Abstract
A pair of linear codes (C, D) of length n over \(\mathbb {F}_q\) is called a linear complementary pair (LCP) if their direct sum yields the full space \(\mathbb {F}_q^n\). By a result of Carlet et al. (2019), the best security parameters of binary LCPs of codes are left open. Motivated by this, we study binary LCPs of codes. We describe a sufficient condition for binary LCPs of codes which are not optimal. We carry out an exhaustive search to determine the best security parameters for binary LCPs of codes up to length 18. We also obtain results on optimal binary LCPs of codes for infinitely many parameters. For any \(k\ge 2\) and length n congruent to 0 or 1 mod \((2^k-1)\), we prove that binary [n, k] LCPs of codes are optimal. Binary LCPs of codes of dimensions 2, 3, and 4 are also optimal for all lengths except for two instances, when \((n,k)=(4,3)\) and (8, 4). We provide explicit constructions of these infinite families of optimal LCPs. Our results also indicate that many security parameters coming from binary LCPs of codes exceed those from binary LCD codes by 1 or 2.
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References
Araya, M., Harada, M.: On the minimum weights of binary linear complementary dual codes. Cryptogr. Commun. 12, 285–300 (2020)
Araya, M., Harada, M., Saito, K.: Characterization and classification of optimal LCD codes. Des. Codes Cryptogr. 89, 617–640 (2021)
Araya, M., Harada, M., Saito, K.: On the minimum weights of binary LCD codes and ternary LCD codes. Finite Fields Appl. 76, 101925 (2021)
Borello, M., de la Cruz, J., Willems, W.: A note on linear complementary pairs of group codes. Discrete Math. 343, 111905 (2020)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997)
Bouyuklieva, S.: Optimal binary LCD codes. Des. Codes Cryptogr. 89, 2445–2461 (2021)
Bringer, J., Carlet, C., Chabanne, H., Guilley, S., Maghrebi, H.: Orthogonal direct sum masking: A smartcard friendly computation paradigm in a code, with builtin protection against side-channel and fault attacks, WISTP, Heraklion, Crete, LNCS, vol. 8501, pp. 40–56. Springer, Berlin, Heidelberg (2014)
Carlet, C., Mesnager, S., Tang, C., Qi, Y.: On \(\sigma\)-LCD codes. IEEE Trans. Inf. Theory. 65, 1694–1704 (2019)
Carlet, C., Mesnager, S., Tang, C., Qi, Y., Pellikaan, R.: Linear codes over \(\mathbb{F}_q\) are equivalent to LCD codes for \(q>3\). IEEE Trans. Inf. Theory. 64, 3010–3017 (2018)
Carlet, C., Güneri, C., Özbudak, F., Özkaya, B., Solé, P.: On linear complementary pairs of codes. IEEE Trans. Inf. Theory. 64, 6583–6589 (2018)
Dougherty, S.T., Kim, J.-L., Özkaya, B., Sok, L., Solé, P.: The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices. Int. J. Information and Coding Theory. 4, 116–128 (2017)
Galvez, L., Kim, J.-L., Lee, N., Roe, Y.G., Won, B.-S.: Some bounds on binary LCD codes. Cryptogr. Commun. 10, 719–728 (2018)
Güneri, C., Özkaya, B., Sayıcı, S.: On linear complementary pair of \(nD\) cyclic codes. IEEE Commun. Lett. 22, 2404–2406 (2018)
Harada, M.: Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes. Des. Codes Cryptogr. 89, 2295–2312 (2021)
Harada, M., Saito, K.: Binary linear complemetary dual codes. Cryptogr. Commun. 11, 677–696 (2019)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Jaffe, D.: Optimal binary linear codes of length \(\le 30\). Discrete Math. 223, 135–155 (2000)
Ngo, X.T., Bhasin, S., Danger, J.-L., Guilley, S., Najm, Z.: Linear complementary dual code improvement to strengthen encoded circuit against hardware Trojan horses, pp. 82–87. IEEE International Symposium on Hardware Oriented Security and Trust (HOST), Washington (2015)
Acknowledgements
The authors are supported by a bilateral cooperation program between Korea and Turkey. Choi and Kim are supported by NRF under the project code 2020K2A9A1A06108874. Güneri and Özbudak are supported by TÜBİTAK under the project code 120N932.
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Choi, WH., Güneri, C., Kim, JL. et al. Optimal Binary Linear Complementary Pairs of Codes. Cryptogr. Commun. 15, 469–486 (2023). https://doi.org/10.1007/s12095-022-00612-4
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DOI: https://doi.org/10.1007/s12095-022-00612-4