Cryptography and Communications最新文献

In this work, we establish a minimal set of generators for the dual code of a cyclic code having arbitrary length over a finite chain ring. It is observed that this set of generators forms a minimal strong Grobner basis for the dual code. Using this structure for the dual of a cyclic code, we obtain sufficient as well as necessary conditions for a cyclic code to be a self dual code over a finite chain ring. We enumerate the distinct non-trivial self dual cyclic codes over these rings. Further, we determine all MHDR dual cyclic codes. We provide a few examples of self dual and MHDR dual cyclic codes over various finite chain rings.

Dillon observed that an APN function F over ({{mathbb {F}}_{2}^{n}}) with n greater than 2 must satisfy the condition ({F(x) + F(y) + F(z) + F(x + y + z) :, x,y,z in {mathbb {F}}_{2}^{n}}= {mathbb {F}}_{2}^{n}). Recently, Taniguchi (Cryptogr. Commun. 15, 627–647 2023) generalized this condition to functions defined from ({{mathbb {F}}_{2}^{n}}) to ({{mathbb {F}}_{2}^{m}}), with (m>n), calling it the D-property. Taniguchi gave some characterizations of APN functions satisfying the D-property and provided some families of APN functions from ({{mathbb {F}}_{2}^{n}}) to ({{mathbb {F}}_{2}^{n+1}}) satisfying this property. In this work, we further study the D-property for (n, m)-functions with (mge n). We give some combinatorial bounds on the dimension m for the existence of such functions. Then, we characterize the D-property in terms of the Walsh transform and for quadratic functions we give a characterization of this property in terms of the ANF. We also give a simplification on checking the D-property for quadratic functions, which permits to extend some of the APN families provided by Taniguchi. We further focus on the class of the plateaued functions, providing conditions for the D-property. To conclude, we show a connection of some results obtained with the higher-order differentiability and the inverse Fourier transform.

Subfield codes of linear codes over finite fields have recently attracted great attention due to their wide applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction of 3-dimensional linear codes (varvec{C}_{varvec{f}}) over finite field ({mathbb {F}_{varvec{2}}}^{varvec{m}}) parameterized by any Boolean function (varvec{f}). Then we determine explicitly the weight distributions of (varvec{C}_{varvec{f}}), the punctured code (widetilde{varvec{C}}_{varvec{f}}), as well as the corresponding subfield codes over (mathbb {F}_{varvec{2}}) for several classes of Boolean functions (varvec{f}). In particular, we determine the weight distributions of subfield codes derived from (varvec{r})-plateaued functions. Moreover, the parameters of their dual codes are investigated, which contain length-optimal and dimension-optimal AMDS codes with respect to the sphere packing bound. We emphasize that the new codes are projective and contain binary self-complementary codes. As applications, some of the projective codes we present can be employed to construct (varvec{s})-sum sets for any odd integer (varvec{s}>varvec{1}).

Using recent results of Keevash et al. [10] and Eberhard et al. [8] together with further new detailed techniques in combinatorics, we present constructions of two concrete families of generalized Maiorana-McFarland bent functions. Our constructions improve the lower bounds on the number of bent functions in n variables over a finite field ({mathbb F}_p) if p is odd and n is odd in the limit as n tends to infinity. Moreover we obtain the asymptotically exact number of two dimensional vectorial Maiorana-McFarland bent functions in n variables over ({mathbb F}_2) as n tends to infinity.

This paper presents a direct construction of type-II Z-complementary pair (ZCP) of q-ary (q is even) for all even lengths with a wide zero-correlation zone (ZCZ). The proposed construction provides type-II (left( N_1times 2^m, N_1times 2^m-left( N_1-1right) /2right) )-ZCP, where (N_1) is an odd positive integer greater than 1, and (mge 1). For (N_1=3), the result produces Z-optimal type-II ZCP of length (3times 2^m). In this paper, we also present a construction of type-II (left( N_2times 2^m, N_2times 2^m-left( N_2-2right) /2right) )-ZCP, where (N_2) is an even positive integer greater than 1, and (mge 1). For (N_2=2) and (N_2=4), the result provides a Golay complementary pair (GCP) of length (2^{m+1}) and Z-optimal type-II ZCP of length (2^{m+2}). Both the proposed constructions are compared with the existing state-of-the-art, and it has been observed that it produces a large ZCZ, which covers all existing work in terms of lengths.

Let R, S be two finite commutative chain rings such that R is the Galois extension of S of degree (r ge 2) and has a self-dual basis over S. Let q be the order of the residue field of S, and let N be a positive integer with (gcd (N,q)=1.) An S-additive cyclic code of length N over R is defined as an S-submodule of (R^N,) which is invariant under the cyclic shift operator on (R^N.) In this paper, we show that each S-additive cyclic code of length N over R can be uniquely expressed as a direct sum of linear codes of length r over certain Galois extensions of the chain ring S, which are called its constituents. We further study the dual code of each S-additive cyclic code of length N over R by placing the ordinary trace bilinear form on (R^N) and relating the constituents of the code with that of its dual code. With the help of these canonical form decompositions of S-additive cyclic codes of length N over R and their dual codes, we further characterize all self-orthogonal, self-dual and complementary-dual S-additive cyclic codes of length N over R in terms of their constituents. We also derive necessary and sufficient conditions for the existence of a self-dual S-additive cyclic code of length N over R and count all self-dual and self-orthogonal S-additive cyclic codes of length N over R by considering the following two cases: (I) both q, r are odd, and (II) q is even and (S=mathbb {F}_{q}[u]/langle u^e rangle .) Besides this, we obtain the explicit enumeration formula for all complementary-dual S-additive cyclic codes of length N over R. We also illustrate our main results with some examples.

It is very hard to construct an infinite family of cyclic codes of rate close to one half whose minimum distances have a good bound. Tang-Ding codes are very interesting, as their minimum distances have a square-root-like bound. Recently, a new generalization of Tang-Ding codes has been presented, Sun constructed several infinite families of binary cyclic codes with length (2^{m}-1) and dimension near (2^{m-1}) whose minimum distances much exceed the square-root bound (Sun, Finite Fields Appl. 89, 102200, 2023). In this paper, we construct several families of q-ary cyclic codes with length (q^{m}-1) and dimension near (frac{q^{m}-1}{2}), where (qge 3) is a prime power and (m ge 3) is an integer. The minimum distances of these codes and their dual codes much exceed the square-root bound.

The concatenation of four Boolean bent functions (f=f_1||f_2||f_3||f_4) is bent if and only if the dual bent condition (f_1^* + f_2^* + f_3^* + f_4^* =1) is satisfied. However, to specify four bent functions satisfying this duality condition is in general quite a difficult task. Commonly, to simplify this problem, certain relations between (f_i) are assumed, as well as functions (f_i) of a special shape are considered, e.g., (f_i(x,y)=xcdot pi _i(y)+h_i(y)) are Maiorana-McFarland bent functions. In the case when permutations (pi _i) of (mathbb {F}_2^m) have the ((mathcal {A}_m)) property and Maiorana-McFarland bent functions (f_i) satisfy the additional condition (f_1+f_2+f_3+f_4=0), the dual bent condition is known to have a relatively simple shape allowing to specify the functions (f_i) explicitly. In this paper, we generalize this result for the case when Maiorana-McFarland bent functions (f_i) satisfy the condition (f_1(x,y)+f_2(x,y)+f_3(x,y)+f_4(x,y)=s(y)) and provide a construction of new permutations with the ((mathcal {A}_m)) property from the old ones. Combining these two results, we obtain a recursive construction method of bent functions satisfying the dual bent condition. Moreover, we provide a generic condition on the Maiorana-McFarland bent functions (f_1,f_2,f_3,f_4) stemming from the permutations of (mathbb {F}_2^m) with the ((mathcal {A}_m)) property, such that the concatenation (f=f_1||f_2||f_3||f_4) does not belong, up to equivalence, to the Maiorana-McFarland class. Using monomial permutations (pi _i) of (mathbb {F}_{2^m}) with the ((mathcal {A}_m)) property and monomial functions (h_i) on (mathbb {F}_{2^m}), we provide explicit constructions of such bent functions; a particular case of our result shows how one can construct bent functions from APN permutations, when m is odd. Finally, with our construction method, we explain how one can construct homogeneous cubic bent functions, noticing that only very few design methods of these objects are known.