Solving Discrete Logarithms in Smooth-Order Groups with CUDA 1
Bibliographic record
Abstract
This paper chronicles our experiences using CUDA to implement a parallelized variant of Pollard’s rho algorithm to solve discrete logarithms in groups with cryptographically large moduli but smooth order using commodity GPUs. We first discuss some key design constraints imposed by modern GPU architectures and the CUDA framework, and then explain how we were able to implement efficient arbitrary-precision modular multiplication within these constraints. Our implementation can execute roughly 51.9 million 768-bit modular multiplications per second — or a whopping 840 million 192-bit modular multiplications per second — on a single Nvidia Tesla M2050 GPU card, which is a notable improvement over all previous results on comparable hardware. We leverage this fast modular multiplication in our implementation of the parallel rho algorithm, which can solve discrete logarithms modulo a 1536-bit RSA number with a 2 55-smooth totient in less than two minutes. We conclude the paper by discussing implications to discrete logarithm-based cryptosystems, and by pointing out how efficient implementations of parallel rho (or related algorithms) lead to trapdoor discrete logarithm groups; we also point out two potential cryptographic applications for the latter. Our code is written in C for CUDA and PTX; it is open source and freely available for download online. 1
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".