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

2026-05-13

AI Summary

This work addresses a key limitation in the construction of locally correctable and private (LCP) algebraic geometry codes, which traditionally rely on Kummer extensions supporting only totally ramified points, thereby restricting the availability of non-special divisors of degrees \(g\) and \(g-1\) and limiting the diversity of admissible function fields. By analyzing Kummer covers with arbitrary ramification and leveraging Galois group actions, invariant divisor theory, and ramification structure, the authors establish necessary and sufficient conditions for non-special divisors without imposing support restrictions—bypassing the conventional reliance on intricate Weierstrass semigroup-based criteria. Building on this theoretical advance, they explicitly construct three new families of LCP codes with precisely determined parameters \([n, k, d]\) that meet or closely approach the Goppa bound, significantly enhancing their flexibility and practicality for cryptographic applications.

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 $y^m = \prod_{i=1}^r (x-α_i)^{λ_i}$ 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.

Problem

• non-special divisors
• Kummer covers
• arbitrary ramification
• LCP codes
• algebraic geometry codes

Innovation

• Kummer covers
• non-special divisors
• arbitrary ramification
• LCP codes
• Galois group actions