How to Watermark Cryptographic Functions

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Abstract

We propose a scheme for watermarking cryptographic functions. Informally speaking, a digital watermarking scheme for cryptographic functions embeds information, called a mark, into functions such as (trapdoor) one-way functions and decryption functions of public-key encryption. It is required that a mark-embedded function is functionally equivalent to the original function and it is difficult for adversaries to remove the embedded mark without damaging the function. In spite of its importance and usefulness, there have only been a few theoretical studies on watermarking for functions (or program), and we do not have rigorous and meaningful definitions of watermarking for cryptographic functions and concrete constructions.

To solve the above problem, we introduce a notion of watermarking for cryptographic functions and define its security. We present a lossy trapdoor function (LTF) based on the decisional linear (DLIN) problem and a watermarking scheme for the LTF. Our watermarking scheme is secure under the DLIN assumption in the standard model. We use the techniques of dual system encryption and dual pairing vector spaces (DPVS) to construct our watermarking scheme. This is a new application of DPVS.