Large Language Models (LLMs) have shown impressive progress in mathematical reasoning. While data augmentation is promising to enhance mathematical problem-solving ability, current approaches are predominantly limited to instance-level modifications—such as rephrasing or generating syntactic variations—which fail to capture and leverage the intrinsic relational structures inherent in mathematical knowledge. Inspired by human learning processes, where mathematical proficiency develops through systematic exposure to interconnected concepts, we introduce MathFusion, a novel framework that enhances mathematical reasoning through cross-problem instruction synthesis. MathFusion implements this through three fusion strategies: (1) sequential fusion, which chains related problems to model solution dependencies; (2) parallel fusion, which combines analogous problems to reinforce conceptual understanding; and (3) conditional fusion, which creates context-aware selective problems to enhance reasoning flexibility. By applying these strategies, we generate a new dataset, MathFusionQA, followed by fine-tuning models (DeepSeekMath-7B, Mistral-7B, Llama3-8B) on it. Experimental results demonstrate that MathFusion achieves substantial improvements in mathematical reasoning while maintaining high data efficiency, boosting performance by 18.0 points in accuracy across diverse benchmarks while requiring only 45K additional synthetic instructions, representing a substantial improvement over traditional single-instruction approaches.

Large language models (LLMs) have demonstrated remarkable capabilities, especially the recent advancements in reasoning, such as o1 and o3, pushing the boundaries of AI. Despite these impressive achievements in mathematics and coding, the reasoning abilities of LLMs in domains requiring cryptographic expertise remain underexplored. In this paper, we introduce CipherBank, a comprehensive benchmark designed to evaluate the reasoning capabilities of LLMs in cryptographic decryption tasks. CipherBank comprises 2,358 meticulously crafted problems, covering 262 unique plaintexts across 5 domains and 14 subdomains, with a focus on privacy-sensitive and real-world scenarios that necessitate encryption. From a cryptographic perspective, CipherBank incorporates 3 major categories of encryption methods, spanning 9 distinct algorithms, ranging from classical ciphers to custom cryptographic techniques. We evaluate state-of-the-art LLMs on CipherBank, e.g., GPT-4o, DeepSeek-V3, and cutting-edge reasoning-focused models such as o1 and DeepSeek-R1. Our results reveal significant gaps in reasoning abilities not only between general-purpose chat LLMs and reasoning-focused LLMs but also in the performance of current reasoning-focused models when applied to classical cryptographic decryption tasks, highlighting the challenges these models face in understanding and manipulating encrypted data. Through detailed analysis and error investigations, we provide several key observations that shed light on the limitations and potential improvement areas for LLMs in cryptographic reasoning.These findings underscore the need for continuous advancements in LLM reasoning capabilities.

Large language models (LLMs) have demonstrated remarkable reasoning capability in solving mathematical problems. However, existing approaches primarily focus on improving the quality of correct training data, e.g., distilling high-quality correct solutions from advanced models, neglecting the value contained in error data, potentially hindering the model’s reflective ability. Though some studies attempted to leverage error data, they often involve complex mechanisms, such as Monte Carlo Tree Search (MCTS) to explore error nodes.In this work, we propose to enhance LLM’s reasoning ability by Learning from Errors for MatheMatical Advancement (LEMMA). LEMMA constructs data consists of an incorrect solution with an erroneous step and a reflection connection to a correct solution for fine-tuning. Specifically, we systematically analyze the model-generated error types and introduce an _error-type grounded mistake augmentation_ method to collect diverse and representative errors. Correct solutions are either from fixing the errors or generating a fresh start. By fine-tuning on the constructed dataset, the model is able to _self-correct errors autonomously_ within the generation process _without relying on external critique models_. Experimental results demonstrate that LEMMA achieves significant performance improvements over other strong models with less than 90k data.