A Tight Lower Bound on the TdScrypt Trapdoor Memory-Hard Function
Bibliographic record
Abstract
A trapdoor Memory-Hard Function is a function that is memory-hard to evaluate for any party who does not have a trapdoor, but is substantially less expensive to evaluate with the trapdoor. Biryukov and Perrin (Asiacrypt 2017) introduced the first candidate trapdoor Memory-Hard Function called Diodon, which modifies a Memory-Hard Function called Scrypt by replacing a hash chain with repeated squaring modulo a composite number N=pq. The trapdoor, which consists of the prime factors p and q, allows one to compute the function with significantly reduced cumulative memory cost (CMC) O(n*log n*(log N)^2) where n denotes the running time parameter, e.g., the length of the hash chain or repeated squaring chain. By contrast, the best-known algorithm to compute Diodon without the trapdoor has the CMC O(n^2*log N). Auerbach et al. (Eurocrypt 2024) provided the first provable lower bound on the CMC of TdScrypt — a specific instantiation of Diodon. In particular, in idealized models, they proved that the CMC of TdScrypt is Omega(n^2*log N/(log n)) which almost matches the upper bound O(n^2*log N) but is off by a multiplicative log n factor. In this work, we show how to tighten the analysis of Auerbach et al. (Eurocrypt 2024) and eliminate the gap. In particular, our results imply that TdScrypt has the CMC at least Omega(n^2*log N).
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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 | 0.005 | 0.037 |
| Meta-epidemiology (narrow) | 0.004 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.004 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.009 | 0.025 |
| Open science | 0.006 | 0.011 |
| Research integrity | 0.004 | 0.019 |
| Insufficient payload (model declined to judge) | 0.024 | 0.010 |
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".