On the rank decoding problem over finite principal ideal rings

Hervé Talé Kalachi; Hermann Tchatchiem Kamche

doi:10.3934/amc.2023003

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2024, Volume 18, Issue 5: 1492-1513. Doi: 10.3934/amc.2023003

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On the rank decoding problem over finite principal ideal rings

Hervé Talé Kalachi^1, , and
Hermann Tchatchiem Kamche^2,

1.
National Advanced School of Engineering of Yaounde, University of Yaounde I, Cameroon
2.
Institute of mathematics, University of Neuchatel, Switzerland

^*Corresponding author: Hervé Talé Kalachi
^*Corresponding author: Hervé Talé Kalachi

Received: August 2022

Early access: February 2023

Published: October 2024

The first author is supported by the UNESCO-TWAS and the German Federal Ministry of Education and Research (BMBF) under the SG-NAPI grant number 4500454079. The Second author is supported by the Swiss Government Excellence Scholarship (ESKAS No. 2022.0689).

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Abstract

The rank decoding problem has been the subject of much attention in this last decade. This problem, which is at the base of the security of public-key cryptosystems based on rank metric codes, is traditionally studied over finite fields. But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings.

In this paper, we show that solving the rank decoding problem over finite principal ideal rings is at least as hard as the rank decoding problem over finite fields. We also show that computing the minimum rank distance for linear codes over finite principal ideal rings is equivalent to the same problem for linear codes over finite fields. Finally, we provide combinatorial type algorithms for solving the rank decoding problem over finite chain rings together with their average complexities.

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Mathematics Subject Classification: Primary: 11T71; Secondary: 13M05.

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