We introduce a new notion called ${\cal Q}$-secure pseudorandom isometries (PRI). A pseudorandom isometry is an efficient quantum circuit that maps an $n$-qubit state to an $(n+m)$-qubit state in an isometric manner. In terms of security, we require that the output of a $q$-fold PRI on $\rho$, for $ \rho \in {\cal Q}$, for any polynomial $q$, should be computationally indistinguishable from the output of a $q$-fold Haar isometry on $\rho$. \par By fine-tuning ${\cal Q}$, we recover many existing notions of pseudorandomness. We present a construction of PRIs and assuming post-quantum one-way functions, we prove the security of ${\cal Q}$-secure pseudorandom isometries (PRI) for different interesting settings of ${\cal Q}$. \par We also demonstrate many cryptographic applications of PRIs, including, length extension theorems for quantum pseudorandomness notions, message authentication schemes for quantum states, multi-copy secure public and private encryption schemes, and succinct quantum commitments. }

翻译：我们引入了一个新概念，称为 ${\cal Q}$-安全的伪随机等距（PRI）。伪随机等距是一种高效的量子电路，能够将一个 $n$ 量子比特状态以等距方式映射为一个 $(n+m)$ 量子比特状态。在安全性方面，我们要求对于任意多项式 $q$，在 $\rho \in {\cal Q}$ 上应用 $q$ 次 PRI 的输出，应与在 $\rho$ 上应用 $q$ 次 Haar 等距的输出在计算上不可区分。通过精细调整 ${\cal Q}$，我们恢复了多种现有的伪随机性概念。我们给出了 PRI 的一种构造，并假设后量子单向函数存在，证明了对于 ${\cal Q}$ 的不同有趣设定，${\cal Q}$-安全的伪随机等距（PRI）的安全性。我们还展示了 PRI 的多种密码学应用，包括量子伪随机性概念的延拓定理、量子态的消息认证方案、多副本安全的公钥和私钥加密方案，以及简洁的量子承诺。