Gate bootstrapping is a core primitive that enables arbitrary circuit evaluation in fully homomorphic encryption (FHE), where blind rotation remains the dominant performance bottleneck. In this work, we present a sub-millisecond NTRU-based gate bootstrapping scheme that achieves state-of-the-art performance through coordinated algorithmic, software, and hardware-level optimizations.

Abstract Fully Homomorphic Encryption (FHE) enables arbitrary computation on encrypted data without decryption, making it a cornerstone of privacy-preserving outsourcing, such as cloud computing. However, homomorphic operations cause ciphertext noise to grow until decryption fails. The efficient solution is bootstrapping, which refreshes the noise in FHE ciphertexts to sustain arbitrary deep homomorphic evaluation. But in practice, bootstrapping consumes over 50% of total execution time, posing a serious obstacle to FHE adoption. This paper presents a systematic survey of FHE bootstrapping algorithms and their optimizations. We organize existing works into three main paradigms: word-wise bootstrapping for BGV, BFV, and CKKS schemes; bit-wise bootstrapping for FHEW and TFHE schemes; and hybrid bootstrapping, which leverages both word-wise schemes and bit-wise schemes. We analyze the evolution of crucial techniques, highlight latest advances in reducing latency, enhancing parallelism, and controlling noise growth, and compare the advantages and limitations of different schemes. Finally, we discuss emerging research trends.

TFHE is a fully homomorphic encryption scheme over the torus that supports fast bootstrapping. Its primary evaluation mechanism is based on gate bootstrapping and programmable bootstrapping (PBS), which computes functions while simultaneously refreshing noise. PBS-based evaluation is user-friendly and efficient for small circuits; however, the number of bootstrapping operations increases exponentially with the circuit depth. To address the challenge of efficiently evaluating large-scale circuits, Chillotti et al. introduced a leveled homomorphic evaluation (LHE) mode at Asiacrypt 2017. This mode decouples circuit evaluation from bootstrapping, resulting in a speedup of hundreds of times over PBS-based methods. However, the remaining circuit bootstrapping (CBS) becomes a performance bottleneck, even though its frequency is linear with the circuit depth.

Verifiable Computation over encrypted data (VCoed) has two popular paradigms: SNARK-FHE (applying SNARKs to prove FHE operations) and FHE-SNARK (homomorphically evaluating SNARK proofs). For the existing works, FHE-SNARK has a much better efficiency compared to SNARK-FHE.

Enhancing the mathematical reasoning capabilities of LLMs has garnered significant attention in both the mathematical and computer science communities.Recent works have made substantial progress in both Natural Language (NL) reasoning and Formal Language (FL) reasoning by leveraging the potential of pure Reinforcement Learning (RL) methods on base models.However, RL approaches struggle to impart new capabilities not presented in the base model (Yue et al., 2025), highlighting the need to integrate more knowledge like FL into NL math reasoning effectively.Yet, this integration is challenging due to inherent disparities in problem structure and reasoning format between NL and FL (Wang et al., 2024).To address these challenges, we introduce NL-FL HybridReasoning (NFL-HR), an end-to-end framework designed to incorporate the FL expert into NL math problem-solving.To bridge the NL and FL input format gap, we propose the NL-FL Problem Alignment method, which reformulates the Question-Answering (QA) problems in NL as existence theorems in FL.Subsequently, the Mixed Problem Input technique we provide enables the FL reasoner to handle both QA and existence problems concurrently.Lastly, we mitigate the NL and FL output format gap in reasoning through an LLMbased Answer Extraction mechanism.Comprehensive experiments demonstrate that the NFL-HR framework achieves 89.80% and 84.34% accuracy rates on the MATH-500 and the AMC benchmarks, surpassing the NL baseline by 4.60% and 4.82%, respectively.Notably, some problems resolved by our framework remain unsolved by the NL baseline model even under a larger number of trials.

Large Language Models (LLMs) have displayed astonishing abilities in various tasks, especially in text generation, classification, question answering, etc.However, the reasoning ability of LLMs still faces many debates, especially in math reasoning.The inherent ambiguity of Natural Language (NL) limits LLMs' ability to perform verifiable reasoning, making the answers lack coherence and trustworthy support.To tackle the above challenges, we propose a novel framework named FANS: Formal ANswer Selection for LLM Natural Language Math Reasoning Using Lean4.It is a pioneering framework that utilizes Lean4 to enhance LLMs' NL math reasoning ability.In particular, given an NL math question and LLM-generated answers, FANS first translates it into Lean4 theorem statements.Then it invokes another Lean4 prover LLM to produce proofs, and finally verifies the proofs by Lean4 compiler.Answers are selected based on the verifications.It enhances LLMs' NL math ability in providing a computer-verifiable solution for its correct answer and proposes an alternative method for answer selection beyond the reward model based ones.Our experiments demonstrate the effectiveness of FANS with an improvement of nearly 2% across several math benchmarks, and even higher further based on reward models or in subfields such as algebra and number theory that Lean4 is better at.The code is available in https://github.com/MaxwellJryao/FANS.

Solving mathematical problems using computer-verifiable languages like Lean has significantly impacted the mathematical and computer science communities. State-of-the-art methods utilize a single Large Language Model (LLM) to generate complete proof or perform tree search, but they fail to balance these tasks. We propose MA-LoT: Model-CollAboration Lean-based Long Chain-of-Thought, a comprehensive framework for Lean4 theorem proving to solve this issue. It separates the cognition tasks of general NL for whole-proof generation and error analysis for proof correction using the model-collaboration method. We achieve this by structured interaction of the LLM and Lean4 verifier in Long CoT. To implement the framework, we propose the novel LoT-Transfer Learning training-inference pipeline, which enables the Long CoT thinking capability to LLMs without special data annotation. Extensive experiment shows that our framework achieves a 61.07% accuracy rate on the Lean4 version of the MiniF2F-Test dataset, largely outperforming DeepSeek-V3 (33.61%), single-model tree search (InternLM-Step-Prover, 50.70%), and whole-proof generation (Godel-Prover, 55.33%) baselines. Furthermore, our findings highlight the potential of combining Long CoT with formal verification for a more insightful generation in a broader perspective.

This paper presents a high-speed 8-bit asynchronous successive approximation register (SAR) analog-to-digital converter (ADC) featuring tailored bit intervals (TBI). The design employs built-in SAR logic to set delays autonomously, eliminating the need for digital assistance, and thereby reducing both power and area consumption. This approach also effectively shortens the waiting time before lower-bit comparisons, enabling faster conversions. The ADC is simulated in the 16 nm process, occupying only 0.0012 mm2, with post-simulation conducted under various extreme process and temperature conditions. Compared to prior works, our design exhibits notable performance advantages, achieving an ENOB of 7.38 bits at TT 25°C with a power consumption of 6.94 mW. Furthermore, the designed TBI-ADC attains a sampling rate of 1.6 GS/s at FF -40°C, representing a 33% increase over the fastest previously reported single-channel, 1b/cycle, 8-bit SAR ADC.

Proving mathematical theorems using computer-verifiable Formal Languages (FL) like Lean significantly impacts mathematical reasoning.One approach to formal theorem proving involves generating complete proofs using Large Language Models (LLMs) based on Natural Language (NL) proofs.However, due to the scarcity of aligned NL and FL theorem-proving data, most modern LLMs exhibit suboptimal performance.This scarcity results in a paucity of methodologies for training LLMs and techniques to fully utilize their capabilities in composing formal proofs.To address these challenges, this paper proposes TheoremLlama, an end-to-end framework that trains a general-purpose LLM to be a Lean4 expert.TheoremLlama includes NL-FL dataset generation and bootstrapping method to obtain aligned dataset, curriculum learning and block training techniques to train the model, and iterative proof writing method to write Lean4 proofs that work together synergistically.Using the dataset generation method in TheoremLlama, we provide Open Bootstrapped Theorems (OBT), an NL-FL aligned and bootstrapped dataset.Our novel NL-FL bootstrapping method, where NL proofs are integrated into Lean4 code for training datasets, leverages the NL reasoning ability of LLMs for formal reasoning.The TheoremLlama framework achieves cumulative accuracies of 36.48% and 33.61% on MiniF2F-Valid and Test datasets respectively, surpassing the GPT-4 baseline of 22.95% and 25.41%.Our code, model checkpoints, and the generated dataset is published in GitHub

Session-based recommendation, aiming at making the prediction of the user's next item click based on the information in a single session only, even in the presence of some random user's behavior, is a complex problem. This complex problem requires a high-capability model of predicting the user's next action. Most (if not all) existing models follow the encoder-predictor paradigm where all studies focus on how to optimize the encoder module extensively in the paradigm, but they overlook how to optimize the predictor module. In this paper, we discover the critical issue of the low-capability predictor module among existing models. Motivated by this, we propose a novel framework called Session-based Recommendation with Predictor Add-On (SR-PredictAO). In this framework, we propose a high-capability predictor module which could alleviate the effect of random user's behavior for prediction. It is worth mentioning that this framework could be applied to any existing models, which could give opportunities for further optimizing the framework. Extensive experiments on two real-world benchmark datasets for three state-of-the-art models show that SR-PredictAO outperforms the current state-of-the-art model by up to 2.9% in HR@20 and 2.3% in MRR@20. More importantly, the improvement is consistent across almost all the existing models on all datasets, and is statistically significant, which could be regarded as a significant contribution in the field.

Abstract Stealth address protocol (SAP) is widely used in blockchain to achieve anonymity. In this paper, we formalize a key derivable signature scheme (KDS) to capture the functionality and security requirements of SAP. We then propose a framework to construct key separation KDS, which follows the key separation principle as all existing SAP solutions to avoid the reuse of the master keys in the derivation and signature component. We also study the joint security in KDS and construct a key reusing KDS framework, which implies the first compact stealth address protocol using a single key pair. Finally, we provide instantiations based on the elliptic curve (widely used in cryptocurrencies) and on the lattice (with quantum resistance), respectively.