Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Small Pseudo-Random Families of Matrices: Derandomizing Approximate Quantum Encryption

Andris Ambainis, Adam Smith

TL;DR

The paper addresses derandomizing approximate quantum encryption (private quantum channels) with short keys. It develops three explicit polynomial-time constructions that encrypt $n$ qubits with $n+o(n)$ bits of shared key, running in $O(n^2)$ time, and provides a Fourier-analytic framework that ties small-bias classical constructions to pseudo-random matrix families in a continuous space. By leveraging $\delta$-biased sets and their average-bias generalizations, the authors present a length-preserving scheme, a length-doubling scheme, and a hybrid scheme with improved tradeoffs, all achieving $\epsilon$-approximate security in the trace norm. This work advances practical, deterministic quantum encryption techniques and demonstrates a deep connection between classical derandomization and quantum information processing.

Abstract

A quantum encryption scheme (also called private quantum channel, or state randomization protocol) is a one-time pad for quantum messages. If two parties share a classical random string, one of them can transmit a quantum state to the other so that an eavesdropper gets little or no information about the state being transmitted. Perfect encryption schemes leak no information at all about the message. Approximate encryption schemes leak a non-zero (though small) amount of information but require a shorter shared random key. Approximate schemes with short keys have been shown to have a number of applications in quantum cryptography and information theory. This paper provides the first deterministic, polynomial-time constructions of quantum approximate encryption schemes with short keys. Previous constructions (quant-ph/0307104) are probabilistic--that is, they show that if the operators used for encryption are chosen at random, then with high probability the resulting protocol will be a secure encryption scheme. Moreover, the resulting protocol descriptions are exponentially long. Our protocols use keys of the same length as (or better length than) the probabilistic constructions; to encrypt $n$ qubits approximately, one needs $n+o(n)$ bits of shared key. An additional contribution of this paper is a connection between classical combinatorial derandomization and constructions of pseudo-random matrix families in a continuous space.

Small Pseudo-Random Families of Matrices: Derandomizing Approximate Quantum Encryption

TL;DR

The paper addresses derandomizing approximate quantum encryption (private quantum channels) with short keys. It develops three explicit polynomial-time constructions that encrypt qubits with bits of shared key, running in time, and provides a Fourier-analytic framework that ties small-bias classical constructions to pseudo-random matrix families in a continuous space. By leveraging -biased sets and their average-bias generalizations, the authors present a length-preserving scheme, a length-doubling scheme, and a hybrid scheme with improved tradeoffs, all achieving -approximate security in the trace norm. This work advances practical, deterministic quantum encryption techniques and demonstrates a deep connection between classical derandomization and quantum information processing.

Abstract

A quantum encryption scheme (also called private quantum channel, or state randomization protocol) is a one-time pad for quantum messages. If two parties share a classical random string, one of them can transmit a quantum state to the other so that an eavesdropper gets little or no information about the state being transmitted. Perfect encryption schemes leak no information at all about the message. Approximate encryption schemes leak a non-zero (though small) amount of information but require a shorter shared random key. Approximate schemes with short keys have been shown to have a number of applications in quantum cryptography and information theory. This paper provides the first deterministic, polynomial-time constructions of quantum approximate encryption schemes with short keys. Previous constructions (quant-ph/0307104) are probabilistic--that is, they show that if the operators used for encryption are chosen at random, then with high probability the resulting protocol will be a secure encryption scheme. Moreover, the resulting protocol descriptions are exponentially long. Our protocols use keys of the same length as (or better length than) the probabilistic constructions; to encrypt qubits approximately, one needs bits of shared key. An additional contribution of this paper is a connection between classical combinatorial derandomization and constructions of pseudo-random matrix families in a continuous space.

Paper Structure

This paper contains 14 sections, 1 theorem, 36 equations.

Table of Contents

  1. Introduction
  2. Definitions and Previous Work
  3. Previous Work
  4. Our Results
  5. Preliminaries
  6. Small-Bias Spaces
  7. Small-bias Set Families
  8. Entropy of Quantum States
  9. Pauli matrices
  10. Pauli matrices and Fourier Analysis
  11. State Randomization and Approximate Encryption
  12. Encrypting with a Small-Bias Space
  13. A Scheme with Shorter Key Length
  14. Hybrid Construction

Key Result

Lemma 3.1

If $\left\{ {A_i} \right\}_{i\in{\cal I}}$ is a family of subsets of $\{0,1\}^{2n}$ with average square bias $\delta^2$, then the operator is an approximate encryption scheme for $n$ qubits with leakage $\epsilon$ whenever $\delta \leq \epsilon 2^{-n/2}$.

Theorems & Definitions (9)

  • Definition 1
  • Remark 1
  • Definition 2
  • Lemma 3.1
  • proof
  • Claim 3.2
  • proof
  • Claim 3.3
  • proof