Higher-Order CIS Codes

초록

We introduce complementary information set codes of higher order. A binary linear code of length t k and dimension k is called a complementary information set code of order t (t-CIS code for short) if it has t pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a t-CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order 3 is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length = 12 is given. Nonlinear codes better than linear codes are derived by taking binary images of Z(4)-codes. A general algorithm based on Edmonds' basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate 1/t, it either provides t disjoint information sets or proves that the code is not t-CIS. Using this algorithm, all optimal or best known [t k, k] codes, where t = 3, 4,..., 256 and 1 <= k <= [256/t] are shown to be t-CIS for all such k and t, except for t = 3 with k = 44 and t = 4 with k = 37.

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