Introduction of structure-preserving signatures

Since its invention in late 70’s, digital signatures have been playing central roles both in theory and practice. The most widely used and direct application is a public-key infrastructure that adds authenticity to communication over insecure network. Digital signatures are also used as building blocks in vast number of cryptographic schemes and protocols. Though the authentication is the essential role of digital signatures, those applications often care for privacy of the signed data or anonymity of the singer. Examples include anonymous e-voting, anonymous e-cash, credential systems, and so on. How these seemingly contradictory natures accommodate? This is where another powerful and important building block called zero-knowledge proofs kick in. It is in particular useful when it comes in with an noninteractive form to save communication complexity. Indeed, complex cryptographic systems are often built in modular fashion that combines several cryptographic schemes. The combination of digital signatures and non-interactive proof system is a standard approach to achieve privacy and authenticity at the same time. In theory, these building blocks had been constructed by early 90’s and they are versatile in exchange of poor efficiency. For practical purposes, invention of efficient non-interactive proof system over bilinear groups by Groth and Sahai in 2008 [12, 13] is a breakthrough and it is followed by practical structure-preserving signatures (SPS) and commitments by Abe, Fuschbauer, Groth, Haralambiev and Ohkubo, in 2010 [4]. A structurepreserving signature scheme is a digital signature scheme whose public-keys, messages, and signatures consist only of elements of source groups of bilinear groups, and verification only evaluates pairing product equations. It is called structure-preserving as the construction preserves the group structure among inputs and outputs. Since the success of their combination, other cryptographic objects such as encryption schemes have been pursued, and variety of applications are proposed, e.g., [10, 11, 14].