**Plain versus Randomized Cascading-Based Key-Length Extension for Block Ciphers**

*Peter Gaži*

**Abstract: **Cascading-based constructions represent the predominant approach
to the problem of key-length extension for block ciphers.
Besides the plain cascade, existing works also consider its
modification containing key-whitening steps between the
invocations of the block cipher, called randomized cascade or
XOR-cascade. We contribute to the understanding of the security
of these two designs by giving the following attacks and security
proofs, assuming an underlying ideal block cipher with key length
$k$ and block length $n$:

- For the plain cascade of odd (resp. even) length $l$ we present a generic attack requiring roughly $2^{k+\frac{l-1}{l+1}n}$ (resp. $2^{k+\frac{l-2}{l}n}$) queries, being a generalization of both the meet-in-the-middle attack on double encryption and the best known attack on triple cascade.

- For XOR-cascade of odd (resp. even) length $l$ we prove security up to $2^{k+\frac{l-1}{l+1}n}$ (resp. $2^{k+\frac{l-2}{l}n}$) queries and also an improved bound $2^{k+\frac{l-1}{l}n}$ for the special case $l\in\{3,4\}$ by relating the problem to the security of key-alternating ciphers in the random-permutation model.

- Finally, for a natural class of sequential constructions where block-cipher encryptions are interleaved with key-dependent permutations, we show a generic attack requiring roughly $2^{k+\frac{l-1}{l}n}$ queries. Since XOR-cascades are sequential, this proves tightness of our above result for XOR-cascades of length $l\in\{3,4\}$ as well as their optimal security within the class of sequential constructions.

These results suggest that XOR-cascades achieve a better security/efficiency trade-off than plain cascades and should be preferred.

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

**Publication Info: **A conference version of this paper appears at CRYPTO 2013.

**Date: **received 11 Jan 2013, last revised 21 Jun 2013

**Contact author: **peter gazi at inf ethz ch

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

**Version: **20130621:075615 (All versions of this report)

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