Security of Key-Alternating Ciphers: Quantum Lower Bounds and Quantum Walk Attacks
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arXiv
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2024
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| author | Bai, Chen Esmaili, Mehdi Mantri, Atul |
| author_facet | Bai, Chen Esmaili, Mehdi Mantri, Atul |
| contents | We study the quantum security of key-alternating ciphers (KAC), a natural multi-round generalization of the Even--Mansour construction. KAC abstracts the round structure of practical block ciphers as public permutations interleaved with key XORs. The $1$-round KAC or EM setting already highlights the power of quantum superposition access: EM is secure against classical and Q1 adversaries (quantum access to the public permutation), but insecure in the Q2 model. The security of multi-round KACs remain largely unexplored; in particular, whether the quantum-classical separation extends beyond a single round had remained open.
1) Quantum Lower Bounds. We prove security of the $t$-round KAC against a non-adaptive adversary in both the Q1 and Q2 models. In the Q1 model, any distinguiser requires $Ω(2^{\frac{tn}{2t+1}})$ oracle queries to distinguish the cipher from a random permutation, whereas classically any distinguisher needs $Ω(2^{\frac{tn}{t+1}})$ queries. As a corollary, we obtain a Q2 lower bound of $Ω(2^{\frac{(t-1)n}{2t}})$ quantum queries. Thus, for $t \geq 2$, the exponential Q1-Q2 gap collapses in the non-adaptive setting, partially resolving an open problem posed by Kuwakado and Morii (2012). Our proofs develop a controlled-reprogramming framework within a quantum hybrid argument, sidestepping the lack of quantum recording techniques for permutation-based ciphers; we expect this framework to be useful for analyzing other post-quantum symmetric primitives.
2) Quantum Key-Recovery Attack. We give the first non-trivial quantum key-recovery algorithm for $t$-round KAC in the Q1 model. It makes $O(2^{αn})$ queries with $α= \frac{t(t+1)}{(t+1)^2 + 1}$, improving on the best known classical bound of $O(2^{α' n})$ with $α' = \frac{t}{t+1}$. The algorithm adapts quantum walk techniques to the KAC structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05026 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Security of Key-Alternating Ciphers: Quantum Lower Bounds and Quantum Walk Attacks Bai, Chen Esmaili, Mehdi Mantri, Atul Quantum Physics Cryptography and Security We study the quantum security of key-alternating ciphers (KAC), a natural multi-round generalization of the Even--Mansour construction. KAC abstracts the round structure of practical block ciphers as public permutations interleaved with key XORs. The $1$-round KAC or EM setting already highlights the power of quantum superposition access: EM is secure against classical and Q1 adversaries (quantum access to the public permutation), but insecure in the Q2 model. The security of multi-round KACs remain largely unexplored; in particular, whether the quantum-classical separation extends beyond a single round had remained open. 1) Quantum Lower Bounds. We prove security of the $t$-round KAC against a non-adaptive adversary in both the Q1 and Q2 models. In the Q1 model, any distinguiser requires $Ω(2^{\frac{tn}{2t+1}})$ oracle queries to distinguish the cipher from a random permutation, whereas classically any distinguisher needs $Ω(2^{\frac{tn}{t+1}})$ queries. As a corollary, we obtain a Q2 lower bound of $Ω(2^{\frac{(t-1)n}{2t}})$ quantum queries. Thus, for $t \geq 2$, the exponential Q1-Q2 gap collapses in the non-adaptive setting, partially resolving an open problem posed by Kuwakado and Morii (2012). Our proofs develop a controlled-reprogramming framework within a quantum hybrid argument, sidestepping the lack of quantum recording techniques for permutation-based ciphers; we expect this framework to be useful for analyzing other post-quantum symmetric primitives. 2) Quantum Key-Recovery Attack. We give the first non-trivial quantum key-recovery algorithm for $t$-round KAC in the Q1 model. It makes $O(2^{αn})$ queries with $α= \frac{t(t+1)}{(t+1)^2 + 1}$, improving on the best known classical bound of $O(2^{α' n})$ with $α' = \frac{t}{t+1}$. The algorithm adapts quantum walk techniques to the KAC structure. |
| title | Security of Key-Alternating Ciphers: Quantum Lower Bounds and Quantum Walk Attacks |
| topic | Quantum Physics Cryptography and Security |
| url | https://arxiv.org/abs/2412.05026 |