Elimination algorithms for skew polynomials with applications in cybersecurity

Rasheed, R ORCID: https://orcid.org/0009-0008-4608-0372, Sadiq, AS ORCID: https://orcid.org/0000-0002-5746-0257 and Kaiwartya, O ORCID: https://orcid.org/0000-0001-9669-8244, 2024. Elimination algorithms for skew polynomials with applications in cybersecurity. Mathematics, 12 (20): 3258. ISSN 2227-7390

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Abstract

It is evident that skew polynomials offer promising direction 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 resultant 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 (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.

Item Type:	Journal article
Publication Title:	Mathematics
Creators:	Rasheed, R., Sadiq, A.S. and Kaiwartya, O.
Publisher:	MDPI
Date:	17 October 2024
Volume:	12
Number:	20
ISSN:	2227-7390
Identifiers:	Number Type 10.3390/math12203258 DOI 2254047 Other
Divisions:	Schools > School of Science and Technology
Record created by:	Jonathan Gallacher
Date Added:	16 Oct 2024 08:37
Last Modified:	05 Nov 2024 14:17
URI:	https://irep.ntu.ac.uk/id/eprint/52420

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