International Association
for Cryptologic Research

Substitution-Permutation Networks (SPNs) refer to a family of constructions which build a wn-bit block cipher from n-bit public permutations (often called S-boxes), which alternate keyless and “local” substitution steps utilizing such S-boxes, with keyed and “global” permutation steps which are non-cryptographic. Many widely deployed block ciphers are constructed based on the SPNs, but there are essentially no provable-security results about SPNs.In this work, we initiate a comprehensive study of the provable security of SPNs as (possibly tweakable) wn-bit block ciphers, when the underlying n-bit permutation is modeled as a public random permutation. When the permutation step is linear (which is the case for most existing designs), we show that 3 SPN rounds are necessary and sufficient for security. On the other hand, even 1-round SPNs can be secure when non-linearity is allowed. Moreover, 2-round non-linear SPNs can achieve “beyond-birthday” (up to $$2^{2n/3}$$ 22n/3 adversarial queries) security, and, as the number of non-linear rounds increases, our bounds are meaningful for the number of queries approaching $$2^n$$ 2n. Finally, our non-linear SPNs can be made tweakable by incorporating the tweak into the permutation layer, and provide good multi-user security.As an application, our construction can turn two public n-bit permutations (or fixed-key block ciphers) into a tweakable block cipher working on wn-bit inputs, 6n-bit key and an n-bit tweak (for any $$w\ge 2$$ w≥2); the tweakable block cipher provides security up to $$2^{2n/3}$$ 22n/3 adversarial queries in the random permutation model, while only requiring w calls to each permutation, and 3w field multiplications for each wn-bit input.

We initiate the study of symmetric encryption in a regime where the memory of the adversary is bounded. For a block cipher with n-bit blocks, we present modes of operation for encryption and authentication that guarantee security beyond$$2^n$$ encrypted/authenticated messages, as long as (1) the adversary’s memory is restricted to be less than $$2^n$$ bits, and (2) the key of the block cipher is long enough to mitigate memory-less key-search attacks. This is the first proposal of a setting which allows to bypass the $$2^n$$ barrier under a reasonable assumption on the adversarial resources.Motivated by the above, we also discuss the problem of stretching the key of a block cipher in the setting where the memory of the adversary is bounded. We show a tight equivalence between the security of double encryption in the ideal-cipher model and the hardness of a special case of the element distinctness problem, which we call the list-disjointness problem. Our result in particular implies a conditional lower bound on time-memory trade-offs to break PRP security of double encryption, assuming optimality of the worst-case complexity of existing algorithms for list disjointness.