We present a candidate obfuscator based on composite-order graded encoding schemes (GES), which are a generalization of multilinear maps. Our obfuscator operates on circuits directly without converting them into formulas or branching programs as was done in previous solutions. As a result, the time and size complexity of the obfuscated program, measured by the number of GES elements, is directly proportional to the circuit complexity of the program being obfuscated. This improves upon previous constructions whose complexity was related to the formula or branching program size. Known instantiations of Graded Encoding Schemes allow us to obfuscate circuit classes of polynomial degree, which include for example families of circuits of logarithmic depth. We prove that our obfuscator is secure against a class of generic algebraic attacks, formulated by a generic graded encoding model. We further consider a more robust model which provides more power to the adversary and extend our results to this setting as well. As a secondary contribution, we define a new simple notion of algebraic security (which was implicit in previous works) and show that it captures standard security relative to an ideal GES oracle.

我们提出了一种基于复合顺序分级编码方案（GES）的候选混淆器，该方案是多线性映射的一种概括。我们的混淆器可以直接在电路上运行，而无需像以前的解决方案那样将其转换为公式或分支程序。结果，混淆程序的时间和大小复杂度（由GES元素的数量来衡量）与被混淆程序的电路复杂度成正比。这改进了以前的构造，后者的复杂性与公式或分支程序的大小有关。分级编码方案的已知实例使我们能够模糊多项式的电路类别，例如包括对数深度的电路族。我们证明了混淆器对于一类通用代数攻击是安全的，由通用分级编码模型制定。我们进一步考虑一个更健壮的模型，该模型可以为对手提供更多功能，并将我们的结果也扩展到此设置。作为辅助贡献，我们定义了一个新的简单概念代数安全性（这在以前的工作中是隐含的），并表明相对于理想的GES oracle，它可以捕获标准安全性。