Paper 2025/953

Tight Multi-User Security of CCM and Enhancement by Tag-Based Key Derivation Applied to GCM and CCM

Abstract

$\textsf{GCM}$ and $\textsf{CCM}$ are block cipher (BC) based authenticated encryption modes. In multi-user (mu) security, a total number of BC invocations by all users $\sigma$ and the maximum number of BC invocations per user $\sigma_\mathsf{u}$ are crucial factors. For $\textsf{GCM}$, the tight mu-security bound has been identified as $\frac{\sigma_\mathsf{u} \sigma}{2^n} + \frac{u p + u^2}{2^k}$, where $k$ and $n$ are respectively the key and block sizes, $u$ is the number of users, $p$ is the number of offline queries.In contrast, the $\mathsf{CCM}$'s mu-security bound is still unclear. Two bounds of $\frac{u \sigma_\mathsf{u}^2}{2^n} + \frac{u p + u^2}{2^k}$ and $\frac{\sigma^2}{2^n} + \frac{u p + u \sigma}{2^k}$ have been derived by Luykx~et~al.~(Asiacrypt~2017) and Zhang~et~al.~(CCS~2024), respectively, but both are not tight and worse than the $\textsf{GCM}$'s bound. Moreover, methods to enhance mu security without disruptive changes in the scheme have been considered for $\textsf{GCM}$, namely nonce randomization ($\textsf{NR}$) to improve offline security and nonce-based key derivation ($\textsf{KD}$) to improve online security, but their applicability to $\textsf{CCM}$ has never been discussed. In this paper, we prove an improved mu-security bound of $\textsf{CCM}$, which is tight, and reaches the $\textsf{GCM}$'s bound. We then prove that $\textsf{NR}$ and $\textsf{KD}$ applied to $\textsf{CCM}$ result in the same bounds for the case to $\textsf{GCM}$. An important takeaway is that $\textsf{CCM}$ is now proved to be as secure as $\textsf{GCM}$. Moreover, we argue that $\textsf{NR}$ and $\textsf{KD}$ can be insufficient for some applications with massive data, and propose a new enhancement method called nonce-based and tag-based key derivation ($\textsf{NTKD}$) that is applied to $\textsf{GCM}$ and $\textsf{CCM}$. We prove that the resulting schemes meet such real-world needs.