理想格、格罗布纳基和广义Hash函数
On Ideal Lattices, Gröbner Bases and Generalized Hash Functions
摘要：
本文利用格罗布纳基建立了理想格与整数上多元多项式环之间的联系。理想格是剩余类环$\mathbb{Z}[x]/\langle f \rangle$（这里$f$是一个monic多项式）中的理想，并且密码原语已经基于这些对象构建。由于理想格在单变量的情况下是循环格的推广，我们引入多元循环格的概念，并表明多元理想格确实是他们的推广。基于多元理想格，利用格罗布纳基技术证明了抗碰撞哈希函数的存在性.对于哈希函数的构造，我们定义了一个最坏情况问题，最短替换问题w.r.t. $\mathbb{Z}[x_1,\ldots, x_n]$中的理想值，并使用功能字段建立硬度结果。
英文摘要：
In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gröbner bases. Ideal lattices are ideals in the residue class ring, $\mathbb{Z}[x]/\langle f \rangle$ (here $f$ is a monic polynomial), and cryptographic primitives have been built based on these objects. As ideal lattices in the univariate case are generalizations of cyclic lattices, we introduce the notion of multivariate cyclic lattices and show that multivariate ideal lattices are indeed a generalization of them. Based on multivariate ideal lattices, we establish the existence of collision resistant hash functions using Gröbner basis techniques. For the construction of hash functions, we define a worst case problem, shortest substitution problem w.r.t. an ideal in $\mathbb{Z}[x_1,\ldots, x_n]$, and establish hardness results using functional fields.
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