Construction of Non-special Divisors on Kummer Covers With Arbritary Ramification For LCP Codes
For cryptographers and coding theorists, this work expands the range of function fields usable for LCP codes, which are important for side-channel and fault-injection attack resistance.
This paper removes the restriction that divisors in Kummer extensions must be supported on totally ramified places for constructing LCP AG codes, enabling explicit construction of non-special divisors with arbitrary ramification. The resulting codes meet or approach the Goppa designed distance, providing greater flexibility for cryptographic applications.
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.