Low-Rank Parity-Check Codes over the Ring of Integers Modulo a Prime Power

  title={Low-Rank Parity-Check Codes over the Ring of Integers Modulo a Prime Power},
  author={Julian Renner and Sven Puchinger and Antonia Wachter-Zeh and Camilla Hollanti and Ragnar Freij},
  journal={2020 IEEE International Symposium on Information Theory (ISIT)},
We define and analyze low-rank parity-check (LRPC) codes over extension rings of the finite chain ring ${{\mathbb{Z}}_{{p^r}}}$, where p is a prime and r is a positive integer. LRPC codes have originally been proposed by Gaborit et al. (2013) over finite fields for cryptographic applications. The adaption to finite rings is inspired by a recent paper by Kamche et al. (2019), which constructed Gabidulin codes over finite principle ideal rings with applications to space-time codes and network… Expand
Generalization of low rank parity-check (LRPC) codes over the ring of integers modulo a positive integer
Following the work of Gaborit et al. (in: The international workshop on coding and cryptography (WCC 13), 2013) defining LRPC codes over finite fields, Renner et al. (in: IEEE international symposiumExpand
Solving the Rank Decoding Problem Over Finite Principal Ideal Rings
This paper shows that some combinatorial type algorithms for solving the rank decoding problem over finite fields can be generalized to solve the same problem over infinite principal ideal rings and observes that some recent algebraic attacks are not directly applicable when the finite ring is not a field due to zero divisors. Expand
Low-Rank Parity-Check Codes Over Finite Commutative Rings and Application to Cryptography
This paper first defines LRPC codes over finite commutative local rings with an efficient decoder and derive an upper bound of the failure probability together with the complexity of the decoder, then extends the definition to arbitrary finite Commutative rings and also provides a decoder in this case. Expand
Low-rank parity-check codes over Galois rings
There is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best decoder forGabidulin codes. Expand


Rank-Metric Codes Over Finite Principal Ideal Rings and Applications
The rank metric is generalized and the rank-metric Singleton bound is established and the definition of Gabidulin codes is extended and it is shown that its properties are preserved. Expand
Low Rank Parity Check codes and their application to cryptography
In this paper we introduce a new family of rank metric codes: the Low Rank Parity Check codes for which we propose an e cient probabilistic decoding algorithm. This family of codes can be seen as theExpand
Low Rank Parity Check Codes: New Decoding Algorithms and Applications to Cryptography
We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as theExpand
MRD Codes: Constructions and Connections
This preprint is of a chapter to appear in {\it Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications. Radon Series on Computational and AppliedExpand
Low Rank Parity Check Codes and their application in Power Line Communications smart grid networks
This work proposes a new code design and an efficient probabilistic decoding algorithm based on calculations of vector spaces over a finite field math formula for Low Rank Parity Check Codes, originally designed for cryptography applications in the context of Power Line Communication. Expand
Bilinear Forms over a Finite Field, with Applications to Coding Theory
  • P. Delsarte
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 1978
The characters of the adjacency algebra of Ω, which yield the MacWilliams transform on q-distance enumerators, are expressed in terms of generalized Krawtchouk polynomials. Expand
Efficient multi-source network coding using low rank parity check code
Results show that the proposed modified-low rank parity check (M-LRPC) decoding algorithm significantly improves the decoding probability compared to Gabidulin codes. Expand
Efficient Decoding of Interleaved Low-Rank Parity-Check Codes
An efficient decoding algorithm for horizontally u-interleaved LRPC codes is proposed and analyzed and it is shown that interleaving reduces the decoding failure rate exponentially in the interleaved order whereas the computational complexity grows linearly. Expand
Author's Reply to Comments on 'Maximum-rank array codes and their application to crisscross error correction'
  • R. Roth
  • Mathematics, Computer Science
  • IEEE Trans. Inf. Theory
  • 1992
It is shown that the dimension of such array codes must satisfy the Singleton-like bound k, which is a k-dimensional linear space of n*n matrices over F such that every nonzero matrix in C has rank mu. Expand
Nist post-quantum cryptography standardization proposal: Rank-ouroboros, lake and locker (rollo)
  • 2019.
  • 2019