On the Resilience of Even-Mansour to Invariant Permutations

Symmetric cryptographic primitives are often exposed to invariances: deterministic relations between plaintexts and ciphertexts that propagate through the primitive. Recent invariant subspace attacks have shown that these can be a serious issue. One way to mitigate invariant subspace attacks is at the primitive level, namely by proper use of round constants (Beierle et al., CRYPTO 2017). In this work, we investigate how to thwart invariance exploitation at the mode level, namely by assuring that a mode never evaluates its underlying primitive under any invariance. We first formalize the use of invariant cryptographic permutations from a security perspective, and analyze the Even-Mansour block cipher construction. We further demonstrate how the model composes, and apply it to the keyed sponge construction. The security analyses exactly pinpoint how the presence of linear invariances affects the bounds compared with analyses in the random permutation model. As such, they give an exact indication how invariances can be exploited. From a practical side, we apply the derived security bounds to the case where the Even-Mansour construction is instantiated with the 512-bit ChaCha permutation, and derive a distinguishing attack against Even-Mansour-ChaCha in $$2^{128}$$ queries, faster than the birthday bound. Comparable results are derived for instantiation using the 200-bit Keccak permutation without round constants (attack in $$2^{50}$$ queries), the 1024-bit CubeHash permutation (attack in $$2^{256}$$ queries), and the 384-bit Gimli permutation without round constants (attack in $$2^{96}$$ queries). The attacks do not invalidate the security of the permutations themselves, but rather they demonstrate the tightness of our bounds and confirm that care should be taken when employing a cryptographic primitive that has nontrivial linear invariances.