Elimination Algorithms for Skew Polynomials with Applications in Cybersecurity

Raqeeb Rasheed, Ali Safaa Sadiq () and Omprakash Kaiwartya
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Raqeeb Rasheed: Cyber Security Research Group (CSRG), Department of Computer Science, Nottingham Trent University, Clifton Campus, Nottingham NG11 8NS, UK
Ali Safaa Sadiq: Cyber Security Research Group (CSRG), Department of Computer Science, Nottingham Trent University, Clifton Campus, Nottingham NG11 8NS, UK
Omprakash Kaiwartya: Cyber Security Research Group (CSRG), Department of Computer Science, Nottingham Trent University, Clifton Campus, Nottingham NG11 8NS, UK

Mathematics, 2024, vol. 12, issue 20, 1-29

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.

Keywords: Ore algebra; skew polynomials; elimination; resultant; symbolic computation; modular method; noncommutative algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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