SaTC: CORE: Medium: Collaborative: An Algebraic Approach to Secure Multilinear Maps for Cryptography

该项目是数学家和计算机科学家之间的跨学科合作，致力于解决密码学中的一个核心挑战，即构建一组安全高效的代数多线性映射。多线性映射在密码学中有重要的应用，如多用户非交互式密钥交换、通用功能加密、全同态加密和不可区分混淆。该项目的成果有望开启密码系统的新时代，并为该领域开辟新的方向。该项目将通过参与深入的现代数学研究并将其应用于计算机科学来培养研究生和博士后。第一个候选的多线性映射在实践中是低效的，并且已经被证明对于某些期望的应用是不安全的。这个项目采用了与以前的项目非常不同的方法。起点是观察到算术几何中已经存在许多自然的多线性映射，这些映射自然地产生于算术变量和动机的上同调，以及k理论。它们提供了比有限域上的椭圆曲线更丰富的对象类别，其点群被广泛应用于加密密钥交换和公钥加密。挑战是找到这样的代数结构，它可以有效地计算多线性映射，并且与之相关的密码问题（例如，离散对数问题）预计是困难的。

The project is an interdisciplinary collaboration between mathematicians and computer scientists in an intensive focused research effort to solve a central challenge in cryptography, namely constructing a family of secure and efficient algebraic multilinear maps. Multilinear maps have remarkable applications in cryptography, such as multiuser non-interactive key-exchange, general functional encryption, fully-homomorphic encryption, and indistinguishability obfuscation. The results of the project are expected to enable a new age of cryptographic systems and open new directions in the field. The project will train graduate students and postdoctoral associates through involvement in deep modern mathematics research with applications in computer science.The first candidate multilinear maps are inefficient in practice, and have been shown to be insecure for some of the desired applications. This project takes a very different approach from earlier ones. The starting point is the observation that there already exist many natural multilinear maps in arithmetic geometry, arising naturally from the cohomology of arithmetic varieties and motives, and from K-theory. They give a richer class of objects than elliptic curves over finite fields, whose groups of points are widely used in practice for cryptographic key exchange and public-key encryption. The challenge is to find such algebraic structures for which the multilinear maps can be efficiently computed, and for which the associated cryptographic problems (e.g., discrete logarithm problems) are expected to be hard.