从双线性对到多线性映射

From Bilinear Pairings to Multilinear Maps

自从2000年Sakai等人利用椭圆曲线上的双线性对提出了基于身份的密钥协商方案,特别是2001年Boneh和Franklin利用双线性对实现了基于身份的加密,基于双线性对的密码体制的研究曾一度成了密码研究领域特别是公钥密码研究中的一个热点.这一研究领域所取得的研究成果在密码学研究领域创造了一个不小的奇迹.在这篇文章中,我们首先介绍什么是双线性对,然后介绍双线性对在密码中的应用,从三方一轮密钥协商到IBE,到基于属性的加密(ABE),断言(或谓词)加密(PE),函数(或功能)加密(FE),可搜索的加密等,从短签名到各种各样的签名等.我们介绍双线性对密码系统的实现现状和安全现状.双线性对可以推广到多线性映射.多线性映射可以实现双线性对所实现的所有体制,更强大的是它可以实现电路,从而可以构造任意布尔电路的断言加密和设计任意多项式电路的不可区分的混淆(iO)等.由于多线性映射的强大功能,使得基于多线性映射的密码体制的研究成为了当前的研究热点.文章第二部分介绍多线性映射定义和构造思想,以及在密码中的应用.最后我们给出一些公开问题和一些讨论.

Since 2000 Sakai et al. proposed the identity based key agreement scheme using bilinear pairings on elliptic curves, especially, when Boneh and Franklin implemented the identity based encryption using bilinear pairing in 2001, the research of pairing based cryptography has become a hot topic in the field of public key cryptography. The pairing based research has achieved a great amount of results. In this paper, we firstly introduce the concept of bilinear pairing, then introduce pairing based cryptosystems, from the three party one round key agreement to identity based encryption (IBE), attribute based encryption (ABE), predicate encryption (PE), function encryption (FE) and searchable encryption, from short signature to a variety of signatures. We also introduce the current status of implementation and security of pairing. Bilinear pairing can be generalized to multilinear mapping. Multilinear maps can be used for all kinds of pairing based cryptography, a more powerful feature of multilinear mapping is that it can realize circuits. Multilinear maps can be used to design Attribute-based encryption and indistinguishability obfuscation for all circuits. Due to the strong functionality of multilinear mapping, the research of multilinear mapping based cryptosystems has become a hot research topic. In the second part of this paper, we introduce the definition, construction, and applications of multilinear mapping. Finally, we briefly discuss some open problems and interesting issues for further study in this area.