Paper 2015/397

Relaxing Full-Codebook Security: A Refined Analysis of Key-Length Extension Schemes

Peter Gazi, Jooyoung Lee, Yannick Seurin, John Steinberger, and Stefano Tessaro

Abstract

We revisit the security (as a pseudorandom permutation) of cascading-based constructions for block-cipher key-length extension. Previous works typically considered the extreme case where the adversary is given the entire codebook of the construction, the only complexity measure being the number $q_e$ of queries to the underlying ideal block cipher, representing adversary's secret-key-independent computation. Here, we initiate a systematic study of the more natural case of an adversary restricted to adaptively learning a number $q_c$ of plaintext/ciphertext pairs that is less than the entire codebook. For any such $q_c$, we aim to determine the highest number of block-cipher queries $q_e$ the adversary can issue without being able to successfully distinguish the construction (under a secret key) from a random permutation. More concretely, we show the following results for key-length extension schemes using a block cipher with $n$-bit blocks and $\kappa$-bit keys: - Plain cascades of length $\ell = 2r+1$ are secure whenever $q_c q_e^r \ll 2^{r(\kappa+n)}$, $q_c \ll 2^\ka$ and $q_e \ll 2^{2\ka}$. The bound for $r = 1$ also applies to two-key triple encryption (as used within Triple DES). - The $r$-round XOR-cascade is secure as long as $q_c q_e^r \ll 2^{r(\kappa+n)}$, matching an attack by Gazi (CRYPTO 2013). - We fully characterize the security of Gazi and Tessaro's two-call 2XOR construction (EUROCRYPT 2012) for all values of $q_c$, and note that the addition of a third whitening step strictly increases security for $2^{n/4} \le q_c \le 2^{3/4n}$. We also propose a variant of this construction without re-keying and achieving comparable security levels.

Note: An abridged version appears in the proceedings of FSE 2015. This is the full version.