Core Concepts
Real-valued unitaries can exhibit pseudorandom properties with a simple construction.
Abstract
The content delves into the exploration of real-valued unitaries to achieve pseudorandom properties. It discusses the distribution of unitaries and their statistical indistinguishability from random Haar unitaries. The analysis shows that even simpler constructions can suffice under certain conditions, leading to an efficient cryptographic instantiation. Various works on pseudorandom quantum states and unitaries are referenced, highlighting the challenges and progress in achieving pseudorandomness with real-valued constructions. The technical overview covers flattening states, achieving randomness, and proving main theorems. Notable results include achieving flatness through binary phase application and computational basis permutation, as well as demonstrating closeness to almost invariant states under specific conditions.
1. Introduction
Pseudorandomness in cryptography is fundamental.
PRP construction requires fewer random bits than a random function.
Quantum objects' pseudorandomness is explored.
2. Preliminaries
Notation includes key concepts like concentration bounds.
Definitions of quantum-secure pseudorandom functions and permutations are provided.
Almost invariance under Haar unitaries is discussed.
3. Somewhat Pseudorandom Unitaries
Definition of non-adaptive orthogonal-inputs secure pseudorandom units.
Construction using quantum secure one-way functions.
4. Analysis
Achieving Flatness
Application of random binary phase for state flattening.
Getting from Flat States to Random-Looking Ones
Analysis of applying binary phase and permutation to orthogonal flat vectors.
Focusing on Unique States
Consideration of unique states for closeness to almost invariant states.
Using Orthogonality to Reach Closeness to an Almost Invariant State
Examination of operators summing over unique vs all permutations.