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

*Peter Gazi and Jooyoung Lee and Yannick Seurin and 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.

**Category / Keywords: **secret-key cryptography / block ciphers, key-length extension, provable security, ideal-cipher model

**Original Publication**** (with major differences): **IACR-FSE-2015

**Date: **received 27 Apr 2015, last revised 27 Apr 2015

**Contact author: **yannick seurin at m4x org

**Available format(s): **PDF | BibTeX Citation

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

**Version: **20150501:120607 (All versions of this report)

**Short URL: **ia.cr/2015/397

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