🤖 AI Summary
This work addresses message integrity protection in scenarios where an adversary cannot observe codewords, by leveraging cyclotomic class theory to construct external difference families (EDFs) and generalized strong external difference families over finite fields, their direct products, and integer residue rings. It establishes a theoretical connection between these combinatorial structures and algebraic manipulation detection (AMD) codes. The study introduces explicit EDF existence criteria based on quadratic and biquadratic residues, enabling the first construction of infinite families of G-optimal systematic strong AMD codes with flexible parameters. Furthermore, it provides explicit constructions of R-optimal weak AMD codes for block lengths up to 14. These results substantially enhance the flexibility and applicability of AMD code constructions.
📝 Abstract
External difference families (EDFs) are closely related to algebraic manipulation detection (AMD) codes, a cryptographic primitive that protects messages against additive tampering by an adversary who cannot observe the transmitted codeword. We first study a cyclotomic construction of external partial difference families over finite fields and derive a necessary and sufficient condition under which the resulting families are EDFs. For even block sizes up to $14$, we obtain explicit criteria in terms of quadratic and biquadratic residues, yielding new $R$-optimal weak AMD codes. We then construct bounded generalized strong external difference families using cyclotomic classes over finite fields, direct products of finite fields, and generalized cyclotomic classes over integer residue rings. These constructions give infinite families of systematic $G$-optimal strong AMD codes with flexible parameters.