American Institute of Mathematical Sciences

Some important properties of lattices are packing density and full diversity, which may be good for signal transmission over both Gaussian and Rayleigh fading channel, respectively. The algebraic lattices are constructed through twisted homomorphism of some modules in the ring of integers of a number field \begin{document}$ \mathbb{K} $\end{document}. In this paper, we present a construction of some families of rotated \begin{document}$ A_n- $\end{document}lattices, for \begin{document}$ n = 2^{r-2}-1 $\end{document}, \begin{document}$ r \geq 4 $\end{document}, via totally real subfield of cyclotomic fields. Furthermore, closed-form expressions for the minimum product distance of those lattices are obtained through algebraic properties.

The present work reports progress in discrete logarithm computation for the general medium prime case using the function field sieve algorithm. A new record discrete logarithm computation over a 1051-bit field having a 22-bit characteristic was performed. This computation builds on and implements previously known techniques. Analysis indicates that the relation collection and descent steps are within reach for fields with 32-bit characteristic and moderate extension degrees. It is the linear algebra step which will dominate the computation time for any discrete logarithm computation over such fields.

Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on \begin{document}$ v $\end{document} prime or composite, cyclotomy on a residue class ring \begin{document}$ {\mathbb{Z}}_{v} $\end{document} can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [40], we introduce a generalized cyclotomy of order \begin{document}$ e $\end{document} on the ring \begin{document}$ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $\end{document}, where \begin{document}$ q_i $\end{document} and \begin{document}$ q_j $\end{document} (\begin{document}$ i\neq j $\end{document}) may not be co-prime, which includes classical cyclotomy as a special case. Here, \begin{document}$ q_1 $\end{document}, \begin{document}$ q_2 $\end{document}, \begin{document}$ \cdots $\end{document}, \begin{document}$ q_k $\end{document} are powers of primes with an integer \begin{document}$ e|(q_i-1) $\end{document} for any \begin{document}$ 1\leq i\leq k $\end{document}. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and \begin{document}$ d $\end{document}-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.

In this paper, based on the theory of \begin{document}$ \mathbb{Z}_{4} $\end{document}-valued quadratic forms we propose several classes of generalized Boolean bent functions over \begin{document}$ \mathbb{Z}_{4} $\end{document}, and new families of codebooks are constructed from these functions. The codebooks constructed in this paper are nearly optimal with respect to the Welch bound, and their parameters are new. Furthermore, some Boolean bent functions are also derived.

The theory of linear codes over finite rings has been generalized to linear codes over infinite rings in two special cases; the ring of \begin{document}$ p $\end{document}-adic integers and formal power series ring. These rings are examples of complete discrete valuation rings (CDVRs). In this paper, we generalize the theory of linear codes over the above two rings to linear codes over complete local principal ideal rings. In particular, we obtain the structure of linear and constacyclic codes over CDVRs. For this generalization, first we study linear codes over \begin{document}$ \hat{R}_{ \frak m} $\end{document}, where \begin{document}$ R $\end{document} is a commutative Noetherian ring, \begin{document}$ \frak m = \langle \gamma\rangle $\end{document} is a maximal ideal of \begin{document}$ R $\end{document}, and \begin{document}$ \hat{R}_{ \frak m} $\end{document} denotes the \begin{document}$ \frak m $\end{document}-adic completion of \begin{document}$ R $\end{document}. We call these codes, \begin{document}$ \frak m $\end{document}-adic codes. Using the structure of \begin{document}$ \frak m $\end{document}-adic codes, we present the structure of linear and constacyclic codes over complete local principal ideal rings.

For any odd prime \begin{document}$ p $\end{document}, the structures and duals of \begin{document}$ \lambda $\end{document}-constacyclic codes of length \begin{document}$ 8p^s $\end{document} over \begin{document}$ \mathcal R = \mathbb F_{p^m}+u\mathbb F_{p^m} $\end{document} are completely determined for all unit \begin{document}$ \lambda $\end{document} of the form \begin{document}$ \lambda = \xi^l\in \mathbb F_{p^m} $\end{document}, where \begin{document}$ l $\end{document} is even. In addition, the algebraic structures of all cyclic and negacyclic codes of length \begin{document}$ 8p^s $\end{document} over \begin{document}$ \mathcal R $\end{document} are established in term of their generator polynomials. Dual codes of all cyclic and negacyclic codes of length \begin{document}$ 8p^s $\end{document} over \begin{document}$ \mathcal R $\end{document} are also investigated. Furthermore, we give the number of codewords in each of those cyclic and negacyclic codes. We also obtain the number of cyclic and negacyclic codes of length \begin{document}$ 8p^s $\end{document} over \begin{document}$ \mathcal R $\end{document}.

In this paper, we introduce two classes of MacDonald codes over the finite non-chain ring \begin{document}$ \mathbb{F}_p+v\mathbb{F}_p+v^2\mathbb{F}_p $\end{document} and their torsion codes which are linear codes over \begin{document}$ \mathbb{F}_p $\end{document}, where \begin{document}$ p $\end{document} is an odd prime and \begin{document}$ v^3 = v $\end{document}. We give the complete weight enumerator of two classes of torsion codes. As an application, systematic authentication codes are obtained by these torsion codes.

Uniformity in binary tuples of various lengths in a pseudorandom sequence is an important randomness property. We consider ideal \begin{document}$ t $\end{document}-tuple distribution of a filtering de Bruijn generator consisting of a de Bruijn sequence of period \begin{document}$ 2^n $\end{document} and a filtering function in \begin{document}$ m $\end{document} variables. We restrict ourselves to the family of orthogonal functions, that correspond to binary sequences with ideal 2-level autocorrelation, used as filtering functions. After the twenty years of discovery of Welch-Gong (WG) transformations, there are no much significant results on randomness of WG transformation sequences. In this article, we present new results on uniformity of the WG transform of orthogonal functions on de Bruijn sequences. First, we introduce a new property, called invariant under the WG transform, of Boolean functions. We have found that there are only two classes of orthogonal functions whose WG transformations preserve \begin{document}$ t $\end{document}-tuple uniformity in output sequences, up to \begin{document}$ t = (n-m+1) $\end{document}. The conjecture of Mandal et al. in [29] about the ideal tuple distribution on the WG transformation is proved. It is also shown that the Gold functions and quadratic functions can guarantee \begin{document}$ (n-m+1) $\end{document}-tuple distributions. A connection between the ideal tuple distribution and the invariance under WG transform property is established.

In this paper, we present a practicable chosen ciphertext timing attack retrieving the secret key of HQC. The attack exploits a correlation between the weight of the error to be decoded and the running time of the decoding algorithm of BCH codes. For the 128-bit security parameters of HQC, the attack runs in less than a minute on a desktop computer using roughly 6000 decoding requests and has a success probability of approximately 93 percent. To prevent this attack, we provide an implementation of a constant time algorithm for the decoding of BCH codes. Our implementation of the countermeasure achieves a constant time execution of the decoding process without a significant performance penalty.

In this paper, we generalize the notion of self-orthogonal codes to \begin{document}$ \sigma $\end{document}-self-orthogonal codes over an arbitrary finite ring. Then, we study the \begin{document}$ \sigma $\end{document}-self-orthogonality of constacyclic codes of length \begin{document}$ p^s $\end{document} over the finite commutative chain ring \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}, where \begin{document}$ p $\end{document} is a prime, \begin{document}$ u^2 = 0 $\end{document} and \begin{document}$ \sigma $\end{document} is an arbitrary ring automorphism of \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}. We characterize the structure of \begin{document}$ \sigma $\end{document}-dual code of a \begin{document}$ \lambda $\end{document}-constacyclic code of length \begin{document}$ p^s $\end{document} over \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}. Further, the necessary and sufficient conditions for a \begin{document}$ \lambda $\end{document}-constacyclic code to be \begin{document}$ \sigma $\end{document}-self-orthogonal are provided. In particular, we determine all \begin{document}$ \sigma $\end{document}-self-dual constacyclic codes of length \begin{document}$ p^s $\end{document} over \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}. In the end of this paper, when \begin{document}$ p $\end{document} is an odd prime, we extend the results to constacyclic codes of length \begin{document}$ 2 p^s $\end{document}.