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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...

Incomplete character sums and a special class of permutations

S. D. Cohen, H. Niederreiter, I. E. Shparlinski, M. Zieve (2001)

Journal de théorie des nombres de Bordeaux

We present a method of bounding incomplete character sums for finite abelian groups with arguments produced by a first-order recursion. This method is particularly effective if the recursion involves a special type of permutation called an ℛ -orthomorphism. Examples of ℛ -orthomorphisms are given.

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 ⇔ 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...

On extremal additive 𝔽 4 codes of length 10 to 18

Christine Bachoc, Philippe Gaborit (2000)

Journal de théorie des nombres de Bordeaux

In this paper we consider the extremal even self-dual 𝔽 4 -additive codes. We give a complete classification for length 10 . Under the hypothesis that at least two minimal words have the same support, we classify the codes of length 14 and we show that in length 18 such a code is equivalent to the unique 𝔽 4 -hermitian code with parameters [18,9,8]. We construct with the help of them some extremal 3 -modular lattices.

On the joint 2-adic complexity of binary multisequences

Lu Zhao, Qiao-Yan Wen (2012)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Joint 2-adic complexity is a new important index of the cryptographic security for multisequences. In this paper, we extend the usual Fourier transform to the case of multisequences and derive an upper bound for the joint 2-adic complexity. Furthermore, for the multisequences with pn-period, we discuss the relation between sequences and their Fourier coefficients. Based on the relation, we determine a lower bound for the number of multisequences with given joint 2-adic complexity.

On the joint 2-adic complexity of binary multisequences∗

Lu Zhao, Qiao-Yan Wen (2012)

RAIRO - Theoretical Informatics and Applications

Joint 2-adic complexity is a new important index of the cryptographic security for multisequences. In this paper, we extend the usual Fourier transform to the case of multisequences and derive an upper bound for the joint 2-adic complexity. Furthermore, for the multisequences with pn-period, we discuss the relation between sequences and their Fourier coefficients. Based on the relation, we determine a lower bound for the number of multisequences...

On the number of zero trace elements in polynomial bases for F2n.

Igor E. Shparlinski (2005)

Revista Matemática Complutense

Let Fq denote the finite field of q elements. O. Ahmadi and A. Menezes have recently considered the question about the possible number of elements with zero trace in polynomial bases of F2n over F2. Here we show that the Weil bound implies that there is such a basis with n + O(log n) zero-trace elements.

Operations of Points on Elliptic Curve in Projective Coordinates

Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, Yasunari Shidama (2012)

Formalized Mathematics

In this article, we formalize operations of points on an elliptic curve over GF(p). Elliptic curve cryptography [7], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security. We prove that the two operations of points: compellProjCo and addellProjCo are unary and binary operations of a point over the elliptic curve.

Poids des duaux des codes BCH de distance prescrite 2 a + 1 et sommes exponentielles

Éric Férard (2002)

Bulletin de la Société Mathématique de France

Soit n un entier pair. On considère un code BCH binaire C n de longueur 2 n - 1 et de distance prescrite 2 a + 1 avec a ≥ 3 . Le poids d’un mot non nul du dual de  C n peut s’exprimer en fonction d’une somme exponentielle. Nous montrerons que cette somme n’atteint pas la borne de Weil et nous proposerons une amélioration de celle-ci. En conséquence, nous obtiendrons une amélioration de la borne de Carlitz-Uchiyama sur le poids des mots du dual de C n .

Some new Results for Additive Self-Dual Codes over GF(4)

Varbanov, Zlatko (2007)

Serdica Journal of Computing

* Supported by COMBSTRU Research Training Network HPRN-CT-2002-00278 and the Bulgarian National Science Foundation under Grant MM-1304/03.Additive code C over GF(4) of length n is an additive subgroup of GF(4)n. It is well known [4] that the problem of finding stabilizer quantum error-correcting codes is transformed into problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product. Our purpose is to construct good additive self-dual codes of length 13...

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