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Low-Memory Attacks against Two-Round Even-Mansour using the 3-XOR Problem

Abstract : The iterated Even-Mansour construction is an elegant construction that idealizes block cipher designs such as the AES. In this work we focus on the simplest variant, the 2-round Even-Mansour construction with a single key. This is the most minimal construction that offers security beyond the birthday bound: there is a security proof up to 2 2n/3 evaluations of the underlying permutations and encryption, and the best known attacks have a complexity of roughly 2 n /n operations. We show that attacking this scheme with block size n is related to the 3-XOR problem with element size = 2n, an important algorithmic problem that has been studied since the nineties. In particular the 3-XOR problem is known to require at least 2 /3 queries, and the best known algorithms require around 2 /2 / operations: this roughly matches the known bounds for the 2-round Even-Mansour scheme. Using this link we describe new attacks against the 2-round Even-Mansour scheme. In particular, we obtain the first algorithms where both the data and the memory complexity are significantly lower than 2 n. From a practical standpoint, previous works with a data and/or memory complexity close to 2 n are unlikely to be more efficient than a simple brute-force search over the key. Our best algorithm requires just λn known plaintex-t/ciphertext pairs, for some constant 0 < λ < 1, 2 n /λn time, and 2 λn memory. For instance, with n = 64 and λ = 1/2, the memory requirement is practical, and we gain a factor 32 over brute-force search. We also describe an algorithm with asymptotic complexity O(2 n ln 2 n/n 2), improving the previous asymptotic complexity of O(2 n /n), using a variant of the 3-SUM algorithm of Baran, Demaine, and Pǎtraşcu.
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Contributor : Gaëtan Leurent <>
Submitted on : Saturday, December 28, 2019 - 8:24:29 PM
Last modification on : Thursday, January 7, 2021 - 3:38:03 PM
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Gaëtan Leurent, Ferdinand Sibleyras. Low-Memory Attacks against Two-Round Even-Mansour using the 3-XOR Problem. CRYPTO 2019 - 39th Annual International Cryptology Conference, Aug 2019, Santa Barbara, United States. pp.210-235, ⟨10.1007/978-3-030-26951-7_8⟩. ⟨hal-02424902⟩



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