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Low-rank parity-check codes over Galois rings
Julian Renner1, Alessandro Neri1, Sven Puchinger2
1Institute for Communications Engineering, Technical University of Munich (TUM), Münich, Germany.
Designs, Codes, and Cryptography
|February 19, 2021
Summary
This study introduces low-rank parity-check (LRPC) codes over Galois rings, extending their use beyond finite fields for cryptography. A new decoding algorithm offers faster performance than existing Gabidulin code decoders, with a bounded failure probability.
Area of Science:
- Coding Theory
- Cryptography
- Algebraic Codes
Background:
- Low-rank parity-check (LRPC) codes are rank-metric codes over finite fields, proposed for cryptographic applications.
- Gabidulin codes have been adapted to finite rings, inspiring further research.
Purpose of the Study:
- To define and study LRPC codes over Galois rings, a broad class of finite commutative rings.
- To develop and analyze a decoding algorithm for these new codes.
Main Methods:
- Definition of LRPC codes over Galois rings.
- Development of a decoding algorithm based on linear-algebraic operations.
- Derivation of an upper bound on the decoder's failure probability.
Main Results:
- A decoding algorithm is presented, similar to existing methods but adapted for Galois rings.
- An upper bound on the failure probability is derived, dependent only on error rank.
- A class of LRPC codes over Galois rings demonstrates faster decoding than Gabidulin codes for similar parameters.
Conclusions:
- LRPC codes over Galois rings offer a promising alternative for cryptographic applications.
- The proposed decoding algorithm provides efficient error correction with controllable failure rates.
- This work extends the applicability of rank-metric codes to a wider algebraic structure.
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