Quantum Claw-Finding Algorithm

• Quantum claw-finding algorithm is a quantum approach that efficiently identifies pairs (claws) where f(x) equals g(y) using quantum walks and amplitude amplification.
• It underpins optimal quantum attacks on double encryption and cascaded ciphers by reducing classical search time from O(N) to O(N^(2/3)).
• QCF sets critical security limits for block cipher design, guiding the development of post-quantum cryptographic methods.

Quantum Claw-Finding (QCF) Algorithm

Quantum claw-finding (QCF) refers to a class of quantum algorithms that, given two functions $f:X\to Z$ and $g:Y\to Z$ (typically with $|X|=|Y|=N$), efficiently finds a "claw," i.e., a pair $(x^*,y^*)$ such that $f(x^*) = g(y^*)$. QCF forms the fundamental quantum speedup underlying quantum meet-in-the-middle (MITM) attacks in cryptanalysis, particularly in attacking block cipher key-length extension constructions. The algorithm exploits quantum walks and amplitude amplification to surpass the lower bounds imposed by classical collision- or claw-finding procedures. QCF underlies the optimal quantum attacks on double encryption (2-Encrypt), multi-encryption, and cascaded block ciphers, and it sets the quantum security limits for key-length extension schemes.

1. Formal Definition and Problem Setting

Let $f:X\to Z$ and $g:Y\to Z$ be two injective functions over domains of size $N$ ($|X|=|Y|=N$). A "claw" is a pair $(x,y)$ such that $f(x) = g(y)$. The claw-finding problem is: given oracle access to $f$ and $g$, find such a pair.

The minimal-query quantum complexity of this task is $O(N^{2/3})$, by Ambainis’s quantum walk algorithm. This stands in contrast to the classical complexity $O(N)$, set by the birthday paradox and hash-table-based search. The quantum lower bound for query complexity is $\Theta(N^{2/3})$, as established via the generalized adversary method (Kaplan, 2014).

2. Quantum Claw-Finding Algorithm Construction

The canonical QCF algorithm leverages quantum walks on the Johnson graph, as first proposed by Ambainis for the element distinctness problem. For injective $f,g$, the functions can be evaluated in superposition, yielding a quantum search space for the overlaps of the images:

1. Initialization: Set up a uniform superposition over all $x\in X$ and $y\in Y$.
2. Quantum Walk: Explore the space of image values $z\in Z$, searching for overlaps $f(x)=g(y)=z$ with amplitude amplification.
3. Measurement: Once an overlap is found, measure to reveal the corresponding claw $(x^*,y^*)$.

The expected quantum time (and query) complexity is $O(N^{2/3})$, consuming $O(N^{2/3})$ quantum memory (typically implemented with QRAM).

3. Application to Block Cipher Key-Length Extension and MITM Attacks

QCF forms the foundation of quantum MITM attacks on block cipher extensions such as double encryption (2-Encrypt), triple encryption, and cascaded ciphers:

• Double Encryption (2-Encrypt): Given $C = F_{k_2}(F_{k_1}(P))$ for unknown keys $(k_1,k_2)\in[N]^2$, classical MITM solves for $(k_1,k_2)$ in $O(N)$ time/space via table intersections. The quantum MITM attack reduces to claw-finding on $G_1(k) = F_k(P)$, $G_2(k) = F_k^{-1}(C)$, and finds the unique key pair in $O(N^{2/3})$ time/memory (Kaplan, 2014, Jaeger et al., 2021, Liang et al., 12 Nov 2025).
• r-Round Iterated Encryption: For multi-encryption (e.g., quadruple encryption), QCF and its generalizations (quantum walks combining MITM and Grover search) yield further quantum speed ups, with quantum time $O(N^{7/6})$ for 4-Encrypt (Kaplan, 2014).
• Triple Encryption and Key-Length Extension: For two-key triple encryption (2kTE), QCF under the Q2 adversarial model recovers the key in $O(2^{2\kappa/3})$ time and $O(2^{2\kappa/3})$ QRAM, for key length $\kappa$ (Liang et al., 12 Nov 2025).
• Feistel Construction: Quantum MITM attacks on $r$-round Feistel networks use QCF to match internal states efficiently, leading to attack times of $O(2^{2n/3})$ on 7 rounds ($n$ = block size) (Xu et al., 2021).

4. Time, Space, and Optimality

The resource requirements for QCF-based attacks are as follows:

Attack setting	Time Complexity	Quantum Memory (QRAM)	Optimality Reference
Double encryption	$O(N^{2/3})$	$O(N^{2/3})$	(Kaplan, 2014, Jaeger et al., 2021)
2kTE (Q2 model)	$O(2^{2\kappa/3})$	$O(2^{2\kappa/3})$	(Liang et al., 12 Nov 2025)
2kTE (Grover-based)	$O(2^{\kappa/2})$	$O(2^\kappa)$	(Liang et al., 12 Nov 2025)
Feistel 7-round	$O(2^{2n/3})$	$O(2^{5n/6})$	(Xu et al., 2021)

For two-function claw-finding with $|X|=|Y|=N$, the query lower bound is $\Omega(N^{2/3})$, tight for Ambainis’s walk.

Trading space for time beyond $O(N^{2/3})$ is not possible for QCF-based quantum MITM; the time–space trade-off lies along the line $T \cdot S = O(N^{4/3})$ for standard models (Kaplan, 2014). In contrast, Grover-based brute-force search with unbounded QRAM reaches $O(N^{1/2})$ queries and $O(N)$ memory.

5. Integration into Advanced Quantum Cryptanalytic Techniques

Quantum claw-finding extends to generic sieve-in-the-middle (SITM) attacks, multi-layer (r-round) encryption, and Feistel networks:

• Generalized MITM for Advanced Constructions: QCF is central in cryptanalysis of 3XOR-cascade (3XCE) and Feistel-based KLE schemes, where the attack is cast as finding inputs to two different layers matching at an internal state (Liang et al., 12 Nov 2025, Xu et al., 2021).
• Combined Attacks: For higher-round constructions and longer cascades, QCF is integrated with quantum walks and outer-layer Grover search, producing tight quantum attacks with provable optimal exponents (Kaplan, 2014, Xu et al., 2021).
• Decision and Search Versions: QCF is used both for key-recovery (search version) and distinguishing (decision version) attacks; lower bounds extend to both via the adversary method (Kaplan, 2014, Jaeger et al., 2021).

6. Security Implications for Block Cipher Design

QCF fundamentally constrains the quantum security amplification attainable by simple key-length extension and iteration:

• Amplification Limits: While classical MITM attacks limit the amplification due to double encryption (from $N^2$ to $N$), quantum QCF raises the effective security exponent only to $2/3$ ($O(N^{2/3})$ time) of the classical double-encryption bound, i.e., a $1.5\times$ security gain rather than quadratic (Kaplan, 2014).
• Cascade and Dissection Attacks: As the number of cipher iterations increases, quantum walks with QCF subroutines yield smaller marginal security gains, indicating that security amplification by iteration becomes less effective against quantum adversaries (Kaplan, 2014).
• Design Guidance: Designers must account for QCF attacks when constructing post-quantum secure key-length extension schemes, weighing quantum time, QRAM, and the attack model (Q1 versus Q2) (Liang et al., 12 Nov 2025).

7. Theoretical Foundations and Optimality Proofs

Optimality of the QCF algorithm in query complexity is established via the generalized adversary bound for quantum query complexity:

$$Q_\text{CF} = \Theta(N^{2/3}) = \min_\ell \frac{\|\Gamma_\text{CF}\|}{ \|\Gamma_\text{CF} \circ \Delta_\ell \| }$$

where $\Gamma_\text{CF}$ is the adversary matrix for the claw-finding problem, and $\Delta_\ell$ are the difference operators corresponding to oracle queries (Kaplan, 2014). This result matches the upper bound achieved by Ambainis’s walk, confirming that QCF is the quantum-optimal algorithm for this class of cryptanalytic problems.

• (Kaplan, 2014) J.-J. Boutros, A. Chailloux, A. Naya-Plasencia, and M. Naya-Plasencia. "Quantum attacks against iterated block ciphers."
• (Liang et al., 12 Nov 2025) J. Jaeger, M. Nandi, S. Tessaro. "Quantum Meet-in-the-Middle Attacks on Key-Length Extension Constructions."
• (Jaeger et al., 2021) J. Jaeger, F. Song, S. Tessaro. "Quantum Key-length Extension."
• (Xu et al., 2021) S. Xu, H. Yuan. "Quantum Meet-in-the-Middle Attack on Feistel Construction."