论文标题
广义三角动力系统：用于在有限场上构建加密排列的代数系统
Generalized Triangular Dynamical System: An Algebraic System for Constructing Cryptographic Permutations over Finite Fields
论文作者
论文摘要
近年来，已经出现了对多方计算至关重要的$ \ mathbb {f} _p $上的一类新类的对称键原语。为了提高此类原语的效率，提出了许多新的块密码和哈希函数，超过$ \ mathbb {f} _p $。这些新的原语还表明，遵循经典替代 - 珀斯特网络（SPN）和Feistel网络的替代设计策略，可导致超过$ \ mathbb {f} _p $的更有效的密码和哈希功能设计，专门针对大型奇数primes $ p $。 鉴于这些努力，在这项工作中，我们构建了一个\ emph {代数框架}，该框架可以系统地探索在$ \ mathbb {f} _p $上构建对称键（迭代）排列的可行有效的设计策略。我们首先将有限领域的迭代多项式动力系统确定为几乎所有块密码设计策略的中心构建块。我们提出了一个广义的三角多项式动力学系统（GTD），并基于GTD，我们提供了$ \ Mathbb {f} _p^n $的迭代（键）置换的通用定义。 我们基于GTD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，根据我们的通用定义，也可以实例化遵循这些设计策略的块密码。此外，我们发现，最近提出的\ texttt {griffin}设计既不遵循feistel and spn设计，也无法使用基于通用GTD的定义来描述。我们还表明，可以从基于GTD的定义实例化新的广义Lai-Massey结构。 我们进一步提供了GTD的通用分析，包括对差异均匀性和相关性的上限。
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.
