Optimized quantum implementation of elliptic curve arithmetic over binary fields
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Shor's quantum algorithm for discrete logarithms applied to elliptic curve groups forms the basis of a ``quantum attack'' of elliptic curve cryptosystems. To implement this algorithm on a quantum computer requires the efficient implementation of the elliptic curve group operation. Such an implementation requires we be able to compute inverses in the underlying field. In \cite{PZ03}, Proos and Zalka show how to implement the extended Euclidean algorithm to compute inverses in the prime field $\GF(p)$. They employ a number of optimizations to achieve a running time of $O(n^2)$, and a space-requirement of $O(n)$ qubits, where $n$ is the number of bits in the binary representation of $p$ (there are some trade-offs that they make, sacrificing a few extra qubits to reduce running-time). In practice, elliptic curve cryptosystems often use curves over the binary field $\GF(2^m)$. In this paper, I show how to implement the extended Euclidean algorithm for polynomials to compute inverses in $\GF(2^m)$. Working under the assumption that qubits will be an `expensive' resource in realistic implementations, I optimize specifically to reduce the qubit space requirement, while keeping the running-time polynomial. The implementation here differs from that in $\cite{PZ03}$ for $\GF(p)$, and we are able to take advantage of some properties of the binary field $\GF(2^m)$. I also optimize the overall qubit space requirement for computing the group operation for elliptic curves over $\GF(2^m)$ by decomposing the group operation to make it ``piecewise reversible'' (similar to what is done in \cite{PZ03} for curves over $\GF(p)$).
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| 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 it