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，旨在解决密码学中的一个核心挑战，即构建一个安全有效的代数多线性映射家族。 多线性映射在密码学中有着重要的应用，如多用户非交互密钥交换、一般函数加密、全同态加密、不可否认性混淆等。该项目的成果有望开启密码系统的新时代，并开辟该领域的新方向。 该项目将培养研究生和博士后助理通过参与深入的现代数学研究与计算机科学中的应用。第一个候选多线性映射在实践中是低效的，并已被证明是不安全的一些所需的应用。这个项目采取了与以前非常不同的方法。出发点是观察到算术几何中已经存在许多自然的多重线性映射，这些映射自然地产生于算术簇和动机的上同调以及K-理论。它们给出了比有限域上的椭圆曲线更丰富的一类对象，椭圆曲线的点群在实践中广泛用于密钥交换和公钥加密。挑战在于找到这样的代数结构，对于该代数结构，可以有效地计算多线性映射，并且对于该代数结构，相关联的密码学问题（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.