Provable Security of Substitution-Permutation Networks

Abstract

Many modern block ciphers are constructed based on the paradigm of substitution-permutation networks (SPNs). But, somewhat surprisingly---especially in comparison with Feistel networks, which have been analyzed by dozens of papers going back to the seminal work of Luby and Rackoff---there are essentially no provable-security results about SPNs. In this work, we initiate a comprehensive study of the security of SPNs as strong pseudorandom permutations when the underlying $S$-box is modeled as a public random permutation. We show that 3~rounds of S-boxes are necessary and sufficient for secure linear SPNs, but that even 1-round SPNs can be secure when non-linearity is allowed. Additionally, our results imply security in settings where an SPN structure is used for domain extension of a block cipher, even when the attacker has direct access to the small-domain block cipher