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

## Peter Gaži

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```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.