Efficient Methods for Simultaneous Homomorphic Inversion
Bibliographic record
Abstract
Efficient implementation of some privacy-preserving algorithms and applications rely on efficient implementation of homomorphic inversion. For example, a recently proposed homomorphic image filtering algorithm and the privacy-preserving body mass index (BMI) calculations repetitively use homomorphic inversion. In this paper, inspired by Montgomery's trick to perform simultaneous plaintext inversion, we tackle the simultaneous homomorphic inversion problem to compute s inverses simultaneously over ciphertexts. The advantage of Montgomery's trick for plaintext arithmetic is well-known. We first observe that the advantage can quickly vanish when homomorphic encryption is employed because of the increased depth of the circuits. Therefore, we propose three algorithms (Montgomery's trick and two other variants) that reduce the number of homomorphic inversions from s to 1 and that offer different levels of trade-offs between the number of multiplications and the circuit depth. We provide a theoretical complexity analysis of our algorithms and implement them using the CKKS scheme in the OpenFHE library. Our experiments show that, for some cases, the run time of homomorphic s-inversion can be improved up to 35 percent while in some other cases, regular inversion seems to outperform Montgomery-based inversion algorithms.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".