This paper presents explicit constructions of bases for Riemann-Roch spaces associated with arbitrary divisors on elliptic curves. In the context of algebraic geometry codes, the knowledge of an explicit basis for arbitrary divisors is especially valuable, as it enables efficient code construction. From a cryptographic point of view, codes associated with arbitrary divisors with many points are closer to Goppa codes, making them attractive for embedding in the McEliece cryptosystem. Using the results obtained in this work, it is also possible to efficiently construct quasi-cyclic subfield subcodes of elliptic codes. These codes enable a significant reduction in public key size for the McEliece cryptosystem and, consequently, represent promising candidates for integration into post-quantum code-based schemes.

翻译：本文提出了与椭圆曲线上任意除子相关的黎曼-罗赫空间的显式基构造方法。在代数几何码的背景下，获取任意除子的显式基知识尤为重要，因为它能够实现高效的码构造。从密码学角度来看，与具有多个点的任意除子相关的码更接近Goppa码，这使得它们对于嵌入McEliece密码系统具有吸引力。利用本工作中获得的结果，还可以高效构造椭圆码的拟循环子域子码。这些码能够显著减小McEliece密码系统的公钥尺寸，因此代表了在基于后量子码方案中集成的有前景候选方案。