Construction of Non-special Divisors on Kummer Covers With Arbritary Ramification For LCP Codes

TL;DR

This paper develops a new framework for constructing non-special divisors on Kummer covers with arbitrary ramification, enabling the creation of more versatile algebraic geometry codes for cryptography.

Contribution

It generalizes existing methods by removing restrictions on divisor support, using Galois actions and invariant divisors, and provides explicit constructions for LCP AG codes.

Findings

Constructed new families of LCP AG codes with parameters close to Goppa bounds.

Established necessary and sufficient conditions for non-special divisors with arbitrary ramification.

Replaced complex semigroup machinery with a more direct, efficient approach.

Abstract

Linear Complementary Pairs (LCP) of algebraic geometry (AG) codes offer strong resistance against side-channel and fault-injection attacks, but their construction depends critically on the explicit identification of non-special divisors of degree $g$ and $g-1$. Existing constructions are restricted to Kummer extensions where divisors are supported exclusively on totally ramified places, significantly limiting the range of applicable function fields and codes. We remove this restriction by developing a framework for general Kummer extensions $y^m = \prod_{i=1}^r (x-\alpha_i)^{\lambda_i}$ over finite fields with arbitrary ramification. Using Galois group actions and invariant divisor techniques, we establish necessary and sufficient conditions for non-speciality with no constraint on the support, yielding explicit constructions where previous methods fail. Our approach replaces the computationally intensive Weierstrass semigroup machinery with a more direct and efficient framework. As an application, we construct new explicit families of LCP AG codes with determined parameters $[n,k,d]$, covering three ramification regimes. The resulting codes meet or approach the Goppa designed distance, offering greater flexibility for cryptographic applications.