Showing papers in "Cryptography and Communications in 2020"

Recent results and problems on constructions of linear codes from cryptographic functions

Abstract: Linear codes have a wide range of applications in the data storage systems, communication systems, consumer electronics products since their algebraic structure can be analyzed and they are easy to implement in hardware. How to construct linear codes with excellent properties to meet the demands of practical systems becomes a research topic, and it is an efficient way to construct linear codes from cryptographic functions. In this paper, we will introduce some methods to construct linear codes by using cryptographic functions over finite fields and present some recent results and problems in this area.

New constructions of involutions over finite fields

Abstract: Involutions over finite fields are permutations whose compositional inverses are themselves. Involutions especially over $ \mathbb {F}_{q} $ with q is even have been used in many applications, including cryptography and coding theory. The explicit study of involutions (including their fixed points) has started with the paper (Charpin et al. IEEE Trans. Inf. Theory, 62(4), 2266–2276 2016) for binary fields and since then a lot of attention had been made in this direction following it; see for example, Charpin et al. (2016), Coulter and Mesnager (IEEE Trans. Inf. Theory, 64(4), 2979–2986, 2018), Fu and Feng (2017), Wang (Finite Fields Appl., 45, 422–427, 2017) and Zheng et al. (2019). In this paper, we study constructions of involutions over finite fields by proposing an involutory version of the AGW Criterion. We demonstrate our general construction method by considering polynomials of different forms. First, in the multiplicative case, we present some necessary conditions of f(x) = xrh(xs) over $\mathbb {F}_{q}$ to be involutory on $\mathbb {F}_{q}$, where s∣(q − 1). Based on this, we provide three explicit classes of involutions of the form xrh(xq− 1) over $\mathbb {F}_{q^{2}}$. Recently, Zheng et al. (Finite Fields Appl., 56, 1–16 2019) found an equivalent relationship between permutation polynomials of $g(x)^{q^{i}} - g(x) + cx +(1-c)\delta $ and $g\left (x^{q^{i}} - x + \delta \right ) +c x$. The other part work of this paper is to consider the involutory property of these two classes of permutation polynomials, which fall into the additive case of the AGW criterion. On one hand, we reveal the relationship of being involutory between the form $ g(x)^{q^{i}} - g(x) + cx +(1-c)\delta $ and the form $ g\left (x^{q^{i}} - x + \delta \right ) +c x $ over $ \mathbb {F}_{q^{m}} $ ; on the other hand, the compositional inverses of permutation polynomials of the form $ g\left (x^{q^{i}} - x + \delta \right ) + cx $ over $ \mathbb {F}_{q^{m}} $ are computed, where $ \delta \in \mathbb {F}_{q^{m}} $, $ g(x) \in \mathbb {F}_{q^{m}}[x] $ and integers m, i satisfy 1 ≤ i ≤ m − 1. In addition, a class of involutions of the form $ g\left (x^{q^{i}} - x + \delta \right ) + cx $ is constructed. Finally, we study the fixed points of constructed involutions and compute the number of all involutions with any given number of fixed points over $ \mathbb {F}_{q} $.

On the boomerang uniformity of some permutation polynomials

Abstract: The boomerang attack, introduced by Wagner in 1999, is a cryptanalysis technique against block ciphers based on differential cryptanalysis. In particular it takes into consideration two differentials, one for the upper part of the cipher and one for the lower part, and it exploits the dependency of these two differentials. At Eurocrypt’18, Cid et al. introduced a new tool, called the Boomerang Connectivity Table (BCT), that permits to simplify this analysis. Next, Boura and Canteaut introduced an important parameter for cryptographic S-boxes called boomerang uniformity, that is the maximum value in the BCT. Very recently, the boomerang uniformity of some classes of permutations (in particular quadratic functions) have been studied by Li, Qu, Sun and Li, and by Mesnager, Tang and Xiong. In this paper we further study the boomerang uniformity of some non-quadratic differentially 4-uniform functions. In particular, we consider the case of the Bracken-Leander cubic function and three classes of 4-uniform functions constructed by Li, Wang and Yu, obtained from modifying the inverse functions.

Several new infinite families of bent functions via second order derivatives

Abstract: Inspired by a recent work of Tang et al. on constructing bent functions [14, IEEE TIT, 63(1): 6149-6157, 2017], we introduce a property (Pτ) of any Boolean function that its second order derivatives vanish at any direction (ui,uj) for some τ-subset {u1,…,uτ} of $\mathbb {F}_{2^{n}}$ , and then establish a link between this property and the construction of Tang et al. (IEEE Trans. Inf. Theory 63(10), 6149–6157 2017). It enables us to find more bent functions efficiently. We construct (at least) five new infinite families of bent functions from some known functions: the Gold’s bent functions and some quadratic non-monomial bent functions, Leander’s monomial bent functions, Canteaut-Charpin-Kyureghyan’s monomial bent functions, and the Maiorana-McFarland class of bent functions, respectively. Our result generalizes some recent works on bent functions. We also provide the corresponding dual functions in all our constructions except the quadratic non-monomial one. It also turns out that we can get new bent functions outside the Maiorana-McFarland completed class.

A class of constacyclic BCH codes

Abstract: Constacyclic codes are a subclass of linear codes and have been well studied. Constacyclic BCH codes are a family of constacyclic codes and contain BCH codes as a subclass. Compared with the in-depth study of BCH codes, there are relatively little study on constacyclic BCH codes. The objective of this paper is to determine the dimension and minimum distance of a class of q-ary constacyclic BCH codes of length $\frac {q^{m}-1}{q-1}$ with designed distances $\delta _{i}=q^{m-1}-\frac {q^{\lfloor \frac {m-3}2 \rfloor +i }-1}{q-1}$ for $1\leq i\leq \min \limits \{\lceil \frac {m+1}2 \rceil -\lfloor \frac {m}{q+1} \rfloor , \lceil \frac {m-1}2 \rceil \}$. As will be seen, some of these codes are optimal.

Quadruple bordered constructions of self-dual codes from group rings

Abstract: In this paper, we introduce a new bordered construction for self-dual codes using group rings. We consider constructions over the binary field, the family of rings Rk and the ring $\mathbb {F}_{4}+u\mathbb {F}_{4}$. We use groups of order 4, 12 and 20. We construct some extremal self-dual codes and non-extremal self-dual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal self-dual codes of length 68.

The 2-adic complexity of a class of binary sequences with optimal autocorrelation magnitude

Abstract: Recently, a class of binary sequences with optimal autocorrelation magnitude has been presented by Su et al. based on Ding-Helleseth-Lam sequences and interleaving technique (Designs, Codes and Cryptography 86, 1329–1338, 2018). The linear complexity of this class of sequences has been proved to be large enough to resist the B-M Algorithm by Fan (Designs, Codes and Cryptography 86, 2441–2450, 2018). In this paper, we study the 2-adic complexities of these sequences with period 4p and show they are no less than 2p, i.e., its 2-adic complexity is large enough to resist the Rational Approximation Algorithm.

The subfield codes of several classes of linear codes

Abstract: Let $\mathbb {F}_{2^{m}}$ be the finite field with 2m elements, where m is a positive integer. Recently, Heng and Ding in (Finite Fields Appl. 56:308–331, 2019) studied the subfield codes of two families of hyperovel codes and determined the weight distribution of the linear code $$ \mathcal{C}_{a,b}=\left\{((\text{Tr}_{1}^{m}(a f(x)+bx)+c)_{x \in \mathbb{F}_{2^{m}}}, \text{Tr}_{1}^{m}(a), \text{Tr}_{1}^{m}(b)) : a,b \in \mathbb{F}_{2^{m}}, c \in \mathbb{F}_{2}\right\}, $$ for f(x) = x2 and f(x) = x6 with odd m. Let v2(⋅) denote the 2-adic order function. This paper investigates more subfield codes of linear codes and obtains the weight distribution of $\mathcal {C}_{a,b}$ for $f(x)=x^{2^{i}+2^{j}}$ , where i, j are nonnegative integers such that v2(m) ≤ v2(i − j)(i ≥ j). In addition to this, we further investigate the punctured code of $\mathcal {C}_{a,b}$ as follows: $$ \mathcal{C}_{a}=\left\{((\text{Tr}_{1}^{m}(a x^{2^{i}+2^{j}}+bx)+c)_{x \in \mathbb{F}_{2^{m}}}, \text{Tr}_{1}^{m}(a)) : a,b \in \mathbb{F}_{2^{m}}, c \in \mathbb{F}_{2}\right\}, $$ and determine its weight distribution for any nonnegative integers i, j. The parameters of these binary linear codes are new in most cases. Some of the codes and their duals obtained are optimal or almost optimal.

On self-dual and LCD double circulant and double negacirculant codes over $\mathbb {F}_{q}+u\mathbb {F}_{q}$

Abstract: Double circulant codes of length 2n over the non-local ring $R=\mathbb {F}_{q}+u\mathbb {F}_{q}, u^{2}=u,$ are studied when q is an odd prime power, and − 1 is a square in $\mathbb {F}_{q}$. Double negacirculant codes of length 2n are studied over R when n is even, and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over $\mathbb {F}_{q}$ are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below for n →∞. The parameters of examples of modest lengths are computed. Several such codes are optimal.

Combinatorial t-designs from special functions

Abstract: A special function is a function either of special form or with a special property. Special functions have interesting applications in coding theory and combinatorial t-designs. The main objective of this paper is to survey t-designs constructed from special functions, including quadratic functions, almost perfect nonlinear functions, almost bent functions, bent functions, bent vectorial functions, and planar functions. These combinatorial designs are not constructed directly from such functions, but come from linear codes which are constructed with such functions. As a byproduct, this paper also surveys linear codes from certain special functions.

On decoding additive generalized twisted Gabidulin codes

Abstract: In this paper, we consider an interpolation-based decoding algorithm for a large family of maximum rank distance codes, known as the additive generalized twisted Gabidulin codes, over the finite field $\mathbb {F}_{q^{n}}$ for any prime power q. This paper extends the work of the conference paper Li and Kadir (2019) presented at the International Workshop on Coding and Cryptography 2019, which decoded these codes over finite fields in characteristic two.

On relations between CCZ- and EA-equivalences

Abstract: In the present paper we introduce some sufficient conditions and a procedure for checking whether, for a given function, CCZ-equivalence is more general than EA-equivalence together with taking inverses of permutations. It is known from Budaghyan et al. (IEEE Trans. Inf. Theory 52.3, 1141–1152 2006; Finite Fields Appl. 15(2), 150–159 2009) that for quadratic APN functions (both monomial and polynomial cases) CCZ-equivalence is more general. We prove hereby that for non-quadratic APN functions CCZ-equivalence can be more general (by studying the only known APN function which is CCZ-inequivalent to both power functions and quadratics). On the contrary, we prove that for power non-Gold APN functions, CCZ equivalence coincides with EA-equivalence and inverse transformation for n ≤ 8. We conjecture that this is true for any n.

Some new bounds on LCD codes over finite fields

Abstract: In this paper, we show that LCD codes are not equivalent to non-LCD codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD(n,2) over $\mathbb {F}_{3}$ and $\mathbb {F}_{4}$ , where LD(n,2) := max{d∣thereexsitsan [n,2, d] LCD $ code~ over~ \mathbb {F}_{q}\}$ . We study the bound of LCD codes over $\mathbb {F}_{q}$ and generalize a conjecture proposed by Galvez et al. about the minimum distance of binary LCD codes.

$\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}$-additive cyclic codes are asymptotically good

Abstract: We construct a class of $\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}$-additive cyclic codes generated by pairs of polynomials, where p is a prime number. The generator matrix of this class of codes is obtained. By establishing the relationship between the random $\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}$-additive cyclic code and random quasi-cyclic code of index 2 over $\mathbb {Z}_{p}$, the asymptotic properties of the rates and relative distances of this class of codes are studied. As a consequence, we prove that $\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}$-additive cyclic codes are asymptotically good since the asymptotic GV-bound at $\frac {1+p^{s-1}}{2}\delta $ is greater than $\frac {1}{2}$, the relative distance of the code is convergent to δ, while the rate is convergent to $\frac {1}{1+p^{s-1}}$ for $0< \delta < \frac {1}{1+p^{s-1}}$.

On the EA-classes of known APN functions in small dimensions

Abstract: Recently Budaghyan et al. (Cryptogr. Commun. 12, 85–100, 2020) introduced a procedure for investigating if CCZ-equivalence can be more general than EA-equivalence together with inverse transformation (when applicable). In this paper, we show that it is possible to use this procedure for classifying, up to EA-equivalence, all known APN functions in dimension 6. We also give some discussion for dimension 7, 8 and 9. In particular, in these cases it is possible to give an upper bound on the EA-classes contained in the CCZ-classes of the known APN functions.

Partially APN Boolean functions and classes of functions that are not APN infinitely often

Abstract: In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field $\mathbb {F}_{2}$, extending some earlier results of Leander and Rodier.

One-weight and two-weight $\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]$-additive codes

Abstract: In this paper, a class of additive codes which is referred to as $\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]$-additive codes is introduced. This is a generalization towards another direction of recently introduced $\mathbb {Z}_{2}\mathbb {Z}_{4}$ codes (Doughterty et al., Appl. Algebra Eng. Commun. Comput. 27(2), 123–138, 7). A MacWilliams-type identity that relates the weight enumerator of a code with its dual is proved. Furthermore, the structure and possible weights for all one-weight and two-weight $\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]$-additive codes are described. Additionally, we also construct some one-weight and two-weight $\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]$-additive codes to illustrate our obtained results.

Solving x + x 2 l + ⋯ + x 2 m l = a $x+x^{2^{l}}+\cdots +x^{2^{ml}}=a$ over F 2 n $\mathbb {F}_{2^{n}}$

Abstract: This paper presents an explicit representation for the solutions of the equation ${\sum }_{i=0}^{\frac kl-1}x^{2^{li}} = a \in \mathbb {F}_{2^{n}}$ for any given positive integers k, l with l|k and n, in the closed field ${\overline {\mathbb {F}_{2}}}$ and in the finite field $\mathbb {F}_{2^{n}}$ . As a by-product of our study, we are able to completely characterize the a’s for which this equation has solutions in $\mathbb {F}_{2^{n}}$ .

On the minimum weights of binary linear complementary dual codes

Abstract: Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study the largest minimum weights d(n,k) among all binary linear complementary dual [n,k] codes. We determine d(n,4) for n ≡ 2,3,4,5,6,9,10,13 (mod 15), and d(n,5) for n ≡ 3,4,5,7,11,19,20, 22,26 (mod 31). Combined with known results, d(n,k) are also determined for n ≤ 24.

Double Bordered Constructions of Self-Dual Codes from Group Rings over Frobenius Rings

Abstract: In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings $\mathbb {F}_{2}+u\mathbb {F}_{2}$ and $\mathbb {F}_{4}+u\mathbb {F}_{4}$ . We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables.

Notes on generalized Hamming weights of some classes of binary codes

Abstract: Some classes of binary codes constructed by using some defining sets are studied, and for most defining sets, we will determine the generalized Hamming weight of the corresponding codes completely, and for other defining sets, we will determine part of the generalized Hamming weight of the corresponding codes.

On the Menezes-Teske-Weng conjecture

Abstract: In 2003, Alfred Menezes, Edlyn Teske and Annegret Weng presented a conjecture on properties of the solutions of a type of quadratic equations over the binary extension fields, which had been confirmed by extensive experiments but the proof was unknown until now. We prove that this conjecture is correct. Furthermore, using this proved conjecture, we have completely determined the null space of a class of linearized polynomials.

A family of negacyclic BCH codes of length n=q2m−12$n=\frac {q^{2m}-1}{2}$

Abstract: In this paper, we investigate a family of q2-ary narrow-sense and non-narrow-sense negacyclic BCH codes with length $n=\frac {q^{2m}-1}{2}$, where q is an odd prime power and m ≥ 3 is odd. We propose Hermitian dual-containing conditions for narrow-sense and non-narrow-sense negacyclic BCH codes, and precisely compute the dimensions of these negacyclic BCH codes whose maximal designed distance can achieve $\delta _{max}^{neg}$. Consequently, many new q-ary quantum codes can be derived from these dual-containing negacyclic BCH codes. Moreover, these new quantum codes are presented either with parameters better than or equal to the ones available in the literature, and also have larger designed distance than those from classical BCH codes.

The primitive idempotents of irreducible constacyclic codes and LCD cyclic codes

Abstract: In this paper, we investigate all irreducible factors of $x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a$ over $\mathbb {F}_{q}$ and obtain all primitive idempotents in $\mathbb {F}_{q}[x]/\langle x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a \rangle $, where $a \in \mathbb {F}_{q}^{*}$, l1, l2 are two distinct odd prime divisors of qt − 1 with $\gcd (l_{1}l_{2},q(q-1))= 1$ for prime t. Furthermore, the weight distributions of all irreducible constacyclic codes of length $l_{1}^{m_{1}}l_{2}^{m_{2}}$ are presented for t = 2. As an application, we determine all linear complementary dual cyclic codes of length $l_{1}^{m_{1}}l_{2}^{m_{2}}$ over $\mathbb {F}_{q}$.

Designed distances and parameters of new LCD BCH codes over finite fields

Abstract: Let $\mathbb {F}_{q}$ be the finite field of q elements and n = qm − 1 with m a positive integer. In this paper we construct a class of BCH and LCD BCH codes of length n over $\mathbb {F}_{q}$ and investigate their dimensions and designed distance. Our results show that the designed distances of BCH and LCD BCH codes in this paper are larger than those in [11, Theorems 7, 10, 18, and 22]. It is viewed as a generalized result of [11].

4-uniform permutations with null nonlinearity

Abstract: We consider n-bit permutations with differential uniformity of 4 and null nonlinearity. We first show that the inverses of Gold functions have the interesting property that one component can be replaced by a linear function such that it still remains a permutation. This directly yields a construction of 4-uniform permutations with trivial nonlinearity in odd dimension. We further show their existence for all n = 3 and n ≥ 5 based on a construction in Alsalami (Cryptogr. Commun. 10(4): 611–628, 2018). In this context, we also show that 4-uniform 2-1 functions obtained from admissible sequences, as defined by Idrisova in (Cryptogr. Commun. 11(1): 21–39, 2019), exist in every dimension n = 3 and n ≥ 5. Such functions fulfill some necessary properties for being subfunctions of APN permutations. Finally, we use the 4-uniform permutations with null nonlinearity to construct some 4-uniform 2-1 functions from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{n-1}$ which are not obtained from admissible sequences. This disproves a conjecture raised by Idrisova.

On complementary dual multinegacirculant codes

Abstract: Linear codes with complementary duals intersect with their duals trivially. Multinegacirculant codes that are complementary dual are characterized algebraically and some good codes are found in this family. Exact enumeration is performed for indices 2 and 3, whereas special choices of the co-index and base field size are needed for higher indices. Asymptotic existence results are derived for the special class of such codes that have co-index a power of two by means of Dickson polynomials. This shows that there are infinite families of complementary dual multinegacirculant codes with relative distance satisfying a modified Gilbert-Varshamov bound.

Vectorial bent functions in odd characteristic and their components

Abstract: Bent functions in odd characteristic can be either (weakly) regular or non-weakly regular. Furthermore one can distinguish between dual-bent functions, which are bent functions for which the dual is bent as well, and non-dual bent functions. Whereas a weakly regular bent function always has a bent dual, a non-weakly regular bent function can be either dual-bent or non-dual-bent. The classical constructions (like quadratic bent functions, Maiorana-McFarland or partial spread) yield weakly regular bent functions, but meanwhile one knows constructions of infinite classes of non-weakly regular bent functions of both types, dual-bent and non-dual-bent. In this article we focus on vectorial bent functions in odd characteristic. We first show that most p-ary bent monomials and binomials are actually vectorial constructions. In the second part we give a positive answer to the question if non-weakly regular bent functions can be components of a vectorial bent function. We present the first construction of vectorial bent functions of which the components are non-weakly regular but dual-bent, and the first construction of vectorial bent functions with non-dual-bent components.

Polyphase zero correlation zone sequences from generalised bent functions

Abstract: Sequence families with zero correlation zone (ZCZ) have been extensively studied in recent years due to their important applications in quasi-synchronous code-division multiple-access (QS-CDMA) systems. To accommodate multiuser environments, multiple ZCZ sequence sets with low inter-set cross-correlation are expected. In this paper, we propose a construction of polyphase ZCZ sequences based on generalised bent functions. Moreover, multiple polyphase ZCZ sequence sets with good inter-set cross-correlation are presented. Each generated ZCZ sequence set is optimal with respect to the Tang-Fan-Matsufuji bound.

Integer codes correcting burst asymmetric within a byte and double asymmetric errors

Abstract: This paper presents a class of integer codes capable of correcting l-bit burst asymmetric errors within a b-bit byte (1 ≤ l < b) and double asymmetric errors within a codeword. The presented codes are constructed with the help of a computer and have the potential to be used in unamplified optical networks. In addition, the paper derives the upper bound on code length and shows that the proposed codes are efficient in terms of redundancy.