Elimination Algorithms for Skew Polynomials with Applications in Cybersecurity
Bibliographic record
Abstract
It is evident that skew polynomials offer promising directions for developing cryptographic schemes. This paper focuses on exploring skew polynomials and studying their properties with the aim of exploring their potential applications in fields such as cryptography and combinatorics. We begin by deriving the concept of resultants for bivariate skew polynomials. Then, we employ the derived resultant to incrementally eliminate indeterminates in skew polynomial systems, utilising both direct and modular approaches. Finally, we discuss some applications of the derived resultant, including cryptographic schemes (such as Diffie–Hellman) and combinatorial identities (such as Pascal’s identity). We start by considering a bivariate skew polynomial system with two indeterminates; our intention is to isolate and eliminate one of the indeterminates to reduce the system to a simpler form (that is, relying only on one indeterminate in this case). The methodology is composed of two main techniques; in the first technique, we apply our definition of a (bivariate) resultant via a Sylvester-style matrix directly from the polynomials’ coefficients, while the second is based on modular methods where we compute the resultant by using evaluation and interpolation approaches. The idea of this second technique is that instead of computing the resultant directly from the coefficients, we propose to evaluate the polynomials at a set of valid points to compute its corresponding set of partial resultants first; then, we can deduce the original resultant by combining all these partial resultants using an interpolation technique by utilising a theorem we have established.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| 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".