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

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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 \(2^{\emph{k}+\frac{\ell-1}{\ell+1}n}\) (resp. \(2^{\emph{k}+\frac{\ell-2}{\ell}n}\) ) 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 \(2^{\emph{k}+\frac{\ell-1}{\ell+1}n}\) (resp. \(2^{\emph{k}+\frac{\ell-2}{\ell}n}\) ) queries and also an improved bound \(2^{\emph{k}+\frac{\ell-1}{\ell}n}\) for the special case ℓ ∈ {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 \(2^{\emph{k}+\frac{\ell-1}{\ell}n}\) queries. Since XOR-cascades are sequential, this proves tightness of our above result for XOR-cascades of length ℓ ∈ {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.