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Rasheed, R 
ORCID: https://orcid.org/0009-0008-4608-0372,
2026.
Elimination algorithms for skew polynomials with applications in cybersecurity.
PhD, Nottingham Trent University.
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Abstract
This thesis details a comprehensive investigation of polynomial resultants, advancing algebraic techniques for non-commutative skew (Ore) polynomials and their novel applications in cryptography. It addresses critical needs in symbolic computation for effective non-commutative tools (such as resultant) and in cybersecurity for enhanced cryptographic primitives, including novel secret sharing schemes and key-exchange protocols.
A core component is the development of skew resultants for multivariate (initially bivariate) skew polynomials, extending existing univariate theories. Key algebraic contributions include new formulations, elimination methods, and establishing links between resultants and operator evaluation maps. Two primary computational methodologies are detailed; a direct matrix approach and a modular method using evaluation and interpolation. Applicability is also demonstrated for solving skew polynomial systems and within structures such as almost commutative rings.
Building on this algebraic foundation, the study introduces the application of polynomial resultants in cryptography. It presents a detailed development of two novel (t,n)-threshold secret sharing schemes and also proposes a Diffie-Hellman-like key-exchange protocol that utilises the inherent commutativity of skew resultants. The two schemes offer different trade-offs; one uses numerical shares for efficiency, while the other employs symbolic polynomial shares for enhanced security and potential post-quantum advantages. The basis for these schemes is a specialised algorithm, also developed in this study, for constructing input polynomials with precise resultant properties. Both schemes incorporate built-in verification and adversary detection derived from these fundamental properties.
The theoretical correctness and practical feasibility of all developed methods are validated by thorough proofs and illustrative examples, including adversarial scenarios. This work significantly advances non-commutative resultant theory and its practical application, offering new tools for symbolic computation and cryptography while also suggesting avenues for future research into n-variate generalisations and further cryptographic development.
| Item Type: | Thesis |
|---|---|
| Creators: | Rasheed, R. |
| Contributors: | Name Role NTU ID ORCID |
| Date: | January 2026 |
| Rights: | This work is the intellectual property of the author. You may copy up to 5% of this work for private study, or personal, non-commercial research. Any re-use of the information contained within this document should be fully referenced, quoting the author, title, university, degree level and pagination. Queries or requests for any other use, or if a more substantial copy is required, should be directed in the owner(s) of the Intellectual Property Rights. |
| Divisions: | Schools > School of Science and Technology |
| Record created by: | Laura Borcherds |
| Date Added: | 09 Apr 2026 15:36 |
| Last Modified: | 09 Apr 2026 15:36 |
| URI: | https://irep.ntu.ac.uk/id/eprint/55529 |
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