Abstract
Side-channel attacks and fault injection attacks are nowadays important cryptanalysis methods on the implementations of block ciphers, which represent huge threats. Direct sum masking (DSM) has been proposed to protect the sensitive data stored in registers against both SCA and FIA. It uses two linear codes \({\mathcal {C}}\) and \({\mathcal {D}}\) whose sum is direct and equals \({\mathbb {F}}_q^n\). The resulting security parameter is the pair \((d({\mathcal {C}})-1,d({{\mathcal {D}}}^\perp )-1)\). For being able to protect not only the sensitive input data stored in registers against SCA and FIA but the whole algorithm (which is required at least in software applications), it is useful to change \(\mathcal C\) and \({\mathcal {D}}\) into \({\mathcal {C}}^\prime \), which has the same minimum distance as \({\mathcal {C}}\), and \({\mathcal {D}}^\prime \), which may have smaller dual distance than \({\mathcal {D}}\). Precisely, \(\mathcal D^\prime \) is the linear code obtained by appending on the right of its generator matrix the identity matrix with the same number of rows. It is then highly desired to construct linear codes \({\mathcal {D}}\) such that \(d({{\mathcal {D}}^\prime }^\perp )\) is very close to \(d({{\mathcal {D}}}^\perp )\). In such case, we say that \({\mathcal {D}}\) is almost optimally extendable (and is optimally extendable if \(d({{\mathcal {D}}^\prime }^\perp )= d({\mathcal {D}}^\perp )\)). In general, it is notoriously difficult to determine the minimum distances of the codes \({\mathcal {D}}^\perp \) and \({{\mathcal {D}}^\prime }^\perp \) simultaneously. In this paper, we mainly investigate constructions of (almost) optimally extendable linear codes from irreducible cyclic codes and from the first-order Reed–Muller codes. The minimum distances of the codes \({\mathcal {D}}, {\mathcal {D}}^\prime , \mathcal D^\perp \), and \({{\mathcal {D}}^\prime }^\perp \) are determined explicitly and their weight enumerators are also given. Furthermore, several families of optimally extendable codes are found (for the second time) among such linear codes.
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References
Bringer J., Carlet C., Chabanne H., Guilley S., Maghrebi H.: Orthogonal direct summaskinga smartcard friendly computation paradigm in a code, with builtin protection against side-channel and fault attacks. In: Naccache D., Sauveron D. (eds.) WISTP, Heraklion, LNCS, vol. 8501, pp. 40–56. Springer, Heidelberg (2014).
Carlet C., Daif A., Guilley S., Tavernier C.: Polynomial direct sum masking to protect against both SCA and FIA. To appear in the J. Cryptogr. Eng. (JCEN).
Carlet C., Guilley S.: Satatistical properties of side-channel and fault injection attacks using coding theory. Crypt. Commun. 10, 909–933 (2018).
Carlet C., Güneri C., Mesnager S., Özbudak F.: Construction of some codes suitable for both side channel and fault injection attacks. In: Budaghyan L., Rodrguez-Henrquez F. (eds.) Arithmetic of Finite Fields, WAIFI 2018, LNCS, vol. 11321, pp. 95–107. Springer, Cham (2018).
Carlet C., Güneri C., Özbudak F., Özkaya B., Solé P.: On linear complementary pairs of codes. IEEE Trans. Inf. Theory 64(10), 6583–6589 (2018).
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(4), 3010–3017 (2018).
Carlet C., Mesnager S., Tang C., Qi Y.: Euclidean and Hermitian LCD MDS codes. Des. Codes Cryptogr. 86, 2605–2618 (2018).
Carlet C., Mesnager S., Tang C., Qi Y.: New characterization and parametrization of LCD codes. IEEE Trans. Inf. Theory 65(1), 39–49 (2019).
Chen B., Liu H.: New constructions of MDS codes with complementary duals. IEEE Trans. Inf. Theory 64(8), 5776–5782 (2018).
Delsarte P.: On subfield subcodes of modified Reed-Solomon codes. IEEE Trans. Inf. Theory 21(5), 575–576 (1975).
Ding C., Yang J.: Hamming weights in irreducible cyclic codes. Discret. Math. 313(4), 434–446 (2013).
Ding C., Li C., Li N., Zhou Z.: Three-weight cyclic codes and their weight distributions. Discret. Math. 339(2), 415–427 (2016).
Dougherty S.T., Kim J.-L., Özkaya B., Sok L., Solè P.: The combinatorics of LCD codes: linear programming bound and orthogonal matrices. Int. J. Inf. Coding Theory 4(2/3), 116–128 (2017).
Golomb S.W., Gong G.: Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar. Cambridge University Press, Cambridge (2005).
Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de. Accessed 9 Mar 2019.
Güneri C., Özkaya B., Solé P.: Quasi-cyclic complementary dual codes. Finite Fields Appl. 42, 67–80 (2016).
Jin L.: Construction of MDS codes with complementary duals. IEEE Trans. Inf. Theory 63(5), 2843–2847 (2017).
Li C.: Hermitian LCD codes from cyclic codes. Des. Codes Cryptogr. 86, 2261–2278 (2018).
Li C., Yue Q., Li F.: Weight distributions of cyclic codes with respect to pairwise coprime order elements. Finite Fields Appl. 28, 94–114 (2014).
Li C., Ding C., Li S.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63(7), 4344–4356 (2017).
Li S., Li C., Ding C., Liu H.: Two Families of LCD BCH codes. IEEE Trans. Inf. Theory 63(9), 5699–5717 (2017).
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977).
Massey J.L.: Linear codes with complementary duals. Discret. Math. 106(107), 337–342 (1992).
Mesnager S., Tang C., Qi Y.: Complementary dual algebraic geometry codes. IEEE Trans. Inf. Theory 64(4), 2390–2397 (2018).
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. In: IEEE International Symposium on Hardware Oriented Security and Trust (HOST), pp. 82–87 (2015).
Shi X., Yue Q., Yang S.: New LCD MDS codes constructed from generalized Reed-Solomon codes. J. Alg. Appl. 1950150 (2018).
van Lint J.H.: Introduction to Coding Theory, 3rd edn. Springer, New York (1999).
Wu Y., Yue Q., Zhu X., Yang S.: Weight enumerators of reducible cyclic codes and their dual codes. Discret. Math. 342(3), 671–682 (2019).
Yan H., Liu H., Li C., Yang S.: Parameters of LCD BCH codes with two lengths. Adv. Math. Commun. 12(3), 579–594 (2018).
Yang S., Yao Z.: Complete weight enumerators of a class of linear codes. Discret. Math. 340(4), 729–739 (2017).
Yang S., Yao Z., Zhao C.: The weight enumerator of the duals of a class of cyclic codes with three zeros. Appl. Algebra Eng. Commun. Comput. 26(4), 347–367 (2015).
Yang S., Yao Z., Zhao C.: The weight distributions of two classes of \(p\)-ary cyclic codes with few weights. Finite Fields Appl. 44, 76–91 (2017).
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The work was supported by the National Natural Science Foundation of China under Grant 11701179, the Shanghai Chenguang Program under Grant 18CG22, the Shanghai Sailing Program under Grant 17YF1404300, the Foundation of State Key Laboratory of Integrated Services Networks under Grant ISN20-02, and the SECODE project in the scope of the CHIST-ERA Program.
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Carlet, C., Li, C. & Mesnager, S. Some (almost) optimally extendable linear codes. Des. Codes Cryptogr. 87, 2813–2834 (2019). https://doi.org/10.1007/s10623-019-00652-7
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DOI: https://doi.org/10.1007/s10623-019-00652-7