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 operations.We show that attacking this scheme with block size n is related to the 3-XOR problem with element size e. = 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~(e/3) queries, and the best known algorithms require around 2~(e/2)/e 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. Prom 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 plaintext/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 In~2 n~2), improving the previous asymptotic complexity of O(2~n), using a variant of the 3-SUM algorithm of Baran, Demaine, and Patrascu.
展开▼
