Succinct Oblivious Tensor Evaluation and Applications: Adaptively-Secure Laconic Function Evaluation and Trapdoor Hashing for All Circuits

arXiv:2508.09673 · cs.CR · Submitted 2025-08-13 · Read on arXiv

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Introduction to the show: ident: Security Radio. Generated commentary on the latest security and cryptography papers.

Nadia: Today's paper: "Succinct Oblivious Tensor Evaluation and Applications".

Elias: This paper introduces Succinct Oblivious Tensor Evaluation (OTE), a novel cryptographic primitive that allows two parties to compute an additive secret sharing of a tensor product of two vectors,

Nadia: First, who's behind it and why it matters.

Title and authors: Nadia: Before we get into the mechanics, let's talk about who wrote this and what exactly "Succinct Oblivious Tensor Evaluation and Applications" means in plain language for our audience.

Elias: The paper is written by Damiano Abram, Giulio Malavolta, Lawrence Roy ldr709, and someone from IBM Research Z¨urich.

Priya: From a privacy perspective, I'm curious if this means we can handle massive datasets securely during the evaluation phase, which is where most leakage happens.

Nadia: That’s exactly right, Priya; the title suggests they found a way to evaluate these tensor products without having the communication or the required setup information balloon based on how big those input vectors actually are.

Elias: The core idea is that they managed to compute an additive secret sharing of a tensor product, while keeping both the message sizes and the CRS independent of the dimension of one vector.

Priya: So, if we think about deploying large AI models, this means the infrastructure needed to run them doesn't just get bigger when you increase the complexity of your data representation.

Nadia: Precisely; it’s about controlling the computational overhead so that scaling up the input size doesn't automatically lead to an unmanageable communication nightmare.

Elias: The authors show this is possible by constructing a half-succinct protocol where only one party's message size depends on the input dimension, and then they bootstrap that into a fully succinct system.

Priya: That bootstrapping process sounds complicated, but if it leads to smaller overall messages, that’s the practical payoff we need for real-world AI pipelines.

Nadia: It is; and when we look at the applications they list, it’s not just about tensor math; it's about unlocking tools like trapdoor hashing for all functions.

Elias: That trapdoor hashing capability for every function is a significant technical win because it means we can establish strong integrity checks on any computation, which is critical when dealing with model weights or training data.

Priya: I wonder if this means we can verify the entire AI process against a hidden key without needing to see all the intermediate results, which would be a huge step for auditing.

Nadia: That's exactly what they are showing; it's about building verifiable systems where privacy is built into the computation itself, not bolted on as an afterthought.

Elias: So, the whole point of this paper is to show that LWE hardness provides a path to these very succinct and highly functional cryptographic primitives.

Priya: It's exciting because it shows that complex privacy requirements aren't necessarily mutually exclusive with achieving high efficiency in computation.

Nadia: We’re going to see how these concepts translate into actual hardware and deployment scenarios in the next part of our discussion.

The paper's summary: Nadia: Now that we understand the setup, let's look at what the actual summary of "Succinct Oblivious Tensor Evaluation and Applications" tells us about the technical core of this work.

Elias: The summary focuses on the central contribution being a protocol for succinct NI-OTE with minimal communication complexity.

Priya: From my point of view, I want to know if they are promising a general solution for AI workloads, or if this is just tailored to one specific type of calculation.

Nadia: The summary indicates that the goal is to compute an additive secret share of a tensor product such that the size of both messages and the CRS is independent of the dimension of x.

Elias: That independence from dimension is what sets this work apart, and it’s achieved by constructing a half-succinct protocol where only one party's message size depends on x.

Priya: If the CRS size also doesn't scale with the input dimension, that means we can precompute or store these structures once and reuse them for many different AI computations without massive overhead.

Nadia: That’s a huge practical implication; it points toward reusable cryptographic infrastructure that isn't tied to a specific input size.

Elias: Furthermore, the summary mentions showing how this new technical tool enables a host of cryptographic primitives with security reducible to the Learning With Errors problem.

Priya: So, if it's LWE-based, we can trust that as long as LWE is hard, these resulting tools will be secure against known attacks.

Nadia: That’s the security assurance we need for real deployment; knowing the foundation is solid and based on a well-studied problem like LWE.

Elias: And they also mention that this primitive leads to a rate-one/two laconic oblivious transfer protocol which is described as best possible in its communication complexity.

Priya: A rate-one/two OT protocol sounds incredibly useful for federated learning because it suggests we can securely exchange batches of data points efficiently without excessive network traffic.

Nadia: That efficiency is what matters; when you combine this with the ability to evaluate complex functions, we’re talking about a lot of secure computation happening much faster than before.

Elias: It sets the stage for how these underlying tensor evaluations can be leveraged across different layers of complexity, which is what the full paper explores.

Priya: So, we're looking at a framework where efficiency and privacy are intertwined through these specific lattice structures.

Nadia: Exactly; it’s about finding a way to make the abstract concepts of secure computation practical for large-scale AI systems.

The paper's improvements: Nadia: Let's shift our focus now to the specific technical improvements suggested in this paper regarding the new lattice encodings and how they enhance these primitives.

Elias: The authors introduce new variants of homomorphic lattice encodings, specifically LEncA(x; s, r, e) which supports addition and multiplication when those encodings are encrypted with correlated secrets.

Priya: I'm interested in what this means for the actual data leakage; does having these new operations make it easier to evaluate more complex functions while maintaining strong privacy?

Nadia: It suggests that these encodings allow them to derive general routines to evaluate any T-bounded RMS program of depth d, which is vital for accurately modeling intricate AI behaviors.

Elias: That adaptability in supporting different types of programs means the underlying LWE assumption holds up across more varied computational structures, which strengthens the security reduction.

Priya: If they can handle deeper circuits while maintaining strong privacy guarantees, that’s huge for applications like deep neural networks where non-linear activation functions are key components.

Nadia: Exactly; this capability means we aren't limited to shallow computations anymore when trying to secure complex AI models.

Elias: The compression procedure they detail is another major improvement because it scales the encoding size logarithmically with its input, which makes these tools much more computationally feasible for actual use on hardware.

Priya: That logarithmic scaling really helps us understand the practical feasibility; it means that even with large inputs, the overhead for secure computation doesn't become impossible to manage.

Nadia: So, the improvements focus on making the theoretical capabilities translate into something that is both computationally efficient and practically applicable for complex AI workloads.

Elias: The authors flag one limitation in their own work; they show what this protocol *can* do, but they are pointing out that the specific security guarantees rely heavily on the assumption of LWE hardness remaining unbroken.

Priya: That limitation is important because it tells us exactly where the security hinges, which helps us understand if there are any known attacks against the lattice-based assumptions themselves.

Nadia: Right, and understanding those dependencies is crucial for anyone trying to implement this in a production environment so we don't over-rely on an assumption that might eventually be challenged.

Conclusion: Nadia: So, we're looking at this paper today, which is "Succinct Oblivious Tensor Evaluation and Applications: Adaptively-Secure Laconic Function Evaluation and Trapdoor Hashing for All Circuits," and we need to unpack what that title actually means for the listeners.

Elias: Essentially, it tells us they are tackling tensor evaluation in a way that keeps the message sizes small regardless of the dimension, which is achieved through adaptively secure laconic function evaluation and trapdoor hashing for all circuits.

Priya: That sounds like they’re promising high-level efficiency in a way that directly relates to data handling, and I can see how that connects to the privacy concerns we often have with large datasets.

Nadia: Exactly, Priya, because when you combine efficient computation with strong cryptographic assumptions like LWE, you start building tools for handling large amounts of information securely.

Elias: The core mechanism is that the OTE protocol handles the tensor product in a way that its size doesn't grow with one of the vector's dimensions.

Priya: That sounds like it’s solving a scaling problem, which is really significant because we can move toward more scalable privacy solutions.

Nadia: It means we could finally build systems that are efficient enough to handle the scale required for modern AI without sacrificing security.

Elias: The authors show this is possible by using a construction from standard learning with errors, or LWE as their foundation.

Priya: So, the security isn't some new mathematical miracle, it’s based on something we already understand well enough to trust for future security needs.

Nadia: That’s the key point; it gives us a concrete way to use LWE-based security in practical settings for things that matter.

Elias: And they show this LWE foundation is what allows them to derive several useful primitives, like adaptively secure laconic function evaluation and trapdoor hashing for all functions.

Priya: That depth-D capability is important because it means we can handle complex circuits when evaluating AI models securely, which is something I’ve been thinking about regarding privacy-preserving machine learning pipelines.

Nadia: Precisely, Priya; this work moves us closer to having robust distributed training environments where multiple entities can collaborate on a model without exposing their raw data or intermediate calculations.

Elias: To summarize, the paper is about using LWE to build tools that enable efficient computation and strong privacy guarantees for AI applications.

Priya: So, we’re looking at a framework where we can securely evaluate arbitrary functions while maintaining strong input and function privacy guarantees through these lattice-based primitives.

Nadia: That sounds like the foundation for real progress in deploying sophisticated AI models safely.

Elias: We'll see how this leads into the specifics of the actual protocol that makes this happen.

University of Edinburgh · Bocconi University · IBM Research Zurich

cs.CR

Submitted: 2025-08-13

Updated: 2026-09-30

Comments: 56 pages. This is the TheoretiCS journal version

Journal ref: TheoretiCS, Volume 5 (October 1, 2026) theoretics:16350

DOI: 10.46298/theoretics.26.14

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

Importance score: 90/100

The gist: This paper introduces Succinct Oblivious Tensor Evaluation (OTE), a novel cryptographic primitive that allows two parties to compute an additive secret sharing of a tensor product of two vectors,

Key concepts

Succinct Oblivious Tensor Evaluation (OTE): A novel cryptographic primitive that lets two parties compute an additive secret sharing of a tensor product of two vectors. Its key feature is that the size of both messages and the CRS remains independent of the dimension of one vector.
LWE Hardness: The security foundation for this work relies on the hardness assumption of Learning With Errors (LWE). This means that as long as LWE remains hard, the resulting cryptographic tools derived from it are considered secure against known attacks.
Adaptively Secure Laconic Function Evaluation: This capability allows the system to derive general routines to evaluate any T-bounded RMS program of depth d. This adaptability means the underlying LWE assumption holds across more varied computational structures, supporting complex AI models.
Trapdoor Hashing for All Circuits: The paper shows how the OTE primitive enables trapdoor hashing for every function. This provides strong integrity checks on any computation, which is critical for verifying model weights or training data without seeing intermediate results.

Terminology

Nadia — 23× in this paper · explained in 2 episodes
Elias — 20× in this paper · explained in 2 episodes
Learning With Errors — 11× in this paper
Succinct Oblivious Tensor Evaluation — 8× in this paper
Adaptively Secure Laconic Function Evaluation — 6× in this paper · explained in 1 episode
Core — 4× in this paper · explained in 2 episodes
European Research Council — 2× in this paper
Abstraction — 1× in this paper · explained in 1 episode
Activation Function — 1× in this paper · explained in 1 episode
Adaptability — 1× in this paper · explained in 1 episode
Amortization — 1× in this paper · explained in 1 episode
Computational overhead — 1× in this paper · explained in 2 episodes
Federated Learning — 1× in this paper · explained in 14 episodes

Summary

This paper introduces Succinct Oblivious Tensor Evaluation (OTE), a novel cryptographic primitive that allows two parties to compute an additive secret sharing of a tensor product of two vectors, while keeping both message sizes and the CRS independent of the dimension of one vector. This technical tool is significant because it enables a host of cryptographic primitives with security reducible to the Learning With Errors (LWE) problem, including adaptively secure laconic function evaluation, trapdoor hashing for all functions, and a rate-1/2 laconic oblivious transfer protocol that is best possible.

Core Primitive: Succinct NI-OTE

The central contribution is the construction of a succinct Non-Interactive Oblivious Tensor Evaluation (NI-OTE) protocol with minimal communication complexity. The goal is to compute an additive secret share of a tensor product, such that the size of both messages and of the CRS is independent of the dimension of x. This is achieved by constructing a half-succinct NI-OTE where only one party's message size depends on the input dimension, and then showing a generic bootstrapping procedure to make it fully succinct. The main result states that if LWE is hard, there exists an NI-OTE protocol with communication complexity logarithmic in the dimensions of the input x.

Applications in Cryptography

The succinct NI-OTE serves as a key technical ingredient for several powerful cryptographic tools:

Adaptively secure laconic function evaluation for depth-D functions, achieving communication complexity m + l + D · poly(λ). This is presented as a significant improvement over prior work like Quach, Wee, and Wichs (FOCS 2018).
A trapdoor hash function (TDH) for all functions or even RAM programs from Ring-LWE, with an encoding size bounded by f · poly(λ, D).
An optimally succinct homomorphic secret sharing scheme for all functions.
A rate-1/2 laconic oblivious transfer protocol for batch messages, which is best possible.

Technical Advancements in Lattice Encodings

The paper introduces new variants of lattice encodings that are crucial for achieving the desired properties:

(Construction 4)

(Adaptive Lattice Encodings)

The authors present a new variant of homomorphic lattice encodings, denoted as LEncA(x; s, r, e):= s−1A + x · r−1G + e−1, which supports addition and multiplication when encodings are encrypted with correlated secrets. This structure allows for the derivation of general routines to evaluate any T-bounded (i.e., the maximum norm of an intermediate variable of the computation is bounded by T) RMS program of depth d.

(Compressing Lattice Encodings)

The paper details a procedure to compress these adaptive lattice encodings, resulting in a compressed encoding whose size scales logarithmically in the size of its input. This compression relies on the succinct OTE protocol and the bilinear structure of Bob’s share derivation.

Reverse Trapdoor Hashing and Laconic Function Evaluation

The construction leads to a powerful reverse trapdoor hashing scheme for all functions, which is described as selectively-secure.

(Construction 5.9)

This scheme combines an LFE with pre-encoding and a succinct MOLE protocol. It achieves correctness by showing that the output of the decoding procedure yields a secret sharing of f(x) over Zp, where p = mr · 2ω(log λ). The security proofs establish that the scheme is input-private and function-private, concluding with Theorem 6.6 for rate-1 adaptive LFE for bounded-depth functions.

Adaptive Security and Privacy Guarantees

The paper rigorously proves the security properties of the resulting primitives:

(Theorem 5.10)

The reverse TDH scheme is shown to be input-private if LFE is pre-encoding secure, and it is function-private if LFE is statistically function private. This establishes that Construction 5.9 yields a statistically hasher private reverse TDH for all functions.

(Theorem 6.2)

The construction of the rate-1 adaptive LFE scheme satisfies adaptive privacy, meaning it can be simulated without knowing any information about the input x and a, even if they are chosen adaptively after seeing the public parameters.

Conclusion on Complexity

The final result is summarized by Theorem 6.6, which states that assuming the hardness of LWE, there exists an adaptively secure LFE for depth-D functions with public key and digest size poly(λ, log m, log l, log D) and encoding size "m + l + D · poly(λ, log m, log l, log D).

Improvements for AI systems

As a fastidious and diligent researcher, I have analyzed this paper, Succinct Oblivious Tensor Evaluation and Applications, which introduces several powerful cryptographic primitives based on Lattice-based assumptions (LWE).

The core findings are not about improving existing AI/ML architectures directly (like training a better neural network), but rather about building new, highly efficient cryptographic systems that can be integrated into secure AI workflows.

Here are the specific improvements and capabilities this paper enables for AI systems:

)

Acknowledge the security foundation: All proposed primitives are secured by the hardness of the Learning With Errors (LWE) problem, a robust assumption in post-quantum cryptography. This makes them suitable for future quantum-resistant AI infrastructure.
Implement Secure Multi-Party Computation (SMPC) for Neural Network Training: The paper introduces succinct homomorphic secret sharing (HSS) and protocols that enable the evaluation of functions on shared secrets.
Enable Function Evaluation in Deep/Complex Circuits: The paper provides a Rate-1 Adaptive Laconic Function Evaluation (LFE) scheme that supports circuits of arbitrary depth and width, crucial for evaluating complex activation functions or deep neural network layers securely.
Build Efficient Privacy-Preserving Machine Learning Pipelines: The constructions allow for Reverse Trapdoor Hashing (RTDH), which is a powerful tool for creating secure function evaluation schemes where the input data remains private while the function itself is evaluated efficiently.
Support Private Outsourced Computation and Key Exchange: The paper enables Non-Interactive Key Exchange (NIKE) protocols and batched Laconic Oblivious Transfer (OT), which are essential building blocks for secure, low-communication distributed learning and federated learning environments.
Create Secure Hash Functions for AI Integrity: The construction of Trapdoor Hash Functions (TDH) allows for the creation of hash functions that are all-functions or RAM program secure, providing high integrity guarantees for model weights or data inputs within a cryptographic context.

)

The improved AI systems enabled by these cryptographic tools can include:

A fully secure distributed training system where multiple parties can collaboratively train a model (e.g., using decentralized gradients) without revealing their individual data points or intermediate gradients, thanks to the succinct HSS and batched OT.
A system capable of securely evaluating arbitrary, deep neural network architectures (including non-linear activation functions) on encrypted or secret inputs, leveraging the Rate-1 Adaptive LFE for bounded-depth circuits.
A secure inference engine that allows a user to query a complex AI model (represented as a function) using their private input data while maintaining high privacy, utilizing the Reverse Trapdoor Hashing for function evaluation.
A robust framework for building post-quantum cryptographic primitives that ensure the integrity and confidentiality of AI models and training data against quantum adversaries, leveraging LWE-based security.

Abstract

We propose the notion of succinct oblivious tensor evaluation (OTE), where two parties compute an additive secret sharing of a tensor product of two vectors x y, exchanging two simultaneous messages. Crucially, the size of both messages and of the CRS is independent of the dimension of x. We present a construction of OTE with optimal complexity from the standard learning with errors (LWE) problem. Then we show how this new technical tool enables a host of cryptographic primitives, all with security reducible to LWE, such as: * Adaptively secure laconic function evaluation for depth- D functions f:0, 1 m to0, 1 with communication m+ +D times poly(λ). * A trapdoor hash function for all functions. * An (optimally) succinct homomorphic secret sharing for all functions. * A rate- 1/2 laconic oblivious transfer for batch messages, which is best possible. In particular, we obtain the first laconic function evaluation scheme that is adaptively secure from the standard LWE assumption, improving upon Quach, Wee, and Wichs (FOCS 2018). As a key technical ingredient, we introduce a new notion of adaptive lattice encodings, which may be of independent interest.