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Displaying 41 – 60 of 107

Equivalences between elliptic curves and real quadratic congruence function fields

Andreas Stein (1997)

Journal de théorie des nombres de Bordeaux

In 1994, the well-known Diffie-Hellman key exchange protocol was for the first time implemented in a non-group based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number field. This set does not possess a group structure, but instead exhibits a so-called infrastructure. More recently, the scheme was extended to real quadratic congruence function fields, whose set of reduced principal ideals has a similar infrastructure. As always, the security...

Étude du problème du logarithme discret dans $I F_{p^{3}}$

Ndiaye El Hadji Oumar (1995)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Explicit form for the discrete logarithm over the field $GF (p, k)$

Gerasimos C. Meletiou (1993)

Archivum Mathematicum

For $a$ generator of the multiplicative group of the field $G F (p, k)$ , the discrete logarithm of an element $b$ of the field to the base $a$ , $b \neq 0$ is that integer $z : 1 \leq z \leq p^{k} - 1$ , $b = a^{z}$ . The $p$ -ary digits which represent $z$ can be described with extremely simple polynomial forms.

Factorization Methods in Cryptosystems

N. Alexandris (1992)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

Finite vector spaces and certain lattices.

Cusick, Thomas W. (1998)

The Electronic Journal of Combinatorics [electronic only]

Formalization of Integral Linear Space

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2011)

Formalized Mathematics

In this article, we formalize integral linear spaces, that is a linear space with integer coefficients. Integral linear spaces are necessary for lattice problems, LLL (Lenstra-Lenstra-Lovász) base reduction algorithm that outputs short lattice base and cryptographic systems with lattice [8].

Formalization of the Advanced Encryption Standard. Part I

Kenichi Arai, Hiroyuki Okazaki (2013)

Formalized Mathematics

In this article, we formalize the Advanced Encryption Standard (AES). AES, which is the most widely used symmetric cryptosystem in the world, is a block cipher that was selected by the National Institute of Standards and Technology (NIST) as an official Federal Information Processing Standard for the United States in 2001 [12]. AES is the successor to DES [13], which was formerly the most widely used symmetric cryptosystem in the world. We formalize the AES algorithm according to [12]. We then verify...

Formalization of the Data Encryption Standard

Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

In this article we formalize DES (the Data Encryption Standard), that was the most widely used symmetric cryptosystem in the world. DES is a block cipher which was selected by the National Bureau of Standards as an official Federal Information Processing Standard for the United States in 1976 [15].

Galois rings and algebraic cryptography

Javier Gomez-Calderon, Gary Mullen (1991)

Acta Arithmetica

Generalized Kotov-Ushakov attack on tropical Stickel protocol based on modified tropical circulant matrices

Sulaiman Alhussaini, Craig Collett, Sergeĭ Sergeev (2024)

Kybernetika

After the Kotov-Ushakov attack on the tropical implementation of Stickel protocol, various attempts have been made to create a secure variant of such implementation. Some of these attempts used a special class of commuting matrices resembling tropical circulants, and they have been proposed with claims of resilience against the Kotov-Ushakov attack, and even being potential post-quantum candidates. This paper, however, reveals that a form of the Kotov-Ushakov attack remains applicable and, moreover,...

Generating quasigroups for cryptographic applications

Czesław Kościelny (2002)

International Journal of Applied Mathematics and Computer Science

A method of generating a practically unlimited number of quasigroups of a (theoretically) arbitrary order using the computer algebra system Maple 7 is presented. This problem is crucial to cryptography and its solution permits to implement practical quasigroup-based endomorphic cryptosystems. The order of a quasigroup usually equals the number of characters of the alphabet used for recording both the plaintext and the ciphertext. From the practical viewpoint, the most important quasigroups are of...

GLS: New class of generalized Legendre sequences with optimal arithmetic cross-correlation

Huijuan WANG, Qiaoyan WEN, Jie ZHANG (2013)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

The Legendre symbol has been used to construct sequences with ideal cross-correlation, but it was never used in the arithmetic cross-correlation. In this paper, a new class of generalized Legendre sequences are described and analyzed with respect to their period, distributional, arithmetic cross-correlation and distinctness properties. This analysis gives a new approach to study the connection between the Legendre symbol and the arithmetic cross-correlation. In the end of this paper, possible application...

Improvements in geometry-based secret image sharing approach with steganography.

Ulutas, Mustafa, Nabiyev, Vasif V., Ulutas, Guzin (2009)

Mathematical Problems in Engineering

Incomplete character sums over finite fields and their application to the interpolation of the discrete logarithm by Boolean functions

Tanja Lange, Arne Winterhof (2002)

Acta Arithmetica

Informační bezpečnost

Jaroslav Mlýnek (2006)

Pokroky matematiky, fyziky a astronomie

Kryptologie: eine neuartige Anwendung der Mathematik.

Ueli Maurer (1995)

Elemente der Mathematik

Left MQQs whose left parastrophe is also quadratic

Simona Samardjiska, Danilo Gligoroski (2012)

Commentationes Mathematicae Universitatis Carolinae

A left quasigroup $(Q, q)$ of order $2^{w}$ that can be represented as a vector of Boolean functions of degree 2 is called a left multivariate quadratic quasigroup (LMQQ). For a given LMQQ there exists a left parastrophe operation $q_{∖}$ defined by: $q_{∖} (u, v) = w \Leftrightarrow q (u, w) = v$ that also defines a left multivariate quasigroup. However, in general, $(Q, q_{∖})$ is not quadratic. Even more, representing it in a symbolic form may require exponential time and space. In this work we investigate the problem of finding a subclass of LMQQs whose left parastrophe...

Linear spans of optimal sets of frequency hopping sequences

Gao Juntao, Hu Yupu, Li Xuelian (2012)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Frequency hopping sequences sets are required in frequency hopping code division multiple access systems. For the anti-jamming purpose, frequency hopping sequences are required to have a large linear span. In this paper, by using a permutation polynomial δ(x) over a finite field, we transform several optimal sets of frequency hopping sequences with small linear span into ones with large linear span. The exact values of the linear span are presented by using the methods of counting the terms of the...

Linear spans of optimal sets of frequency hopping sequences∗

Gao Juntao, Hu Yupu, Li Xuelian (2012)

RAIRO - Theoretical Informatics and Applications

Mathematical foundation of digital signatures.

Bojan, Daniela, Vultur, Sidonia (2007)

Acta Universitatis Apulensis. Mathematics - Informatics

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