantum memory O(n). The previous quantum k-collision algorithms can not be converted into a low quantum memory k-collision algorithm directly, because the time complexity of the converted algorithm is larger than the basic k-collision algorithm. To solve this problem, we shall not only divide our low memory quantum k-collision algorithm into several subroutines, but also need to achieve some balances between these subroutines. The time complexity of our k-collision search algorithm is O( 2(2 k −2)n 2 k+1 −3 ), and the classical memory and quantum memory complexities are O( 2(2 k−1 −1)n 2 k+1 −3 ) and O(n) respectively. In addition, we propose an efficient k-claw search algorithm, which can output a k-claw with O(n) qubits. Given 2 s quantum processors, we can construct our quantum k-collision and k-claw parallel algorithm with), while the classical memory and quantum memory complexities are) and O(n), respectively.

show abstract

On Finding Quantum Multi-collisions

Cited by 33 publications

References 34 publications

Improved Quantum Multicollision-Finding Algorithm

Improved Quantum Multicollision-Finding Algorithm

Provably Quantum-Secure Tweakable Block Ciphers

An Efficient Quantum Multi-Collision Search Algorithm

Contact Info

Product

Resources

About