CIS Codes Research Articles

Construction of single-deletion-correcting DNA codes using CIS codes

We find a method for constructing DNA codes with single-deletion-correcting capability. We first present an explicit algorithm for the construction of the q-ary single-deletion-correcting codes (abbreviated as SDC codes) using a class of the complementary information set codes (abbreviated as CIS codes), where q is a power of a prime. We then show that the encoding/decoding scheme of the CIS codes with single-deletion-correcting capability has a simple deterministic algorithm. Finally, applying our algorithm to the generated DNA codes with appropriate modification, we obtain the DNA codes with single-deletion-correcting capability. We present some various examples of such DNA codes, and we also obtain some lower bounds on the maximum size of the single-deletion-correcting DNA codes.

CIS codes over ${\mathbb F}_4$

We study the complementary information set codes (for short, CIS codes) over ${\mathbb F}_4$. They are strongly connected to correlation-immune functions over ${\mathbb F}_4$. Also the class of CIS codes includes the self-dual codes. We find a construction method of CIS codes over ${\mathbb F}_4$ and a criterion for checking equivalence of CIS codes over ${\mathbb F}_4$. We complete the classification of all inequivalent CIS codes of length up to $8$ over ${\mathbb F}_4$.

Long quasi-polycyclic <inline-formula><tex-math id="M1">\begin{document}$t-$\end{document}</tex-math></inline-formula> CIS codes

We study complementary information set codes of length $tn$ and dimension $n$ of order $t$ called ($t-$CIS code for short). Quasi-cyclic (QC) and quasi-twisted (QT) $t$-CIS codes are enumerated by using their concatenated structure. Asymptotic existence results are derived for one-generator and fixed co-index QC and QT codes depending on Artin's primitive root conjecture. This shows that there are infinite families of rate $1/t$ long QC and QT $t$-CIS codes with relative distance satisfying a modified Varshamov-Gilbert bound. Similar results are defined for the new and more general class of quasi-polycyclic codes introduced recently by Berger and Amrani.

Formula omitted]-CIS codes over [formula omitted] and orthogonal arrays

We first show that orthogonal arrays over GF(p) can be explicitly constructed from t-CIS codes over GF(p), where t-CIS codes are CIS codes of order t≥2. With this motivation, we are interested in developing methods of constructing t-CIS codes over GF(p). We present two types of constructions; the first one is a “t-extension method” which is finding t-CIS codes over GF(p) of length tn from given (t−1)-CIS codes over GF(p) of length (t−1)n for t>2, and the second one is a “building-up type construction” which is finding t-CIS codes over GF(p) of length t(n+1) from given t-CIS codes over GF(p) of length tn. Furthermore, we find a criterion for checking equivalence of t-CIS codes over GF(p). We find inequivalent t-CIS codes over GF(p) of length n for t=3,4, n=9,12,16, and p=3,5,7 using our construction and criterion, and corresponding orthogonal arrays are found. We point out that 171t-CIS codes we found are optimal codes.

Complementary information set codes over GF(p)

Complementary information set codes (CIS codes) over a finite field GF(p) are closely connected to correlation-immune functions over GF(p), which are important cryptographic functions, where p is an odd prime. Using our CIS codes over GF(p) of minimum weight $$d+1$$d+1, we can obtain p-ary correlation-immune function of strength d. We find an efficient method for constructing CIS codes over GF(p). We also find a criterion for checking equivalence of CIS codes over GF(p). We complete the classification of all inequivalent CIS codes over GF(p) of lengths up to 8 for $$p = 3,5,7$$p=3,5,7 using our construction and criterion. We also find their weight enumerators and the order of their automorphism groups. The class of CIS codes over GF(p) includes self-dual codes over GF(p) as its subclass, and some CIS codes are formally self-dual codes as well; we sort out our classification results. Furthermore, we show that long CIS codes over GF(p) meet the Gilbert---Vashamov bound.

Higher-Order CIS Codes

We introduce complementary information set codes of higher order. A binary linear code of length tk 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -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 [tk, 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.

The classification of complementary information set codes of lengths $14$ and $16$

In the paper ``A new class of codes for Boolean masking of cryptographic computations,'' Carlet, Gaborit, Kim, and Sole defined a new class of rate one-half binary codes called complementary information set (or CIS) codes. The authors then classified all CIS codes of length less than or equal to 12. CIS codes have relations to classical Coding Theory as they are a generali-zation of self-dual codes. As stated in the paper, CIS codes also have important practical applications as they may improve the cost of masking cryptographic algorithms against side channel attacks. In this paper, we give a complete classification result for length 14 CIS codes using an equivalence relation on $GL(n,\mathbb{F}_2)$. We also give a new classification for all binary $[16,8,3]$ and $[16,8,4]$ codes. We then complete the classification for length 16 CIS codes and give additional classifications for optimal CIS codes of lengths 20 and 26.

A New Class of Codes for Boolean Masking of Cryptographic Computations

We introduce a new class of rate one-half binary codes: {\bf complementary information set codes.} A binary linear code of length $2n$ and dimension $n$ is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune Boolean functions of use in the security of hardware implementations of cryptographic primitives. Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks. In this paper we investigate this new class of codes: we give optimal or best known CIS codes of length $<132.$ We derive general constructions based on cyclic codes and on double circulant codes. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths $\le 12$ by the building up construction. Some nonlinear permutations are constructed by using $\Z_4$-codes, based on the notion of dual distance of an unrestricted code.