International Association for Cryptologic Research (IACR)
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 22n/3 evaluations of the underlying permutations and encryption, and the best known attacks have a complexity of roughly 2n/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 2w/3 queries, and the best known algorithms require around 2w/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 2n .
From a practical standpoint, previous works with a data and/or memory complexity close to 2n are unlikely to be more efficient than a simple brute-force search over the key. Our best algorithm requires just λn known plaintext/ciphertext pairs, for some constant 0<λ<1, 2n/λn time, and 2λn 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(2n(ln2n/n2), improving the previous asymptotic complexity of O(2n/n), using a variant of the 3-SUM algorithm of Baran, Demaine, and Patrascu