Lower bounds on the linear complexity of the discrete logarithm in finite fields

2001 ◽  
Vol 47 (7) ◽  
pp. 2807-2811 ◽  
Author(s):  
W. Meidl ◽  
A. Winterhof
Keyword(s):  
Finite Fields ◽  
Lower Bounds ◽  
Linear Complexity ◽  
Discrete Logarithm

scholarly journals On the k-Error Linear Complexity of Binary Sequences Derived from the Discrete Logarithm in Finite Fields

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Zhixiong Chen ◽  
Qiuyan Wang
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Finite Fields ◽  
Linear Complexity ◽  
Discrete Logarithm ◽  
Binary Sequence ◽  
Binary Sequences ◽  
Quadratic Character ◽  
Legendre Sequence ◽  
Error Linear Complexity

Let Fq be the finite field with q=pr elements, where p is an odd prime. For the ordered elements ξ0,ξ1,…,ξq-1∈Fq, the binary sequence σ=(σ0,σ1,…,σq-1) with period q is defined over the finite field F2={0,1} as follows: σn=0,  if  n=0,  (1-χ(ξn))/2,  if  1≤n<q,  σn+q=σn, where χ is the quadratic character of Fq. Obviously, σ is the Legendre sequence if r=1. In this paper, our first contribution is to prove a lower bound on the linear complexity of σ for r≥2, which improves some results of Meidl and Winterhof. Our second contribution is to study the distribution of the k-error linear complexity of σ for r=2. Unfortunately, the method presented in this paper seems not suitable for the case r>2 and we leave it open.

PERSPECTIVES OF ELLIPTICAL CRYPTOGRAPHY TO ENSURE THE INTEGRITY AND CONFIDENTIALITY OF INFORMATION

2019 ◽  
Author(s):  
Anna ILYENKO ◽  
Sergii ILYENKO ◽  
Yana MASUR
Keyword(s):  
Information Systems ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Computational Efficiency ◽  
Digital Signature ◽  
Discrete Logarithm ◽  
Algebraic Groups ◽  
Schnorr Signature ◽  
Stability Of Algorithms

In this article, the main problems underlying the current asymmetric crypto algorithms for the formation and verification of electronic-digital signature are considered: problems of factorization of large integers and problems of discrete logarithm. It is noted that for the second problem, it is possible to use algebraic groups of points other than finite fields. The group of points of the elliptical curve, which satisfies all set requirements, looked attractive on this side. Aspects of the application of elliptic curves in cryptography and the possibilities offered by these algebraic groups in terms of computational efficiency and crypto-stability of algorithms were also considered. Information systems using elliptic curves, the keys have a shorter length than the algorithms above the finite fields. Theoretical directions of improvement of procedure of formation and verification of electronic-digital signature with the possibility of ensuring the integrity and confidentiality of information were considered. The proposed method is based on the Schnorr signature algorithm, which allows data to be recovered directly from the signature itself, similarly to RSA-like signature systems, and the amount of recoverable information is variable depending on the information message. As a result, the length of the signature itself, which is equal to the sum of the length of the end field over which the elliptic curve is determined, and the artificial excess redundancy provided to the hidden message was achieved.

scholarly journals Expansion complexity and linear complexity of sequences over finite fields

2016 ◽  
Vol 9 (4) ◽  
pp. 501-509 ◽  
Author(s):  
László Mérai ◽  
Harald Niederreiter ◽  
Arne Winterhof
Keyword(s):  
Finite Fields ◽  
Linear Complexity

A new class of quaternary generalized cyclotomic sequences of order 2d and length 2pm with high linear complexity

2018 ◽  
Vol 16 (01) ◽  
pp. 1850006 ◽  
Author(s):  
Longfei Liu ◽  
Xiaoyuan Yang ◽  
Bin Wei ◽  
Liqiang Wu
Keyword(s):  
Finite Fields ◽  
Linear Complexity ◽  
Periodic Sequences ◽  
New Class ◽  
New Family ◽  
Generalized Cyclotomic Classes ◽  
Cyclotomic Sequences

Periodic sequences over finite fields, constructed by classical cyclotomic classes and generalized cyclotomic classes, have good pseudo-random properties. The linear complexity of a period sequence plays a fundamental role in the randomness of sequences. In this paper, we construct a new family of quaternary generalized cyclotomic sequences with order [Formula: see text] and length [Formula: see text], which generalize the sequences constructed by Ke et al. in 2012. In addition, we determine its linear complexity using cyclotomic theory. The conclusions reveal that these sequences have high linear complexity, which means they can resist linear attacks.

Weighted Measures of Pseudorandom Binary Lattices

2021 ◽  
Vol 58 (3) ◽  
pp. 319-334
Author(s):  
Huaning Liu ◽  
Yinyin Yang
Keyword(s):  
Finite Fields ◽  
Lower Bounds ◽  
Binary Sequences ◽  
Quadratic Character ◽  
Pseudorandom Sequences ◽  
Even Order

In cryptography one needs pseudorandom sequences whose short subsequences are also pseudorandom. To handle this problem, Dartyge, Gyarmati and Sárközy introduced weighted measures of pseudorandomness of binary sequences. In this paper we continue the research in this direction. We introduce weighted pseudorandom measure for multidimensional binary lattices and estimate weighted pseudorandom measure for truly random binary lattices. We also give lower bounds for weighted measures of even order and present an example by using the quadratic character of finite fields.

scholarly journals Finding roots in with the successive resultants algorithm

2014 ◽  
Vol 17 (A) ◽  
pp. 203-217 ◽  
Author(s):  
Christophe Petit
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Finite Fields ◽  
Coding Theory ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Polynomial Equations ◽  
Preliminary Results ◽  
Characteristic Case ◽  
Practical Security

AbstractThe problem of solving polynomial equations over finite fields has many applications in cryptography and coding theory. In this paper, we consider polynomial equations over a ‘large’ finite field with a ‘small’ characteristic. We introduce a new algorithm for solving this type of equations, called the successive resultants algorithm (SRA). SRA is radically different from previous algorithms for this problem, yet it is conceptually simple. A straightforward implementation using Magma was able to beat the built-in Roots function for some parameters. These preliminary results encourage a more detailed study of SRA and its applications. Moreover, we point out that an extension of SRA to the multivariate case would have an important impact on the practical security of the elliptic curve discrete logarithm problem in the small characteristic case.Supplementary materials are available with this article.

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