A software watermarking scheme allows one to embed a “mark” into a program without significantly altering the behavior of the program. Moreover, it should be difficult to remove the watermark without destroying the functionality of the program. Recently, Cohen et al. (STOC 2016) and Boneh et al. (PKC 2017) showed how to watermark cryptographic functions such as pseudorandom functions (PRFs) using indistinguishability obfuscation. Notably, in their constructions, the watermark remains intact even against arbitrary removal strategies. A natural question is whether we can build watermarking schemes from standard assumptions that achieve this strong mark-unremovability property. We give the first construction of a watermarkable family of PRFs that satisfies this strong mark-unremovability property from standard lattice assumptions (namely, the learning with errors (LWE) and the one-dimensional short integer solution (SIS) problems). As part of our construction, we introduce a new cryptographic primitive called a translucent PRF. We then give a concrete construction of a translucent PRF family from standard lattice assumptions, which in turn yields a watermarkable family of PRFs from the same assumptions.

一种软件加水印方案允许人们在程序中嵌入“标记”而不会显着改变程序的行为。此外，在不破坏程序功能的情况下删除水印应该很困难。最近，科恩等。（STOC 2016）和Boneh等人。（PKC 2017）展示了如何使用不可区分性混淆给加密函数加水印，例如伪随机函数（PRF）。值得注意的是，在其构造中，即使针对任意内容，水印仍保持完整清除策略。一个自然的问题是，我们是否可以根据实现这种强大的标记不可移除性的标准假设来构建水印方案。我们给出了一个可水印的PRF系列的第一个结构，该结构满足了标准晶格假设（即带错误学习（LWE）和一维短整数解决方案（SIS）问题）中这种强大的标记不可移除性。作为构建的一部分，我们引入了一种称为半透明PRF的新密码原语。然后，我们根据标准晶格假设给出了半透明PRF族的具体构造，而根据相同的假设，这反过来又产生了可水印的PRF族。