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The security parameter for a linear complementary pair (C, D) of codes is defined to be the minimum of the minimum distances d(C) and d(D ⊥ ). Recently, Carlet et al. showed that if C and D are both cyclic or both two-dimensional (2D) cyclic linear complementary pair of codes, then C and D ⊥ are equivalent codes. Hence, the security parameter for cyclic and 2D cyclic linear complementary pair of codes is simply d(C). We extend this result to nD cyclic linear complementary pair of codes. The proof of Carlet et al. for the 2D cyclic case is based on the trace representation of the codes, which is technical and nontrivial to generalize. Our proof for the generalization is based on the zero sets of the ideals corresponding to nD cyclic codes.

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 abelian codes over a finite field Fq, and the length of the codes is relatively prime to q, then C and D ⊥ are equivalent. Hence the security parameter for an LCP of abelian codes (C, D) is simply d(C). In this work, we first extend this result to the non-semisimple case, i.e. the code length is divisible by the characteristic of the field of definition. Then we use the result over the finite fields to prove the same fact for an LCP of abelian codes over any finite chain ring.

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