Pseudorandom Isometries

We introduce a new notion called 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 𝜌, for 𝜌 ∈ 𝒬, for any polynomial 𝑞, should be computationally indistinguishable from the output of a 𝑞-fold Haar isometry on 𝜌.

By fine-tuning 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 𝒬-secure pseudorandom isometries (PRI) for different interesting settings of 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.