In this study, we propose a quantum attack on the two-round iterated Even-Mansour cipher, which consists of two independent permutations and three independent subkeys. Our attack is in the Q1 model, where an adversary makes classical online queries and quantum offline queries. The complexity of our attack is O(2n/2), where n is the block size, while the best known attack for this variant is O(23n/5). Thus, our method significantly improves the computational complexity compared to existing quantum attacks.
In this study, we propose a quantum attack on the two-round iterated Even-Mansour cipher, which consists of two independent permutations and three independent subkeys. Our attack is in the Q1 model, where an adversary makes classical online queries and quantum offline queries. The complexity of our attack is O(2n/2), where n is the block size, while the best known attack for this variant is O(23n/5). Thus, our method significantly improves the computational complexity compared to existing quantum attacks.
It has been said that security of symmetric key schemes is not so much affected by quantum computers, compared to public key schemes. However, recent works revealed that, in some specific situations, symmetric key schemes are also broken in polynomial time by adversaries with quantum computers. These works contain a quantum distinguishing attack on 3-round Feistel ciphers and a quantum key recovery attack on the Even-Mansour cipher by Kuwakado and Morii, in addition to the quantum forgery attack on CBC-MAC which is proposed independently by Kaplan et al., and by Santoli and Schaffner. Iterated Even-Mansour cipher is a simple but important block cipher, which can be regarded as an idealization of AES. Whether there exists an efficient quantum algorithm that can break iterated Even-Mansour cipher with independent subkeys is an important problem from the viewpoint of analyzing post-quantum security of block ciphers. Actually there is an efficient quantum attack on iterated Even-Mansour cipher by Kaplan et al., but their attack can only be applied in the case that all subkeys are the same. This paper shows that there is a polynomial time quantum algorithm that recovers partial keys of the iterated Even-Mansour cipher with independent subkeys, in a related-key setting. The related-key condition is somewhat strong, but our algorithm can recover subkeys with two related oracles. In addition, we also show that our algorithm can recover all keys of the i-round iterated Even-Mansour cipher, if we are allowed to access i related quantum oracles. To realize quantum related-key attacks, we extend Simon's quantum algorithm so that we can recover the hidden period of a function that is periodic only up to constant. Our technique is to take differential of the target function to make a double periodic function, and then apply Simon's algorithm.
Pseudorandom permutations are ubiquitous in theoretical and applied cryptography. PRPs that offer security even against adversaries making quantum queries are of increasing interest, and used in applications ranging from constructing pseudorandom unitaries to separating SZK from BQP.
A successful framework for constructing classically-secure PRPs is the key-alternating Even-Mansour approach, which interleaves applications of public permutations with additions of round keys. The single-round construction is already classically secure in the ideal permutation model (IPM), with added rounds offering improved concrete security. However, in the quantum-query setting, the status of this framework is presently unclear. A simple quantum-query attack based on Simon's algorithm breaks the one-round cipher. For two or more rounds, security is only known against non-adaptive adversaries who must prepare all queries in advance.
In this work, we show that the two-round Even-Mansour cipher is information theoretically secure in the IPM against adversaries making polynomially-many adaptive forward and inverse quantum queries to all available oracles. Our proof uses compressed permutation oracles and a specially crafted isometry relating the ideal and real experiments. We also show that this construction is minimal, in the sense that essentially any cipher constructed via a single call to a public permutation is quantumly insecure.
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(22t+1tn) oracle queries to distinguish the cipher from a random permutation, whereas classically any distinguisher needs Ω(2t+1tn) queries. As a corollary, we obtain a Q2 lower bound of Ω(22t(t−1)n) quantum queries. Thus, for t≥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(2αn) queries with α=(t+1)2+1t(t+1), improving on the best known classical bound of O(2α′n) with α′=t+1t. The algorithm adapts quantum walk techniques to the KAC structure.
The Sum of Even-Mansour (SoEM) construction, proposed by Chen et al. at Crypto 2019, has become the basis for designing some symmetric schemes, such as the nonce-based MAC scheme nEHtMp and the nonce-based encryption scheme CENCPP∗. In this paper, we make the first attempt to study the quantum security of SoEM under the Q1 model where the targeted encryption oracle can only respond to classical queries rather than quantum ones. Firstly, we propose a quantum key recovery attack on SoEM21 with a time complexity of O~(2n/3) along with O(2n/3) online classical queries. Compared with the current best classical result which requires O(22n/3) time, our method offers a quadratic time speedup while maintaining the same number of queries. The time complexity of our attack is less than that observed for quantum exhaustive search by a factor of 2n/6 . We further propose classical and quantum key recovery attacks on the generalized SoEMs1 construction (consisting of s≥2 independent public permutations), revealing that the application of quantum algorithms can provide a quadratic acceleration over the pure classical methods. Our results also imply that the quantum security of SoEM21 cannot be strengthened merely by increasing the number of permutations.
Even-Mansour is one of the most important constructions in symmetric cryptography, both from a theoretical and practical perspective. With the rapid development of quantum computing, the security of Even-Mansour construction in quantum setting needs to be considered. For one round Even-Mansour construction, it is well settled by classical and quantum attacks. While for the iterated scheme, the situation is much more complex. In this paper, we study the next case in line in detail and depth: quantum attacks against two rounds case. We first make an asymptotic comparison with existing classical and quantum attacks. Then we give concrete resource estimation for the proposed quantum attacks on round reduced LED cipher and AES2. The resource estimation allows to deduce the most efficient attacks based on the trade-off of the number of qubits and Toffoli depth.
Due to their simple security assessments, permutation-based pseudo-random functions (PRFs) have become widely used in cryptography. It has been shown that PRFs using a single n-bit permutation achieve n/2 bits of security, while those using two permutation calls provide 2n/3 bits of security in the classical setting. This paper studies the security of permutation-based PRFs within the Q1 model, where attackers are restricted to classical queries and offline quantum computations. We present improved quantum-time/classical-data tradeoffs compared with the previous attacks. Specifically, under the same assumptions/hardware as Grover’s exhaustive search attack, i.e. the offline Simon algorithm, we can recover keys in quantum time Õ(2n/3), with O(2n/3) classical queries and O(n2) qubits. Furthermore, we enhance previous superposition attacks by reducing the data complexity from exponential to polynomial, while maintaining the same time complexity. This implies that permutation-based PRFs become vulnerable when adversaries have access to quantum computing resources. It is pointed out that the above quantum attack can be applied to several cryptographic schemes, including PDMMAC and pEDM, as well as general instantiations like XopEM, EDMEM, EDMDEM, and others.