How to Watermark Cryptographic Functions by Bilinear Maps

Abstract

We introduce a notion of watermarking for cryptographic functions and propose a concrete scheme for watermarking cryptographic functions. Informally speaking, a digital watermarking scheme for cryptographic functions embeds information, called a mark, into functions such as one-way functions and decryption functions of public-key encryption. There are two basic requirements for watermarking schemes. A mark-embedded function must be functionally equivalent to the original function. It must be difficult for adversaries to remove the embedded mark without damaging the original functionality. In spite of its importance and usefulness, there have only been a few theoretical works on watermarking for functions (or programs). Furthermore, we do not have rigorous definitions of watermarking for cryptographic functions and concrete constructions. To solve the problem above, we introduce a notion of watermarking for cryptographic functions and define its security. Furthermore, we present a lossy trapdoor function (LTF) based on the decisional bilinear Diffie-Hellman problem problem and a watermarking scheme for the LTF. Our watermarking scheme is secure under the symmetric external Diffie-Hellman assumption in the standard model. We use techniques of dual system encryption and dual pairing vector spaces (DPVS) to construct our watermarking scheme. This is a new application of DPVS.