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Modular addition is, on its face, a simple operation: given $N$ elements in $\mathbb{Z}_q$, compute their sum modulo $q$. Yet, scalable machine learning solutions to this problem remain elusive: prior work trains ML models that sum $N \le 6$ elements mod $q \le 1000$. Promising applications of ML models for cryptanalysis-which often involve modular arithmetic with large $N$ and $q$-motivate reconsideration of this problem. This work proposes three changes to the modular addition model training pipeline: more diverse training data, an angular embedding, and a custom loss function. With these changes, we demonstrate success with our approach for $N = 256, q = 3329$, a case which is interesting for cryptographic applications, and a significant increase in $N$ and $q$ over prior work. These techniques also generalize to other modular arithmetic problems, motivating future work.

Lattice cryptography schemes based on the learning with errors (LWE) hardness assumption have been standardized by NIST for use as post-quantum cryptosystems, and by HomomorphicEncryption.org for encrypted compute on sensitive data. Thus, understanding their concrete security is critical. Most work on LWE security focuses on theoretical estimates of attack performance, which is important but may overlook attack nuances arising in real-world implementations. The sole existing concrete benchmarking effort, the Darmstadt Lattice Challenge, does not include benchmarks relevant to the standardized LWE parameter choices - such as small secret and small error distributions, and Ring-LWE (RLWE) and Module-LWE (MLWE) variants. To improve our understanding of concrete LWE security, we provide the first benchmarks for LWE secret recovery on standardized parameters, for small and low-weight (sparse) secrets. We evaluate four LWE attacks in these settings to serve as a baseline: the Search-LWE attacks uSVP, SALSA, and Cool & Cruel, and the Decision-LWE attack: Dual Hybrid Meet-in-the-Middle (MitM). We extend the SALSA and Cool & Cruel attacks in significant ways, and implement and scale up MitM attacks for the first time. For example, we recover hamming weight $9-11$ binomial secrets for KYBER ($\kappa=2$) parameters in $28-36$ hours with SALSA and Cool\&Cruel, while we find that MitM can solve Decision-LWE instances for hamming weights up to $4$ in under an hour for Kyber parameters, while uSVP attacks do not recover any secrets after running for more than $1100$ hours. We also compare concrete performance against theoretical estimates. Finally, we open source the code to enable future research., Comment: Accepted at Oakland S&P 2025

Sparse binary LWE secrets are under consideration for standardization for Homomorphic Encryption and its applications to private computation. Known attacks on sparse binary LWE secrets include the sparse dual attack and the hybrid sparse dual-meet in the middle attack which requires significant memory. In this paper, we provide a new statistical attack with low memory requirement. The attack relies on some initial lattice reduction. The key observation is that, after lattice reduction is applied to the rows of a q-ary-like embedded random matrix $\mathbf A$, the entries with high variance are concentrated in the early columns of the extracted matrix. This allows us to separate out the "hard part" of the LWE secret. We can first solve the sub-problem of finding the "cruel" bits of the secret in the early columns, and then find the remaining "cool" bits in linear time. We use statistical techniques to distinguish distributions to identify both the cruel and the cool bits of the secret. We provide concrete attack timings for recovering secrets in dimensions $n=256$, $512$, and $768$. For the lattice reduction stage, we leverage recent improvements in lattice reduction (e.g. flatter) applied in parallel. We also apply our new attack in the RLWE setting for $2$-power cyclotomic rings, showing that these RLWE instances are much more vulnerable to this attack than LWE., Comment: Camera ready version for AFRICACRYPT

Motivated by cryptographic applications, we investigate two machine learning approaches to modular multiplication: namely circular regression and a sequence-to-sequence transformer model. The limited success of both methods demonstrated in our results gives evidence for the hardness of tasks involving modular multiplication upon which cryptosystems are based., Comment: 14 pages, 12 figures. Comments welcome!

Learning with Errors (LWE) is a hard math problem underlying recently standardized post-quantum cryptography (PQC) systems for key exchange and digital signatures. Prior work proposed new machine learning (ML)-based attacks on LWE problems with small, sparse secrets, but these attacks require millions of LWE samples to train on and take days to recover secrets. We propose three key methods -- better preprocessing, angular embeddings and model pre-training -- to improve these attacks, speeding up preprocessing by $25\times$ and improving model sample efficiency by $10\times$. We demonstrate for the first time that pre-training improves and reduces the cost of ML attacks on LWE. Our architecture improvements enable scaling to larger-dimension LWE problems: this work is the first instance of ML attacks recovering sparse binary secrets in dimension $n=1024$, the smallest dimension used in practice for homomorphic encryption applications of LWE where sparse binary secrets are proposed., Comment: 8 pages (main text)