Bounds in Shallows and in Miseries

Proving bounds on the expected differential probability (EDP) of a characteristic over all keys has been a popular technique of arguing security for both block ciphers and hash functions. In fact, to a large extent, it was the clear formulation and elegant deployment of this very principle that helped Rijndael win the AES competition. Moreover, most SHA-3 finalists have come with explicit upper bounds on the EDP of a characteristic as a major part of their design rationale. However, despite the pervasiveness of this design approach, there is no understanding of what such bounds actually mean for the security of a primitive once a key is fixed — an essential security question in practice. In this paper, we aim to bridge this fundamental gap. Our main result is a quantitative connection between a bound on the EDP of differential characteristics and the highest number of input pairs that actually satisfy a characteristic for a fixed key. This is particularly important for the design of permutation-based hash functions such as sponge functions, where the EDP value itself is not informative for the absence of rekeying. We apply our theoretical result to revisit the security arguments of some prominent recent block ciphers and hash functions. For most of those, we have good news: a characteristic is followed by a small number of pairs only. For Keccak, though, currently much more rounds would be needed for our technique to guarantee any reasonable maximum number of pairs. Thus, our work — for the first time — sheds light on the fixed-key differential behaviour of block ciphers in general and substitution-permutation networks in particular which has been a long-standing fundamental problem in symmetric-key cryptography.