Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Real-Valued Somewhat-Pseudorandom Unitaries

Zvika Brakerski, Nir Magrafta

TL;DR

The paper presents a minimal, real-valued construction for pseudorandom unitaries that remain indistinguishable from Haar random unitaries when acting on polynomial-sized, orthogonal input sets. The key idea is to combine a random binary phase with a Hadamard transform to flatten input states, followed by a random permutation to induce pseudorandom unitary behavior on flat, orthogonal inputs, with a rigorous information-theoretic analysis bounding the deviation from Haar statistics. The results rely on quantum-secure cryptographic primitives (PRFs and PRPs) and yield both a PRU and PRSS guarantee under non-adaptive, orthogonal-input constraints. The work contributes a simple, robust pathway to real-valued pseudorandomness and opens questions about scalability and broader input classes beyond orthogonal states. The findings have potential implications for quantum cryptography and quantum information tasks where efficient, real-valued pseudorandom unitaries are desirable.

Abstract

We explore a very simple distribution of unitaries: random (binary) phase -- Hadamard -- random (binary) phase -- random computational-basis permutation. We show that this distribution is statistically indistinguishable from random Haar unitaries for any polynomial set of orthogonal input states (in any basis) with polynomial multiplicity. This shows that even though real-valued unitaries cannot be completely pseudorandom (Haug, Bharti, Koh, arXiv:2306.11677), we can still obtain some pseudorandom properties without giving up on the simplicity of a real-valued unitary. Our analysis shows that an even simpler construction: applying a random (binary) phase followed by a random computational-basis permutation, would suffice, assuming that the input is orthogonal and \emph{flat} (that is, has high min-entropy when measured in the computational basis). Using quantum-secure one-way functions (which imply quantum-secure pseudorandom functions and permutations), we obtain an efficient cryptographic instantiation of the above.

Real-Valued Somewhat-Pseudorandom Unitaries

TL;DR

The paper presents a minimal, real-valued construction for pseudorandom unitaries that remain indistinguishable from Haar random unitaries when acting on polynomial-sized, orthogonal input sets. The key idea is to combine a random binary phase with a Hadamard transform to flatten input states, followed by a random permutation to induce pseudorandom unitary behavior on flat, orthogonal inputs, with a rigorous information-theoretic analysis bounding the deviation from Haar statistics. The results rely on quantum-secure cryptographic primitives (PRFs and PRPs) and yield both a PRU and PRSS guarantee under non-adaptive, orthogonal-input constraints. The work contributes a simple, robust pathway to real-valued pseudorandomness and opens questions about scalability and broader input classes beyond orthogonal states. The findings have potential implications for quantum cryptography and quantum information tasks where efficient, real-valued pseudorandom unitaries are desirable.

Abstract

We explore a very simple distribution of unitaries: random (binary) phase -- Hadamard -- random (binary) phase -- random computational-basis permutation. We show that this distribution is statistically indistinguishable from random Haar unitaries for any polynomial set of orthogonal input states (in any basis) with polynomial multiplicity. This shows that even though real-valued unitaries cannot be completely pseudorandom (Haug, Bharti, Koh, arXiv:2306.11677), we can still obtain some pseudorandom properties without giving up on the simplicity of a real-valued unitary. Our analysis shows that an even simpler construction: applying a random (binary) phase followed by a random computational-basis permutation, would suffice, assuming that the input is orthogonal and \emph{flat} (that is, has high min-entropy when measured in the computational basis). Using quantum-secure one-way functions (which imply quantum-secure pseudorandom functions and permutations), we obtain an efficient cryptographic instantiation of the above.
Paper Structure (24 sections, 13 theorems, 63 equations)

This paper contains 24 sections, 13 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Technical Overview
  4. Flattening
  5. Cross-Term Removal and Symmetrization
  6. Bounding The Difference
  7. Other Previous Works
  8. Future Directions
  9. Paper Organization
  10. Preliminaries
  11. Notation
  12. Vectors, Functions, and Permutations.
  13. Concentration Bounds
  14. Pseudorandomness
  15. Almost Invariance Under Haar Unitaries
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $Z_x$, $x\in {\{0,1\}^{n}}$, be independent random variables with zero expectation such that $a_x \le Z_x \le b_x$ with probability $1$. Then for all $\epsilon>0$,

Theorems & Definitions (35)

  • Theorem 2.1: Hoeffding's Inequality
  • Definition 2.2: Quantum-Secure Pseudorandom Function
  • Definition 2.3: Quantum-Secure Pseudorandom Permutation
  • Definition 2.4
  • Claim 2.5
  • Lemma 2.6
  • Definition 3.1: Non-Adaptive Orthogonal-Inputs Pseudorandom Unitary
  • Theorem 3.2
  • Claim 3.3
  • proof
  • ...and 25 more