Non-interactive zero-knowledge ( $$\mathsf {NIZK}$$ NIZK ) is a fundamental primitive that is widely used in the construction of cryptographic schemes and protocols. Our main result is a reduction from constructing $$\mathsf {NIZK}$$ NIZK proof systems for all of $$\mathbf {NP}$$ NP based on $$\mathsf {LWE}$$ LWE , to constructing a $$\mathsf {NIZK}$$ NIZK proof system for a particular computational problem on lattices, namely a decisional variant of the bounded distance decoding ( $$\mathsf {BDD}$$ BDD ) problem. That is, we show that assuming $$\mathsf {LWE}$$ LWE , every language $$L \in \mathbf {NP}$$ L ∈ NP has a $$\mathsf {NIZK}$$ NIZK proof system if (and only if) the decisional $$\mathsf {BDD}$$ BDD problem has a $$\mathsf {NIZK}$$ NIZK proof system. This (almost) confirms a conjecture of Peikert and Vaikuntanathan (CRYPTO, 2008). To construct our $$\mathsf {NIZK}$$ NIZK proof system, we introduce a new notion that we call prover-assisted oblivious ciphertext sampling ( $$\mathsf {POCS}$$ POCS ), which we believe to be of independent interest. This notion extends the idea of oblivious ciphertext sampling , which allows one to sample ciphertexts without knowing the underlying plaintext. Specifically, we augment the oblivious ciphertext sampler with access to an (untrusted) prover to help it accomplish this task. We show that the existence of encryption schemes with a $$\mathsf {POCS}$$ POCS procedure, as well as some additional natural requirements, suffices for obtaining $$\mathsf {NIZK}$$ NIZK proofs for $$\mathbf {NP}$$ NP . We further show that such encryption schemes can be instantiated based on $$\mathsf {LWE}$$ LWE , assuming the existence of a $$\mathsf {NIZK}$$ NIZK proof system for the decisional $$\mathsf {BDD}$$ BDD problem.

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从 LWE 走向 NP 的非交互式零知识证明

非交互式零知识 ( $$\mathsf {NIZK}$$ NIZK ) 是一种基本原语，广泛用于构建密码方案和协议。我们的主要结果是从基于 $$\mathsf {LWE}$$ LWE 为所有 $$\mathbf {NP}$$ NP 构建 $$\mathsf {NIZK}$$ NIZK 证明系统减少到构建 $ $\mathsf {NIZK}$$ NIZK 证明系统用于格子上的特定计算问题，即有界距离解码（ $$\mathsf {BDD}$$ BDD ）问题的决策变体。也就是说，我们证明假设 $$\mathsf {LWE}$$ LWE ，每一种语言 $$L \in \mathbf {NP}$$ L ∈ NP 都有一个 $$\mathsf {NIZK}$$ NIZK 证明系统，如果（并且仅当）决策性 $$\mathsf {BDD}$$ BDD 问题具有 $$\mathsf {NIZK}$$ NIZK 证明系统。这（几乎）证实了 Peikert 和 Vaikuntanathan 的猜想（CRYPTO，2008 年）。为了构建我们的 $$\mathsf {NIZK}$$ NIZK 证明系统，我们引入了一个新的概念，我们称之为证明者辅助的不经意密文采样（ $$\mathsf {POCS}$$ POCS ），我们认为它是独立的兴趣。这一概念扩展了不经意密文采样的思想，它允许在不知道底层明文的情况下对密文进行采样。具体来说，我们通过访问（不受信任的）证明者来增强不经意的密文采样器，以帮助它完成这项任务。我们证明了具有 $$\mathsf {POCS}$$ POCS 过程的加密方案的存在，以及一些额外的自然要求，足以获得 $$\mathsf {NIZK}$$ NIZK 证明 $$\mathbf { NP}$$ NP。我们进一步表明，这种加密方案可以基于 $$\mathsf {LWE}$$ LWE 进行实例化，