ISSN:

1930-5346

eISSN:

1930-5338

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## Advances in Mathematics of Communications

February 2020 , Volume 14 , Issue 1

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**Abstract:**

Orthogonal sequences can be assigned to a regular tessellation of hexagonal cells, typical for synchronised code-division multiple-access (S-CDMA) systems. In this paper, we first construct a new class of orthogonal sequences with increasing the number of users per cell to be

*+*[Abstract](2040)

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**Abstract:**

In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings

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**Abstract:**

As maximal, nonlinear Boolean functions, bent functions have many theoretical and practical applications in combinatorics, coding theory, and cryptography. In this paper, we present a construction of bent function

*+*[Abstract](1551)

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**Abstract:**

Good integers introduced in 1997 form an interesting family of integers that has been continuously studied due to their rich number theoretical properties and wide applications. In this paper, we have focused on classes of

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**Abstract:**

We introduce skew constacyclic codes over the local Frobenius non-chain rings of order 16 by defining non-trivial automorphisms on these rings. We study the Gray images of these codes, obtaining a number of binary and quaternary codes with good parameters as images of skew cyclic codes over some of these rings.

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**Abstract:**

We generalize the definition of partial MDS codes to locality blocks of various length and show that these codes are maximally recoverable. Then we focus on partial MDS codes with exactly one global parity. We derive a general construction for such codes by describing a suitable parity check matrix. Then we give a construction of generator matrices of such codes. Afterwards we show that all partial MDS codes with one global parity have a generator matrix (or parity check matrix) of this form. This gives a complete classification and hence also a sufficient and necessary condition on the underlying field size for the existence of such codes. This condition is related to the classical MDS conjecture. Moreover, we investigate the decoding of such codes and give some comments on partial MDS codes with more than one global parity.

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**Abstract:**

In this note, we construct new doubly even self-dual codes having minimum weight 20 for lengths 112,120 and 128. This implies that there are at least three inequivalent extremal doubly even self-dual codes of length 112.

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**Abstract:**

A construction of differentially 4-uniform permutations by modifying the values of the inverse function on a union of some cosets of a multiplication subgroup of

*+*[Abstract](1464)

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**Abstract:**

Boolean functions used as nonlinear filters and/or combiners in LFSR-based stream ciphers should satisfy several desired cryptographic properties simultaneously, to withstand all known cryptographic attacks. In the past decade, the algebraic and fast algebraic immunities are the most infusive criteria on the design of cryptographic Boolean functions, due to the high efficiency of the algebraic and fast algebraic attacks on stream ciphers. Up to now, Boolean functions with optimal algebraic immunity have been built in several ways, but there are not many known results on their fast algebraic immunities. In this paper, we first derive a relation on the fast algebraic immunity between a Boolean function *f* and it’s modifications *f* + *s*, which shows that if *f* has low fast algebraic immunity and s has low algebraic immunity then *f* + *s* may also have low fast algebraic immunity in general. Thanks to this relation, we obtain some upper bounds on the fast algebraic immunity of several known classes of modified majority functions.

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**Abstract:**

We first give a brief survey of the results on highly nonlinear single-output Boolean functions and bijective S-boxes that are symmetric under some permutations. After that, we perform a heuristic search for the symmetric (and involution) S-boxes which are bijective in dimension 8 and identify corresponding permutations yielding rich classes in terms of cryptographically desirable properties.

*+*[Abstract](3262)

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**Abstract:**

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra–Lenstra–Lovász reduction algorithm (so-called LLL) has been improved in many ways through the past decades and remains one of the central methods used for reducing *integral* lattice basis. In particular, its floating-point variants---where the rational arithmetic required by Gram–Schmidt orthogonalization is replaced by floating-point arithmetic---are now the fastest known. However, the systematic study of the reduction theory of *real* quadratic forms or, more generally, of real lattices is not widely represented in the literature. When the problem arises, the lattice is usually replaced by an integral approximation of (a multiple of) the original lattice, which is then reduced. While practically useful and proven in some special cases, this method doesn't offer any guarantee of success in general. In this work, we present an adaptive-precision version of a generalized LLL algorithm that covers this case in all generality. In particular, we replace floating-point arithmetic by Interval Arithmetic to certify the behavior of the algorithm. We conclude by giving a typical application of the result in algebraic number theory for the reduction of ideal lattices in number fields.

Letters for Post-Quantum Cryptography (PQC)

*+*[Abstract](2958)

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**Abstract:**

This article introduces the NIST post-quantum cryptography standardization process. We highlight the challenges, discuss the mathematical problems in the proposed post-quantum cryptographic algorithms and the opportunities for mathematics researchers to contribute.

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**Abstract:**

In this paper, we attack the recent NIST submission Giophantus, a public key encryption scheme. We find that the complicated structure of Giophantus's ciphertexts leaks information via a correspondence from a low dimensional lattice. This allows us to distinguish encrypted data from random data by the LLL algorithm. This is a more efficient attack than previous proposed attacks.

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