Paper 2023/1242

Cascading Four Round LRW1 is Beyond Birthday Bound Secure

Abstract

In CRYPTO'02, Liskov et al. have introduced a new symmetric key primitive called tweakable block cipher. They have proposed two constructions of designing a tweakable block cipher from block ciphers. The first proposed construction is called $\mathsf{LRW1}$ and the second proposed construction is called $\mathsf{LRW2}$. Although, $\mathsf{LRW2}$ has been extended in later works to provide beyond birthday bound security (e.g., cascaded $\mathsf{LRW2}$ in CRYPTO'12 by Landecker et al.), but extension of the $\mathsf{LRW1}$ has received no attention until the work of Bao et al. in EUROCRYPT'20, where the authors have shown that one round extension of $\mathsf{LRW1}$, i.e., masking the output of $\mathsf{LRW1}$ with the given tweak and then re-encrypting it with the same block cipher, gives security up to $2^{2n/3}$ queries. Recently, Khairallah has shown a birthday bound distinguishing attack on the construction and hence invalidated the security claim of Bao et al. This has led to the open research question, that {\em how many round are required for cascading $\mathsf{LRW1}$ to achieve beyond birthday bound security ?} In this paper, we have shown that cascading $\mathsf{LRW1}$ up to four rounds is sufficient for ensuring beyond the birthday bound security. In particular, we have shown that $\mathsf{CLRW1}^4$ provides security up to $2^{3n/4}$ queries. Security analysis of our construction is based on the recent development of the mirror theory technique for tweakable random permutations under the framework of the Expectation Method.

Note: The previous version of this work provides $2n/3$-bit security of $\mathsf{CLRW1}^4$. In this version, we have improved the security up to $3n/4$-bit.