Bases of Riemann-Roch spaces associated with arbitrary elliptic curve divisors and their application in constructing various elliptic Codes families

Abstract: In this paper, we determine explicit bases for Riemann--Roch spaces associated with various families of elliptic codes. We establish the feasibility and provide exact algorithms for constructing bases of Riemann--Roch spaces corresponding to arbitrary divisors on elliptic curves. These results are subsequently applied to derive bases for quasi-cyclic elliptic codes and their subfield subcodes as well as for the class of Goppa-like elliptic codes. For algebraic geometry code applications, having an explicit description of Riemann--Roch space bases for arbitrary divisors is particularly valuable as it simultaneously enables efficient code construction and reveals structural properties of the codes leading to the new cryptanalysis methods when these codes are employed in cryptographic schemes