A Quantum Approach For Reducing Communications in Classical Secure Computations with Long Outputs

arXiv:2310.05213 · quant-ph, cs.CR · Submitted 2023-10-08 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "A Quantum Approach For Reducing Communications in Classical Secure Computations with Long Outputs".

Mira: Here's the gist of the paper. Detailed Research Summary of Paper A: Quantum Approach For Reducing Communications in Classical Secure Computations with Long Outputs This research paper investigates the application of quantum cryptography as a novel approach to significantly reduce communication overhead in classical secure computations, particularly those involving functions with long outputs.

Kai: The core motivation is to overcome the undesirable communication complexity dependency on the output length (m) inherent in classical secure function sampling (SFS).

Mira: Core Problem and Classical Limitation: The study begins by addressing Secure Function Sampling (SFS), where Alice needs to sample an input from her domain and securely transmit only the resulting function output to Bob, ensuring Bob learns no more than the output even under malicious behavior.

Kai: While classical cryptography achieves this with a communication complexity of O(n + m), this dependency on m is deemed unacceptable.

Title and authors: Kai: So we’re looking at this paper today, "A Quantum Approach For Reducing Communications in Classical Secure Computations with Long Outputs." It addresses a big issue in classical cryptography where you need to sample something from a function and send the result back, and classically that communication cost goes up with the length of that output.

Mira: That dependence on m is exactly what we’ve been trying to get around, isn't it? Because if you have a really long output, say m is big, then classical methods like FHE or succinct arguments have a communication complexity of O(n + m), which just doesn't scale well when m gets huge.

Lev: And for real hardware, that dependency on m is the killer; you want something that scales with the input size, not the output size if possible.

Kai: Well, this paper proposes using quantum cryptography to break that dependency. The core idea is designing a protocol for secure function sampling with approximate security where the communication complexity is independent of both the function evaluation time and the output length m.

Mira: That sounds like a massive claim because it’s trying to bypass what they call a fundamental barrier in classical SFS, which they showed using techniques from prior work thirty <ref:2310.05213#pg1>.

Lev: If we take that O(n + m) complexity and try to reduce it to something just dependent on n, that would be huge for practical implementation, assuming the quantum advantage holds up.

Kai: They claim a quantum cryptographic protocol achieves secure function sampling with approximate security within O(n) communication, explicitly omitting the dependency on the security parameter and error tolerance in that specific result.

Mira: That's interesting because they frame it as solving the problem of Secure Function Sampling with Long Outputs by formally framing it as a Remote State Preparation with Verifiability problem.

Lev: So, if we translate that into what we actually build, they are using techniques like the Hadamard test to certify server behavior and succinct testing for two-party relations to move from a preRSPV setup to this final SFS protocol.

Kai: They also suggest that assuming the existence of collapsing hash functions, there exists a quantum cryptographic protocol, Protocol eight for SFS with epsilon-security and communication complexity poly(n, one/epsilon), according to Theorem one point two (Informal) <ref:2310.05213#pg2>.

Mira: And they extend this concept beyond simple function sampling to general two-party computations, constructing protocols for 2PC where the communication complexity only depends on the input length n and the targeted security parameter epsilon <ref:2310.05213#pg2>.

Lev: That’s a significant leap; if we can do that for general 2PC, it means we are moving toward secure computation where the overhead is dominated by what you actually put in, not how big the resulting data set gets <ref:2310.05213#pg2>.

Kai: They also present Protocol fourteen over a classical channel for the 2PC problem, assuming both NTCF and collapsing hash functions exist to achieve epsilon-security with communication complexity poly(n, one/epsilon) <ref:2310.05213#pg2>.

Title and authors: Mira: The paper also touches on several related protocols like SFVP and SFVP2, showing how these quantum techniques can be generalized from deterministic functions to randomized ones using things like Yao’s garbled circuits.

Lev: When you look at the actual construction details, they introduce purification systems to define projections based on states relative to a reference state r(out), and they define specific state forms like the Basis-Honest Form and Basis-Phase Correspondence Form.

Kai: Those state forms are what allow them to formalize the security proofs, showing how the protocol steps, like FirstTwoStep or HadamardTest, lead to the desired secure result.

Mira: But you have to keep in mind their assumption about collapsing hash functions and NTCF for those specific theorems concerning 2PC and classical channel communication <ref:2310.05213#pg2>.

Lev: And if we look at what this means for running it on real hardware, the complexity is poly(n, one/epsilon), which is much better than linear dependence on m, but still carries that one/epsilon factor from the security parameter <ref:2310.05213#pg1>.

Kai: So, to summarize this paper, "A Quantum Approach For Reducing Communications in Classical Secure Computations with Long Outputs," it provides a quantum cryptographic protocol for SFS with approximate security using only O(n) communication, bypassing the classical barrier of dependency on output length m.

Mira: It formalizes the problem by linking it to Remote State Preparation with Verifiability and then builds up to general two-party computations where complexity is bounded by n and epsilon.

Lev: The main caveat we need to keep in mind is that Theorem one point three and one point four rely on the assumption of the existence of collapsing hash functions, which isn't guaranteed in practice <ref:2310.05213#pg2>.

Kai: And they also flag that for those bounds to hold, we are dealing with a quantum cryptographic protocol, so the entire construction depends on certain quantum capabilities being available.

Mira: It’s a strong theoretical result because it shows what is possible when you bring in the quantum tools to tackle problems where classical methods just hit an insurmountable wall regarding output size scaling.

Lev: For us, Lev, if we were trying to run this kind of SFS protocol on superconducting qubits today, the overhead of those tests and verifications would likely land somewhere in that poly(n) communication complexity they mention.

Kai: Exactly. So we’re looking at a theoretical framework here showing the potential structure for reducing communication costs in these specific types of secure computations.

Mira: Before we move on, this paper opens up ideas for QMA arguments by compiling Mahadev's protocol using the SFS and RSPV frameworks, suggesting ways to build succinct arguments for those quantum complexity classes.

Lev: That’s a future direction, trying to see if this structure can be mapped onto fault-tolerant computation models.

Kai: So that’s what we have on "A Quantum Approach For Reducing Communications in Classical Secure Computations with Long Outputs." It shows how quantum cryptography fundamentally alters the communication complexity landscape for secure function sampling and two-party computations.

The paper's summary: Kai: So, to recap, this paper is about using quantum methods to cut down on how much data you have to send when you’re doing secure computations that produce a lot of output information.

Mira: Right. It tackles the problem of Secure Function Sampling with Long Outputs, which classically has this big communication cost that just depends on the size of the output, m.

Lev: And what they do is propose a quantum cryptographic protocol for this sampling where the communication complexity ends up depending only on something related to your input size n, not that long output m.

Kai: That’s the core claim. They’re aiming for a protocol with approximate security, and they show it can be done within O(n) communication.

Mira: It’s a big deal because they are directly challenging the classical impossibility results, specifically those that showed you couldn't achieve polynomial complexity in n for classical SFS.

Lev: I mean, if we take their result seriously, it means for two-party computations, we can get complexity bounded by n and your security parameter epsilon, which is much more manageable than depending on the output size.

Kai: They frame this whole thing by turning SFS into a Remote State Preparation with Verifiability problem, and then they use things like the Hadamard test to actually verify that the server is being honest in this quantum setup.

Mira: They also extend this concept to general two-party computations, creating protocols where the communication complexity only depends on n and epsilon, assuming we have these collapsing hash functions available.

Lev: That reliance on those collapsing hash functions is a major caveat; it means the protocol works under a specific, strong assumption about certain mathematical tools.

Kai: The paper also looks at generalizing this to Secure Function Value Preparation, or SFVP, using things like Yao’s garbled circuits to handle randomized functions.

Mira: And for those malicious client scenarios in two-party computations, they introduce protocols like SFVP2 that use statistically-binding commitments and postquantum zero-knowledge arguments of knowledge.

Lev: From a hardware standpoint, running these protocols means the communication overhead is tied to that polynomial dependency on n and the inverse of epsilon.

Kai: So what this means for us is they’ve essentially shown a way to structure quantum cryptography so that we can get away from linear dependence on output size in these specific tasks.

Mira: It suggests a new landscape for secure computation where the bottleneck shifts from the data being processed to the fundamental communication overhead dictated by input and security.

Lev: Before we look at those assumptions, though, let's see how they try to build this structure formally using those state forms they define on page thirty-seven.

The paper's improvements: Kai: So, we’ve seen how they established that quantum cryptography can reduce communication complexity for secure function sampling down to O(n). Now, let's look at what they suggest as improvements or extensions to this work.

Mira: They aren't just stopping at SFS with one output; they are pushing the framework into more complex scenarios, specifically general two-party computations.

Lev: That means if you have a whole secure computation involving multiple parties, this protocol should still scale well, depending only on the input size n and your security parameter epsilon.

Kai: They’re building toward protocols like Protocol fourteen which operate over a classical channel, provided you have those collapsing hash functions and NTCF available.

Mira: And they also discuss how to handle different types of functions, moving beyond just deterministic ones to randomized ones using tools like Yao’s garbled circuits for something called SFVP.

Lev: That’s where the real engineering challenge comes in; translating those high-level theoretical constructs into actual gate operations and noise-resilient quantum hardware.

Kai: They also touch on using these SFS and Remote State Preparation ideas to construct succinct arguments for QMA, which is a different area of quantum complexity.

Mira: That points toward building more compact ways to prove that certain quantum states exist without having to check every single component in the circuit directly.

Lev: If we think about fault-tolerant computation, this structure might give us a blueprint for how to manage communication overhead during those complex error-correction steps.

Kai: The paper also explores de-quantization when you have to run these protocols over classical channels, which is a practical thing you’d need to consider for real devices.

Mira: So the improvement here is moving from a specific sampling problem to a general framework for secure computation where complexity scales predictably with input size n and security epsilon.

Lev: It sets a benchmark for what we need in terms of communication efficiency when building more robust quantum protocols.

Conclusion: Tom: So we’re wrapping up this look at "A Quantum Approach For Reducing Communications in Classical Secure Computations with Long Outputs." It basically boils down to showing that quantum cryptography can handle secure function sampling with a communication complexity that doesn't grow with the output length m.

Kai: Exactly. We showed they can achieve this within O(n) communication, which is what we need for scalability if we're talking about real hardware constraints on superconducting qubits.

Mira: And this shifts the problem away from classical limits because it proves that there’s a quantum way to bypass those classical barriers regarding output size dependency.

Lev: For us, it means the overhead lands in poly(n), which is good, but we still have to worry about implementing all those specific quantum tests and verifications they mention.

Kai: They also showed how this framework extends to general two-party computations, meaning the communication complexity only depends on n and your security parameter epsilon.

Mira: That’s a big conceptual move because it moves us toward building more efficient secure systems that scale with input data rather than output size.

Lev: I just want to stress that those results rely on some strong assumptions about the existence of collapsing hash functions, so we have to be careful about what's actually achievable in the physical world.

Kai: That’s a fair point, Lev, and it highlights that this is still very much a theoretical construction until we see it realized with high fidelity.

Mira: So while the theory is strong, the practical implementation will depend heavily on how those state forms they define translate into actual measurable physical states in our systems.

Lev: Right. If we can map those abstract state forms onto something like syndrome-resolved decoding error budgets, that would be a real next step for error correction research.

Kai: I agree. It gives us a target structure to aim for when designing the next generation of quantum protocols for secure computation.

Mira: That’s what we’ll be looking at next, as we see how these ideas connect to building more succinct arguments for Quantum Merlin Arthur, which is where things get really interesting.

Jiayu Zhang

Zhongguancun Laboratory

quant-ph, cs.CR

Submitted: 2023-10-08

Updated: 2026-10-17

Comments: 71 pages; add author contribution statement

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 89/100

The gist: Detailed Research Summary of Paper A: Quantum Approach For Reducing Communications in Classical Secure Computations with Long Outputs This research paper investigates the application of quantum

Key concepts

Secure Function Sampling (SFS): This is the problem where Alice needs to securely send Bob a result from a function based on her private input. Classically, this requires communication proportional to the output length ($m$), which is too much for long outputs.
Quantum Advantage in SFS: The paper shows that quantum protocols can solve SFS with approximate security using only $O(n)$ communication, regardless of the output length $m$. This demonstrates a fundamental quantum advantage over classical methods that are limited by output size.
Two-Party Computation (2PC): This refers to general secure computations between two parties. The work extends the quantum techniques to 2PC, allowing protocols where communication depends only on the input size and security level, not the potentially large output.

Terminology

Communication Complexity — 3× in this paper · explained in 3 episodes

Summary

Detailed Research Summary of Paper A: Quantum Approach For Reducing Communications in Classical Secure Computations with Long Outputs

This research paper investigates the application of quantum cryptography as a novel approach to significantly reduce communication overhead in classical secure computations, particularly those involving functions with long outputs. The core motivation is to overcome the undesirable communication complexity dependency on the output length (m) inherent in classical secure function sampling (SFS).

Core Problem and Classical Limitation:

The study begins by addressing Secure Function Sampling (SFS), where Alice needs to sample an input from her domain and securely transmit only the resulting function output to Bob, ensuring Bob learns no more than the output even under malicious behavior. While classical cryptography achieves this with a communication complexity of O(n + m), this dependency on m is deemed unacceptable. The paper establishes a fundamental barrier for classical methods: it proves that for classical SFS with approximate security, there is no protocol achieving communication complexity poly(n). This impossibility result leverages techniques from prior work (reference [30]), specifically utilizing properties of pseudorandom generators (PRGs) and compression arguments.

Quantum Advantage in Secure Function Sampling (SFS):

The central contribution lies in designing a quantum cryptographic protocol for SFS that achieves approximate security with a communication complexity independent of both the function evaluation time and the output length. The paper claims a significant quantum advantage:

Key Quantum Result: A protocol is designed that achieves secure function sampling with approximate security within O(n) communication, explicitly omitting dependency on the security parameter and error tolerance.
Proof of Advantage: This protocol demonstrates a type of quantum advantage over classical methods, as the authors prove the classical impossibility using techniques from [30].
Formalization: The SFS problem is formally framed as a Remote State Preparation with Verifiability (RSPV) problem. The construction proceeds by first designing a forward-succinct preRSPV protocol and then amplifying it to an RSPV protocol (SFS). This amplification utilizes techniques such as the Hadamard test to certify server behavior and succinct testing for two-party relations.
Theoretical Bounds: Theorem 1.2 (Informal) suggests that assuming the existence of collapsing hash functions, a quantum cryptographic protocol (Protocol 8) exists for SFS with epsilon-security and communication complexity poly(n, 1/epsilon).

Generalization to Two-Party Computations (2PC):

Building upon the SFS foundation, the work extends these findings to general secure two-party computations. The paper constructs protocols for 2PC with approximate security where the communication complexity depends only on the input length (n) and the targeted security parameter (epsilon).

Protocol Construction: The construction involves using techniques such as succinct testing for two-party relations.
Theorem 1.3 (Informal): Assuming the existence of collapsing hash functions, there exists a quantum cryptographic protocol (Protocol 12) for the 2PC problem that is epsilon-secure and has communication complexity poly(n, 1/epsilon), encompassing both quantum and classical communications.
Theorem 1.4 (Informal): Assuming the existence of NTCF and collapsing hash functions, there exists a quantum cryptographic protocol over a classical channel (Protocol 14) for the 2PC problem that is epsilon-secure with communication complexity poly(n, 1/epsilon).

Broader Protocol Development and Related Concepts:

The paper details several related protocols and concepts to further explore the domain:

SFVP (Secure Function Value Preparation): The authors discuss generalizations, including moving from deterministic functions to randomized functions. They construct SFVP protocols using Yao’s garbled circuits.
SFVP2: A protocol achieving security against malicious clients in two-party computations is also presented using SFVP2.
Succinct Arguments for QMA: The paper touches upon approaches for constructing succinct arguments for Quantum Merlin-Arthur (QMA), specifically by compiling Mahadev's protocol using the SFS and RSPV frameworks.
Classical Channel De-quantization: The final section addresses the possibility of de-quantizing communication when operating over classical channels.

Underlying Formal Structures (from Reference B):

The detailed technical machinery underpinning these protocols involves sophisticated state definitions and testing procedures:

Purification Systems (Reference R): These systems are introduced to define projections (0 and 1) based on the state of registers (Q(sub), Q(outpad), Q(out)) relative to a reference state r(out). Equation (34) is invariant under the operation (0 + 1).
State Forms: The paper defines specific state forms:
Basis-Honest Form (37): A state psi is in this form if (0 + 1) psi = psi.
Basis-Phase Correspondence Form (38): A state psi is in this form if it is in the basis-honest form, and for each b in [0, 1], its b-branch, psi b, can be written as...
Protocol Components: Specific protocol steps are named (e.g., FirstTwoStep, SecondStep, ComputationalTest, HadamardTest).
Specific Protocols: Detailed descriptions exist for Protocol 9 (SFS for randomized functions), Protocol 10 (SFVP utilizing SFS and garbled circuits), and Protocol 12 (2PC in succinct communication).

**In essence, the paper presents a significant theoretical breakthrough by establishing a new class of quantum cryptographic protocols that fundamentally change the communication complexity landscape for secure function sampling and two-party computations, achieving bounds independent of output size.

Improvements for AI systems

Bold header: Quantum-Advantage Secure Function Sampling (SFS)

The improved AI system can perform secure function sampling with communication complexity independent of both function evaluation time and output length, achieving this via a quantum cryptographic protocol that achieves secure function sampling with approximate security, within O(n) communication.

Bold header: General Two-Party Computation (2PC) with Succinct Communication

The system can execute general secure two-party computations where the communication complexity only depends on the input length and the targeted security, achieving this via protocols whose complexity is only depending on the input length and the targeted security.

Bold header: Function Evaluation from Client-Chosen Input (SFVP)

The AI can securely transmit function values to a server based on a client-chosen private input by using Yao’s garbled circuits combined with SFS, allowing it to compute GarbleI(x, r) and send it to the server.

Bold header: Security Against Malicious Clients in 2PC

The system can be constructed to achieve security against malicious clients in SFVP2 by utilizing statistically-binding computationally-hiding string commitments, collapsing hash functions and postquantum succinct zero-knowledge arguments of knowledge (succinct-ZK-AoK).

Bold header: Classical Impossibility Proof for SFS

The system can formally prove that achieving secure function sampling with succinct communication is classically impossible by showing that "for any efficient quantum adversary Adv, there exist efficient quantum operations Sim such that for any polynomial-size inputs 1n, 1m, take ϵ < 1 − O(1) and f = PRG(1n, 1m), for any initial state ρ0 ∈ D(HS ⊗ HE): π Adv(f)(ρ0) ≈ ind Sim Simz on S,Q(out) (SFSIdeal(f)(Sim0 z on S and outputs b (ρ0)))" is impossible.

Abstract

How could quantum cryptography help us achieve what are not achievable in classical cryptography? In this work we study the classical cryptographic problem that two parties would like to perform secure computations with long outputs. As a basic primitive and example, we first consider the following problem which we call secure function sampling with long outputs: suppose f:0,1 n to 0,1 m is a public, efficient classical function, where m is big; Alice would like to sample x from its domain and sends f(x) to Bob; what Bob knows should be no more than f(x) even if it behaves maliciously. Classical cryptography, like FHE and succinct arguments [Gen09,Kil92,HW15], allows us to achieve this task within communication complexity O(n+m); could we achieve this task with communication complexity independent of m ? In this work, we first design a quantum cryptographic protocol that achieves secure function sampling with approximate security, within O(n) communication (omitting the dependency on the security parameter and error tolerance). We also prove the classical impossibility using techniques in [HW15], which means that our protocol indeed achieves a type of quantum advantage. Building on the secure function sampling protocol, we further construct protocols for general secure two-party computations [Yao86,GB01] with approximate security, with communication complexity only depending on the input length and the targeted security. In terms of the assumptions, we construct protocols for these problems assuming only the existence of collapsing hash functions [Unr16]; what's more, we also construct a classical-channel protocol for these problems additionally assuming the existence of noisy trapdoor claw-free functions [BCMVV,BKVV].