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On Practical Aspects of Constructing Quasi-Cyclic Subfield Subcodes of Dual Elliptic Codes and Their Application in McEliece-type Cryptosystems
In this work we study the applicability of Quasi-Cyclic Subfield Subcodes of Dual Elliptic (QC-SSDE) codes for integration into code-based cryptographic schemes. Detailed algorithms are provided for constructing parity-check matrices as well as block-circulant parity-check matrices for this family of codes, accompanied by empirical results that enable the construction of QC-SSDE codes with predetermined dimensions. We establish an improved decoding complexity bound, representing the current state-of-the-art estimate for decoding algebraic geometry codes and their subfield subcodes beyond half the designed distance. The obtained results and other recent works highlight the cryptographic potential of subfield subcodes of algebraic geometry codes, positioning them as promising alternative for McEliece-type schemes on Goppa codes.