International Association
for Cryptologic Research

This paper discusses provable security of two types of cascade encryptions. The first construction $\\CE^l$, called $l$-cascade encryption, is obtained by sequentially composing $l$ blockcipher calls with independent keys. The security of $\\CE^l$ has been a longstanding open problem until Ga\\v{z}i and Maurer~\\cite{GM09} proved its security up to $2^{\\ka+\\min\\{\\frac{n}{2},\\ka\\}}$ query complexity for large cascading length, where $\\ka$ and $n$ denote the key size and the block size of the underlying blockcipher, respectively. We improve this limit by proving the security of $\\CE^l$ up to $2^{\\ka+\\min\\left\\{\\ka,n\\right\\}-\\frac{16}{l}\\left(\\frac{n}{2}+2\\right)}$ query complexity: this bound approaches $2^{\\ka+\\min\\left\\{\\ka,n\\right\\}}$ with increasing cascade length $l$.

The second construction $\\XCE^l$ is a natural cascade version of the DESX scheme with intermediate keys xored between blockcipher calls. This can also be viewed as an extension of double XOR-cascade proposed by Ga\\v{z}i and Tessaro~\\cite{GT12}. We prove that $\\XCE^l$ is secure up to $2^{\\ka+n-\\frac{8}{l}\\left(\\frac{n}{2}+2\\right)}$ query complexity. As cascade length $l$ increases, this bound approaches $2^{\\ka+n}$.

In the ideal cipher model, one can obtain all the evaluations of the underlying blockcipher by making $2^{\\ka+n}$ queries, so the $(\\ka+n)$-bit security becomes the maximum that key-length extension based on a single $\\ka$-bit key $n$-bit blockcipher is able to achieve. Cascade encryptions $\\CE^l$~(with $n\\leq\\ka$) and $\\XCE^l$ provide almost optimal security with large cascade length.