Mathematics

Research

16 pages, 816 KiB

Open AccessArticle

Minimal State-Space Representation of Convolutional Product Codes

by Joan-Josep Climent, Diego Napp, Raquel Pinto and Verónica Requena

Mathematics 2021, 9(12), 1410; https://doi.org/10.3390/math9121410 - 17 Jun 2021

Cited by 1 | Viewed by 1601

In this paper, we study product convolutional codes described by state-space representations. In particular, we investigate how to derive state-space representations of the product code from the horizontal and vertical convolutional codes. We present a systematic procedure to build such representation with minimal [...] Read more.

In this paper, we study product convolutional codes described by state-space representations. In particular, we investigate how to derive state-space representations of the product code from the horizontal and vertical convolutional codes. We present a systematic procedure to build such representation with minimal dimension, i.e., reachable and observable. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

15 pages, 288 KiB

Open AccessArticle

Three Authentication Schemes without Secrecy over Finite Fields and Galois Rings

by Juan Carlos Ku-Cauich and Miguel Angel Márquez-Hidalgo

Mathematics 2021, 9(9), 942; https://doi.org/10.3390/math9090942 - 23 Apr 2021

Viewed by 1216

Abstract

We provide three new authentication schemes without secrecy. The first two on finite fields and Galois rings, using Gray map for this link. The third construction is based on Galois rings. The main achievement in this work is to obtain optimal impersonation and [...] Read more.

We provide three new authentication schemes without secrecy. The first two on finite fields and Galois rings, using Gray map for this link. The third construction is based on Galois rings. The main achievement in this work is to obtain optimal impersonation and substitution probabilities in the schemes. Additionally, in the first and second scheme, we simplify the source space and obtain a better relationship between the size of the message space and the key space than the one given in a recent paper. Finally, we provide a third scheme on Galois rings. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

17 pages, 365 KiB

Open AccessArticle

An Application of p-Fibonacci Error-Correcting Codes to Cryptography

by Emanuele Bellini, Chiara Marcolla and Nadir Murru

Mathematics 2021, 9(7), 789; https://doi.org/10.3390/math9070789 - 06 Apr 2021

Cited by 1 | Viewed by 2259

Abstract

In addition to their usefulness in proving one’s identity electronically, identification protocols based on zero-knowledge proofs allow designing secure cryptographic signature schemes by means of the Fiat–Shamir transform or other similar constructs. This approach has been followed by many cryptographers during the NIST [...] Read more.

In addition to their usefulness in proving one’s identity electronically, identification protocols based on zero-knowledge proofs allow designing secure cryptographic signature schemes by means of the Fiat–Shamir transform or other similar constructs. This approach has been followed by many cryptographers during the NIST (National Institute of Standards and Technology) standardization process for quantum-resistant signature schemes. NIST candidates include solutions in different settings, such as lattices and multivariate and multiparty computation. While error-correcting codes may also be used, they do not provide very practical parameters, with a few exceptions. In this manuscript, we explored the possibility of using the error-correcting codes proposed by Stakhov in 2006 to design an identification protocol based on zero-knowledge proofs. We showed that this type of code offers a valid alternative in the error-correcting code setting to build such protocols and, consequently, quantum-resistant signature schemes. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

► Show Figures

Figure 1

23 pages, 345 KiB

Open AccessArticle

Interleaving Shifted Versions of a PN-Sequence

by Sara Díaz Cardell, Amparo Fúster-Sabater and Verónica Requena

Mathematics 2021, 9(6), 687; https://doi.org/10.3390/math9060687 - 23 Mar 2021

Cited by 4 | Viewed by 1954

Abstract

The output sequence of the shrinking generator can be considered as an interleaving of determined shifted versions of a single PN -sequence. In this paper, we present a study of the interleaving of a PN-sequence and shifted versions of itself. We analyze some [...] Read more.

The output sequence of the shrinking generator can be considered as an interleaving of determined shifted versions of a single PN -sequence. In this paper, we present a study of the interleaving of a PN-sequence and shifted versions of itself. We analyze some important cryptographic properties as the period and the linear complexity in terms of the shifts. Furthermore, we determine the total number of the interleaving sequences that achieve each possible value of the linear complexity. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

► Show Figures

Figure 1

14 pages, 292 KiB

Open AccessArticle

Ternary Menger Algebras: A Generalization of Ternary Semigroups

by Anak Nongmanee and Sorasak Leeratanavalee

Mathematics 2021, 9(5), 553; https://doi.org/10.3390/math9050553 - 05 Mar 2021

Cited by 7 | Viewed by 1997

Abstract

Let n be a fixed natural number. Menger algebras of rank n, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups [...] Read more.

Let n be a fixed natural number. Menger algebras of rank n, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups is. In this article, we first introduce the notion of ternary Menger algebras of rank n, which is a canonical generalization of arbitrary ternary semigroups, and discuss their related properties. In the second part, we establish the so-called a diagonal ternary semigroup which its operation is induced by the operation on ternary Menger algebras of rank n and then investigate their interesting properties. Moreover, we introduce the concept of homomorphism and congruences on ternary Menger algebras of rank n. These lead us to study the quotient ternary Menger algebras of rank n and to investigate the homomorphism theorem for ternary Menger algebra of rank n with respect to congruences. Furthermore, the characterization of reduction of ternary Menger algebra into Menger algebra is presented. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

10 pages, 238 KiB

Open AccessArticle

Factorization and Malleability of RSA Moduli, and Counting Points on Elliptic Curves Modulo N

by Luis V. Dieulefait and Jorge Urroz

Mathematics 2020, 8(12), 2126; https://doi.org/10.3390/math8122126 - 27 Nov 2020

Cited by 2 | Viewed by 1710

Abstract

In this paper we address two different problems related with the factorization of an RSA (Rivest–Shamir–Adleman cryptosystem) modulus N. First we show that factoring is equivalent, in deterministic polynomial time, to counting points on a pair of twisted Elliptic curves modulo N [...] Read more.

In this paper we address two different problems related with the factorization of an RSA (Rivest–Shamir–Adleman cryptosystem) modulus N. First we show that factoring is equivalent, in deterministic polynomial time, to counting points on a pair of twisted Elliptic curves modulo N. The second problem is related with malleability. This notion was introduced in 2006 by Pailler and Villar, and deals with the question of whether or not the factorization of a given number N becomes substantially easier when knowing the factorization of another one

N^{'}

relatively prime to N. Despite the efforts done up to now, a complete answer to this question was unknown. Here we settle the problem affirmatively. To construct a particular

N^{'}

that helps the factorization of N, we use the number of points of a single elliptic curve modulo N. Coppersmith’s algorithm allows us to go from the factors of

N^{'}

to the factors of N in polynomial time. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

18 pages, 787 KiB

Open AccessArticle

Novel Parametric Solutions for the Ideal and Non-Ideal Prouhet Tarry Escott Problem

by Srikanth Raghavendran and Veena Narayanan

Mathematics 2020, 8(10), 1775; https://doi.org/10.3390/math8101775 - 14 Oct 2020

Cited by 2 | Viewed by 1305

Abstract

The present study aims to develop novel parametric solutions for the Prouhet Tarry Escott problem of second degree with sizes 3, 4 and 5. During this investigation, new parametric representations for integers as the sum of three, four and five perfect squares in [...] Read more.

The present study aims to develop novel parametric solutions for the Prouhet Tarry Escott problem of second degree with sizes 3, 4 and 5. During this investigation, new parametric representations for integers as the sum of three, four and five perfect squares in two distinct ways are identified. Moreover, a new proof for the non-existence of solutions of ideal Prouhet Tarry Escott problem with degree 3 and size 2 is derived. The present work also derives a three parametric solution of ideal Prouhet Tarry Escott problem of degree three and size two. The present study also aimed to discuss the Fibonacci-like pattern in the solutions and finally obtained an upper bound for this new pattern. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

10 pages, 269 KiB

Open AccessArticle

First-Degree Prime Ideals of Biquadratic Fields Dividing Prescribed Principal Ideals

by Giordano Santilli and Daniele Taufer

Mathematics 2020, 8(9), 1433; https://doi.org/10.3390/math8091433 - 26 Aug 2020

Viewed by 1394

Abstract

We describe first-degree prime ideals of biquadratic extensions in terms of the first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. The correspondence between these ideals in the larger [...] Read more.

We describe first-degree prime ideals of biquadratic extensions in terms of the first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. The correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of specially-shaped principal ideals in their respective rings, with some exceptions that we explicitly characterize. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

11 pages, 311 KiB

Open AccessArticle

Magnifiers in Some Generalization of the Full Transformation Semigroups

by Thananya Kaewnoi, Montakarn Petapirak and Ronnason Chinram

Mathematics 2020, 8(4), 473; https://doi.org/10.3390/math8040473 - 30 Mar 2020

Cited by 5 | Viewed by 1911

Abstract

An element a of a semigroup S is called a left [right] magnifier if there exists a proper subset M of S such that

a M = S

(M a = S) .

Let

T (X)

denote the semigroup [...] Read more.

An element a of a semigroup S is called a left [right] magnifier if there exists a proper subset M of S such that

a M = S

(M a = S) .

Let

T (X)

denote the semigroup of all transformations on a nonempty set X under the composition of functions,

P = {X_{i} ∣ i \in Λ}

be a partition, and

ρ

be an equivalence relation on the set X. In this paper, we focus on the properties of magnifiers of the set

T_{ρ} (X, P) = {f \in T (X) ∣ \forall (x, y) \in ρ, (x f, y f) \in ρ and X_{i} f \subseteq X_{i} for all i \in Λ},

which is a subsemigroup of

T (X),

and provide the necessary and sufficient conditions for elements in

T_{ρ} (X, P)

to be left or right magnifiers. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

Journal Menu

Journal Browser

Algebra and Number Theory

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Published Papers (9 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI