广义三角动力系统：一种在有限域上构建密码置换的代数系统
Generalized Triangular Dynamical System: An Algebraic System for Constructing Cryptographic Permutations over Finite Fields
AI总结：
本文提出广义三角多项式动力系统(GTDS)代数框架，用于在有限域上系统地构建对称密钥迭代置换，能统一描述SPN、Feistel、Lai-Massey及Griffin等设计策略，并提供了差分均匀性和相关性的上限分析。
摘要：
近年来，一类在 $\mathbb{F}_p$ 上的对称密钥原语已经出现，这对于基于多方计算和零知识证明的协议至关重要。为了提高此类原语的效率，人们提出了许多在 $\mathbb{F}_p$ 上的新分组密码和哈希函数。这些新原语还表明，对于大的奇素数 $p$，遵循替代经典 Substitution-Permutation Network (SPN) 和 Feistel Networks 的替代设计策略，可以在 $\mathbb{F}_p$ 上实现更高效的密码和哈希函数设计。鉴于这些努力，在本工作中，我们构建了一个代数框架，允许系统地探索可行的、高效的设计策略，以在 $\mathbb{F}_p$ 上构建对称密钥（迭代）置换。我们首先将有限域上的迭代多项式动力系统确定为几乎所有分组密码设计策略的核心构建块。我们提出了广义三角多项式动力系统 (GTDS)，并基于 GTDS 提供了在 $\mathbb{F}_p^n$ 上的迭代（带密钥）置换的通用定义。我们基于 GTDS 的通用定义能够描述三种最著名的设计策略，即 SPNs、Feistel networks 和 Lai--Massey。因此，遵循这些设计策略构建的分组密码也可以从我们的通用定义中实例化。此外，我们发现最近提出的 Griffin 设计，既不遵循 Feistel 也不遵循 SPN 设计，可以使用基于 GTDS 的通用定义来描述。我们还展示了新的广义 Lai--Massey 构造可以从基于 GTDS 的定义中实例化。我们进一步提供了 GTDS 的通用分析，包括差分均匀性和相关性的上限。
英文摘要：
In recent years a new class of symmetric-key primitives over $\mathbb{F}_p$ that are essential to Multi-Party Computation and Zero-Knowledge Proofs based protocols have emerged. Towards improving the efficiency of such primitives, a number of new block ciphers and hash functions over $\mathbb{F}_p$ were proposed. These new primitives also showed that following alternative design strategies to the classical Substitution-Permutation Network (SPN) and Feistel Networks leads to more efficient cipher and hash function designs over $\mathbb{F}_p$ specifically for large odd primes $p$. In view of these efforts, in this work we build an \emph{algebraic framework} that allows the systematic exploration of viable and efficient design strategies for constructing symmetric-key (iterative) permutations over $\mathbb{F}_p$. We first identify iterative polynomial dynamical systems over finite fields as the central building block of almost all block cipher design strategies. We propose a generalized triangular polynomial dynamical system (GTDS), and based on the GTDS we provide a generic definition of an iterative (keyed) permutation over $\mathbb{F}_p^n$. Our GTDS-based generic definition is able to describe the three most well-known design strategies, namely SPNs, Feistel networks and Lai--Massey. Consequently, the block ciphers that are constructed following these design strategies can also be instantiated from our generic definition. Moreover, we find that the recently proposed \texttt{Griffin} design, which neither follows the Feistel nor the SPN design, can be described using the generic GTDS-based definition. We also show that a new generalized Lai--Massey construction can be instantiated from the GTDS-based definition. We further provide generic analysis of the GTDS including an upper bound on the differential uniformity and the correlation.
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