LCD Codes as a Counter-Measure for Relevant Security Threats: A Survey

Linear codes with certain special properties have received renewed attention in recent years due to their practical applications. Among them, binary linear complementary dual (LCD) codes play an important role in implementations against side-channel attacks and fault injection attacks. Self-orthogonal codes can be used to construct quantum codes. In this paper, four classes of binary linear codes are constructed via a generic construction which has been intensively investigated in the past decade. Simple characterizations of these linear codes to be LCD or self-orthogonal are presented. Resultantly, infinite families of binary LCD codes and self-orthogonal codes are obtained. Infinite families of binary LCD codes from the duals of these four classes of linear codes are produced. Many LCD codes and self-orthogonal codes obtained in this paper are optimal or almost optimal in the sense that they meet certain bounds on general linear codes. In addition, the weight distributions of two sub-families of the proposed linear codes are established in terms of Krawtchouk polynomials.

Linear complementary dual(LCD) codes are being utilized to protect cryptographic implementations from side-channel and fault attacks. In this work, LCD codes over the ring Z 4 are examined. It is observed that an LCD cyclic code over Z 4 of odd length is always reversible. However, the converse of this result does not hold. A suitable example to validate this statement is presented. Further, a necessary and sufficient condition for a reversible cyclic code of odd length to be an LCD code is obtained. Finally, it is found that among cyclic codes of length 2 k , no non-trivial LCD code exists.

For an odd prime p, we obtain algebraic structure of cyclic codes of length n over a finite commutative non-chain semi-local ring \(\mathfrak {R}=\mathbb {F}_{p}[u,v,w]/\langle u^{2}-u,v^{2}-1,w^2-1,uv-vu,vw-wv,wu-uw\rangle \). These codes of length n can be viewed as principal ideals of the quotient ring \(\mathfrak {R}[x]/\langle x^n-1\rangle \). Here, a Gray map is defined to obtain p-ary quasi-cyclic codes of length 8n with index 8 as \(\mathbb {F}_p\)-images of cyclic codes of length n over \(\mathfrak {R}\). Also, we present necessary and sufficient conditions for a cyclic code to be an LCD (linear complementary dual) code over \(\mathfrak {R}\). Moreover, it is shown that the Gray image of an LCD code of length n over \(\mathfrak {R}\) is an LCD code of length 8n over \(\mathbb {F}_{p}\). Finally, a few non-trivial examples are given in support of our derived results.KeywordsCyclic codeNon-chain ringSemi-local ringGray mapLCD code

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study binary linear complementary dual $[n,k]$ codes with the largest minimum weight among all binary linear complementary dual $[n,k]$ codes. We characterize binary linear complementary dual codes with the largest minimum weight for small dimensions. A complete classification of binary linear complementary dual $[n,k]$ codes with the largest minimum weight is also given for $1 \le k \le n \le 16$.

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study the largest minimum weight $d(n,k)$ among all binary linear complementary dual $[n,k]$ codes. We determine $d(n,4)$ for $n \equiv 2,3,4,5,6,9,10,13 \pmod{15}$, and $d(n,5)$ for $n \equiv 3,4,5,7,11,19,20,22,26 \pmod{31}$. Combined with known results, the values $d(n,k)$ are also determined for $n \le 24$.

A q-aryl linear code C is called a linear complementary dual (LCD) code if it meets its dual trivially. Binary LCD codes play a significant role for their advantage of low complexity for implementations against side-channel attacks. In this paper, we construct some LCD codes over 2 F by puncturing, shortening and extending. Then using some cyclic codes, we propose several series of optimal LCD codes with d=4, 5, 6, respectively.

Maximum distance separable (MDS) codes and near MDS (NMDS) codes play a fundamental role in coding theory due to their maximal error-correcting capabilities and rich algebraic structure. This paper constructs a new class of linear codes by twice extending TGRS codes, denoted as $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $. First, we establish the necessary and sufficient conditions for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ to be MDS. By analyzing the AMDS properties of both $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and their duals $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty)^{\bot} $, we determine the necessary and sufficient conditions for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ to be NMDS. Besides, we provide the form of the parity check matrix for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and characterize the weight distributions of both the codes $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and their duals. Additionally, using the Schur method, the non-generalized Reed-Solomon (non-GRS) properties are proven. Finally, several classes of linear complementary dual (LCD) codes are constructed.

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure a Boolean function's resistance to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. Some of the binary LCD codes presented in this paper are optimal. These binary LCD codes have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems.

In this chapter, we describe self-dual codes over Frobenius rings. We give constructions of self-dual codes over any Frobenius ring. We describe connections to unimodular lattices, binary self-dual codes and to designs. We also describe linear complementary dual codes and make a new definition of a broad generalization encompassing both self-dual and linear complementary codes.

A Brillouin optical time-domain analysis (BOTDA) sensor that combines the conventional complementary coding with the pulse prepump technique for high-accuracy and long-range distributed sensing is implemented and analyzed. The employment of the complementary coding provides an enhanced signal-to-noise ratio (SNR) of the sensing system and an extended sensing distance, and the measurement time is also reduced compared with a BOTDA sensor using linear coding. The combination of pulse prepump technique enables the establishment of a preactivated acoustic field in each pump pulse of the complementary codeword, which ensures measurements of high spatial resolution and high frequency accuracy. The feasibility of the prepumped complementary coding is analyzed theoretically and experimentally. The experiments are carried out beyond 50-km single-mode fiber, and experimental results show the capabilities of the proposed scheme to achieve 1-m spatial resolution with temperature and strain resolutions equal to ∼1.6°C and ∼32 μϵ, and 2-m spatial resolution with temperature and strain resolutions equal to ∼0.3°C and ∼6 μϵ, respectively. A longer sensing distance with the same spatial resolution and measurement accuracy can be achieved through increasing the code length of the prepumped complementary code.

The $\ell$-Galois hull $h_{\ell}(C)$ of an $[n,k]$ linear code $C$ over a finite field $\mathbb{F}_q$ is the intersection of $C$ and $C^{{\bot}_{\ell}}$, where $C^{\bot_{\ell}}$ denotes the $\ell$-Galois dual of $C$ which introduced by Fan and Zhang (2017). The $\ell$- Galois LCD code is a linear code $C$ with $h_{\ell}(C) = 0$. In this paper, we show that the dimension of the $\ell$-Galois hull of a linear code is invariant under permutation equivalence and we provide a method to calculate the dimension of the $\ell$-Galois hull by the generator matrix of the code. Moreover, we obtain that the dimension of the $\ell$-Galois hulls of ternary codes are also invariant under monomial equivalence. %The dimension of $l$-Galois hull of a code is not invariant under monomial equivalence if $q>4$. We show that every $[n,k]$ linear code over $\mathbb F_{q}$ is monomial equivalent to an $\ell$-Galois LCD code for any $q>4$. We conclude that if there exists an $[n,k]$ linear code over $\mathbb F_{q}$ for any $q>4$, then there exists an $\ell$-Galois LCD code with the same parameters for any $0\le \ell\le e-1$, where $q=p^e$ for some prime $p$. As an application, we characterize the $\ell$-Galois hull of matrix product codes over finite fields.

This paper studies linear coding (LC) techniques in the setting of computing functions of correlated memoryless sources. Instead of linear mappings over finite fields, we consider using linear mappings over finite rings as encoders. It is shown that generally the region c×R, where c ≥ 1 is a constant and R is the Slepian-Wolf (SW) region, is achievable with LC over ring (LCoR) when the function to compute is the identity function. c = 1 if the ring used is a field. Hence, LCoR could be suboptimal in terms of achieving the best coding rates (the SW region) for computing the identity function. In spite of that, the ring version shows several advantages. It is demonstrated that there exists a function that is neither linear nor can be linearized over any finite field. Thus, LC over field (LCoF) does not apply directly for computing such a function unless the polynomial approach [1], [2] is used. On the contrary, such a function is linear over some ring. Using LCoR, an achievable region containing the SW region can be obtained for computing this function. In addition, the alphabet sizes of the encoders are strictly smaller than using LCoF. More interestingly, LCoF is not useful if some special requirement is imposed.

Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f. Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polynomials. Munemasa (Finite Fields Appl 4(3):252---260, 1998) uses trinomials over $${\mathbb{F}_2}$$ to construct orthogonal arrays of guaranteed strength 2 (and almost strength 3). That result was extended by Dewar et al. (Des Codes Cryptogr 45:1---17, 2007) to construct orthogonal arrays of guaranteed strength 3 by considering divisibility of trinomials by pentanomials over $${\mathbb{F}_2}$$ . Here we first simplify the requirement in Munemasa's approach that the characteristic polynomial of the sequence must be primitive: we show that the method applies even to the much broader class of polynomials with no repeated roots. Then we give characterizations of divisibility for binomials and trinomials over $${\mathbb{F}_3}$$ . Some of our results apply to any finite field $${\mathbb{F}_q}$$ with q elements.

The structure of binary LCD codes invariant under a permutation of odd prime order p is described. A systematic construction of such codes when 2 is a multiplicative root modulo p is presented. As an example, binary linear codes with complementary dual (or LCD codes) whose automorphism groups have orders divisible by 5 are constructed.

Grassmann codes are linear codes associated with the Grassmann variety $G(\ell,m)$ of $\ell$-dimensional subspaces of an $m$ dimensional vector space $\mathbb{F}_{q}^{m}.$ They were studied by Nogin for general $q.$ These codes are conveniently described using the correspondence between non-degenerate $[n,k]_{q}$ linear codes on one hand and non-degenerate $[n,k]$ projective systems on the other hand. A non-degenerate $[n,k]$ projective system is simply a collection of $n$ points in projective space $\mathbb{P}^{k-1}$ satisfying the condition that no hyperplane of $\mathbb{P}^{k-1}$ contains all the $n$ points under consideration. In this paper we will determine the weight of linear codes $C(3,8)$ associated with Grassmann varieties $G(3,8)$ over an arbitrary finite field $\mathbb{F}_{q}$. We use a formula for the weight of a codeword of $C(3,8)$, in terms of the cardinalities certain varieties associated with alternating trilinear forms on $\mathbb{F}_{q}^{8}.$ For $m=6$ and $7,$ the weight spectrum of $C(3,m)$ associated with $G(3,m),$ have been fully determined by Kaipa K.V, Pillai H.K and Nogin Y. A classification of trivectors depends essentially on the dimension $n$ of the base space. For $n\leq 8 $ there exist only finitely many trivector classes under the action of the general linear group $GL(n).$ The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from $\mathbb{\bar{F}}$ to $\mathbb{F}.$ This program is partially determined by Noui L. and Midoune N. and the classification of trilinear alternating forms on a vector space of dimension $8$ over a finite field $\mathbb{F}_{q}$ of characteristic other than $2$ and $3$ was solved by Noui L. and Midoune N. We describe the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$, where char $\mathbb{F}_{q}\neq 3.$ This fact we use to determine the weight of the $\mathbb{F}_{q}$-forms.