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 ${\cal Q}$. 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：我们引入了一种新概念，称为${\cal Q}$-安全伪随机等距映射（PRI）。伪随机等距映射是一种高效的量子电路，能够将$n$量子比特态以等距方式映射为$(n+m)$量子比特态。在安全性方面，我们要求对于任意多项式$q$，将$q$重PRI作用于$\rho$（其中$\rho \in {\cal Q}$）的输出，应与$q$重Haar等距映射作用于$\rho$的输出在计算上不可区分。\par 通过精细调整${\cal Q}$，我们恢复了多种现有的伪随机性概念。我们提出了PRI的一种构造，并假设量子抗单向函数存在，证明了在${\cal Q}$的不同有趣设置下${\cal Q}$-安全伪随机等距映射（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长度扩展定理、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，以及简洁量子承诺。