The power of a single Haar random state: constructing and separating quantum pseudorandomness

TL;DR

This work investigates the cryptographic implications of access to a single Haar random quantum state by introducing the common Haar random state (CHRS) model. It proves that single-copy pseudorandom states (1PRS) exist in CHRS and can yield statistically hiding and binding quantum commitments, while standard multi-copy pseudorandom states (PRS) do not necessarily follow from the same assumptions. The authors establish a black-box separation between 1PRS and PRS using both isometry and unitary oracles, and develop a lifting framework that transfers state-based oracle separations to unitary-oracle settings via weak simulations and a quantum OR lemma. A key technical driver is a quantum one-time-pad strategy applied to a portion of the Haar state, along with a stretching/amplification argument to achieve full-state pseudorandomness from partial scrambling. Collectively, these results introduce a new framework for black-box separations among quantum pseudorandom primitives, highlight intrinsic differences between single-copy and multi-copy notions, and connect them to quantum one-wayness concepts and commitments in Microcrypt.

Abstract

In this work, we focus on the following question: what are the cryptographic implications of having access to an oracle that provides a single Haar random quantum state? We find that the study of such a model sheds light on several aspects of the notion of quantum pseudorandomness. Pseudorandom states (PRS) are a family of states for which it is hard to distinguish between polynomially many copies of either a state sampled uniformly from the family or a Haar random state. A weaker notion, called single-copy pseudorandom states (1PRS), satisfies this property with respect to a single copy. We obtain the following results: 1. First, we show, perhaps surprisingly, that 1PRS (as well as bit-commitments) exist relative to an oracle that provides a single Haar random state. 2. Second, we build on this result to show the existence of an isometry oracle relative to which 1PRS exist, but PRS do not. Taken together, our contributions yield one of the first black-box separations between central notions of quantum pseudorandomness, and introduce a new framework to study black-box separations between various inherently quantum primitives.

Figures (7)

• Figure 1: Diagram of the known relations and applications in Microcrypt, as of September 2024. Regular arrows indicate implications, and dotted arrows indicate black-box separations. Nodes that share a color are equivalent. An interactive version of this diagram is available at https://sattath.github.io/microcrypt-zoo/, with additional features, such as "mouseover a node" reveals additional details, and "mouseover an edge" shows a clickable source for that relation. The website is updated periodically, therefore, the online version may differ from the one above as new results are published.
• Figure 2: A construction that satisfies the statistical pseudorandomness property of a ${\sf{1PRS}}$ in the CHRS model, but not the length-stretching requirement.
• Figure 3: Construction of a ${\sf{1PRS}}$ in the CHRS model
• Figure 4: Algorithm 1
• Figure 5: Algorithm 2

Theorems & Definitions (70)

• Theorem 1.1: Informal
• Theorem 1.2: Informal
• Theorem 1.3: Informal
• Lemma 2.1: Harrow harrow2023approximate, informal
• Theorem 2.2: Informal
• Corollary 2.3: Informal
• Remark 2.4
• Remark 2.5
• Theorem 2.6
• Corollary 2.7