标题： 二阶具有线性位移的单向多线性函数 显示英文标题

标题： One-way multilinear functions of the second order with linear shifts

摘要： 我们引入并分析了一类在域K上的有限维向量空间上的二元运算，这些运算由带有线性移位的二阶多重线性表达式定义。 这些运算生成的多项式的次数随着每次迭代应用线性增加，而不同单项式的数量则呈组合增长。 我们证明了，尽管这些运算通常不具备结合性和交换性，但在对单个向量进行迭代时，它们表现出幂结合性和内部交换性。 这确保了a^n的幂运算定义明确且无歧义。 关键的是，a^n没有闭合表达式，这暗示了一种单向性：从a和n计算a^n是高效的，而从a^n恢复n（离散迭代问题）似乎在计算上是困难的。 我们提出了一种基于此原理的类似Diffie-Hellman的密钥交换协议，并引入代数Diffie-Hellman问题（ADHP）作为安全性的基础假设。 除了代数基础外，我们还通过实验研究了这些运算在有限域上的轨道结构，观察到在参数集上频繁出现长循环和高度规律的行为。 受这些动态的启发，我们进一步提出了一种基于多元素乘法模式的伪随机数生成（PRNG）策略。 这种方法在实验中实现了接近最大循环长度和出色的统计均匀性，突显了这些运算在密码学和组合应用中的潜力。 显示英文摘要

摘要： We introduce and analyze a novel class of binary operations on finite-dimensional vector spaces over a field K, defined by second-order multilinear expressions with linear shifts. These operations generate polynomials whose degree increases linearly with each iterated application, while the number of distinct monomials grows combinatorially. We demonstrate that, despite being non-associative and non-commutative in general, these operations exhibit power associativity and internal commutativity when iterated on a single vector. This ensures that exponentiation a^n is well-defined and unambiguous. Crucially, the absence of a closed-form expression for a^n suggests a one-way property: computing a^n from a and n is efficient, while recovering n from a^n (the Discrete Iteration Problem) appears computationally hard. We propose a Diffie-Hellman-like key exchange protocol based on this principle, introducing the Algebraic Diffie-Hellman Problem (ADHP) as an underlying assumption of security. In addition to the algebraic foundations, we empirically investigate the orbit structure of these operations over finite fields, observing frequent emergence of long cycles and highly regular behavior across parameter sets. Motivated by these dynamics, we further propose a pseudorandom number generation (PRNG) strategy based on multi-element multiplication patterns. This approach empirically achieves near-maximal cycle lengths and excellent statistical uniformity, highlighting the potential of these operations for cryptographic and combinatorial applications.