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

*Gaëtan Leurent and Ferdinand Sibleyras*

**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 $w = 2n$, an important algorithmic problem that has been studied since the nineties. In particular the 3-XOR problem is known to require at least $2^{w/3}$ queries, and the best known algorithms require around $2^{w/2} / w$ 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 $\lambda n$ known plaintext/ciphertext pairs, for some constant $0 < \lambda < 1$, $2^n/\lambda n$ time, and $2^{\lambda n}$ memory. For instance, with $n = 64$ and $\lambda = 1/2$, 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 Patrascu.

**Category / Keywords: **secret-key cryptography / Even-Mansour, Cryptanalysis, 3-XOR

**Original Publication**** (in the same form): **IACR-CRYPTO-2019
**DOI: **10.1007/978-3-030-26951-7_8

**Date: **received 17 Aug 2019

**Contact author: **gaetan leurent at inria fr, ferdinand sibleyras at inria fr

**Available format(s): **PDF | BibTeX Citation

**Version: **20190818:160157 (All versions of this report)

**Short URL: **ia.cr/2019/938

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