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Study of cyclic codes from low differentially uniform functions and its consequences

Published: 01 July 2024 Publication History

Abstract

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles (Ding and Zhou 2014 [11]) and (Ding, 2013 [7]), Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall answer some open problems presented in the literature. The first concerns Open Problem 1, proposed by Ding and Zhou (2014) [11]. The others are Open Problems 5.16 and 5.25, raised by Ding (2013) [7].

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Published In

cover image Discrete Mathematics
Discrete Mathematics  Volume 347, Issue 7
Jul 2024
658 pages

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Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 July 2024

Author Tags

  1. Cyclic code
  2. Linear span
  3. Sequence
  4. Differential uniformity
  5. Differential cryptanalysis
  6. APN function

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