Linear complementary pair of group codes over finite chain rings
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Linear complementary pair of group codes over finite chain rings Cem Güneri1 · Edgar Martínez-Moro2
· Selcen Sayıcı1
Received: 6 December 2019 / Revised: 26 May 2020 / Accepted: 17 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Linear complementary dual (LCD) codes and linear complementary pair (LCP) of codes over finite fields have been intensively studied recently due to their applications in cryptography, in the context of side channel and fault injection attacks. The security parameter for an LCP of codes (C, D) is defined as the minimum of the minimum distances d(C) and d(D ⊥ ). It has been recently shown that if C and D are both 2-sided group codes over a finite field, then C and D ⊥ are permutation equivalent. Hence the security parameter for an LCP of 2-sided group codes (C, D) is simply d(C). We extend this result to 2-sided group codes over finite chain rings. Keywords LCP of codes · Group codes · Finite chain rings · Code equivalence Mathematics Subject Classification 94B05 · 94B15 · 94B60 · 94B99
1 Introduction A pair of linear codes (C, D) of length n over a finite field Fq is called a linear complementary pair (LCP) of codes if C ∩ D = {0} and C + D = Fqn (i.e. C ⊕ D = Fqn ). In the case D = C ⊥ , C is referred as a linear complementary dual (LCD) code. LCD codes were introduced by Massey [10] in 1992. There has been a revived interest in LCD and LCP of codes due to their application in protection against side channel and fault injection attacks [1,4]. In this context, the security parameter of an LCP (C, D) is defined to
Communicated by J.-L. Kim.
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Edgar Martínez-Moro [email protected] Cem Güneri [email protected] Selcen Sayıcı [email protected]
1
Faculty of Engineering and Natural Sciences, Sabancı University, Istanbul, Turkey
2
Institute of Mathematics, University of Valladolid, Castilla, Spain
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C. Güneri et al.
be min{d(C), d(D ⊥ )}, where d(C) stands for the minimum distance of the code C. In the LCD case, this parameter is simply d(C), since D ⊥ = C. Carlet et al. [5] showed that if (C, D) is LCP, where C and D are both cyclic codes over a finite field Fq , then C is equivalent to D ⊥ . They showed that the same result holds if C and D are 2D cyclic codes, under the assumption that the length of the codes is relatively prime to the characteristic of the finite field (semisimple case). Cyclic and 2D cyclic codes are special abelian codes, which are defined as ideals of the group algebra Fq [G] for a finite abelian group G. In the case gcd(q, |G|) = 1, Güneri et al. extended this result to LCP of abelian codes in Fq [G] [6]. If G is any finite group (not necessarily abelian), a right ideal of Fq [G] is called a group code. In [3], Borello et al. obtained the most general statement for any finite group (also without a restriction on the order of the group) by showing that if (C, D) is LCP of 2-sided group codes (ideals) in Fq [G], then C and D ⊥ are permutation equivalent. Note in particular that this implies d(C) = d
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