The Discrete Logarithm Problem in the ElGamal Cryptosystem over the Abelian Group U(n) Where n= pm,or 2pm

In most existing authentication schemes users are authenticated by the server one by one which results in lower efficiency of authentication when the number of users is large. Aiming at the drawback, this paper propose an efficient trusted multi-party authentication scheme based on threshold secret sharing, the discrete logarithm problem and ElGamal cryptosystem. Using (t,n) threshold secret sharing idea, the paper propose a (t,n) threshold authentication scheme which can not only simultaneously authenticate t users satisfying some specific conditions, but also authenticate new users dynamically by distributing a new authentication key for the new user which was produced by t old users and the new user together. Finally, the security and efficiency of the proposed scheme are analyzed.

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Composite Discrete Logarithm Problem and a Reconstituted ElGamal Cryptosystem Based on the Problem

Implementing Computational Intelligence Techniques for Security Systems Design - Advances in Computational Intelligence and Robotics ◽

10.4018/978-1-7998-2418-3.ch008 ◽

2020 ◽

pp. 163-181

Author(s):

Çağla Özyılmaz ◽

Ayşe Nallı

Keyword(s):

Discrete Logarithm ◽

Fibonacci Sequence ◽

The Other ◽

Discrete Logarithm Problem ◽

The Third ◽

Elgamal Cryptosystem ◽

New System

In this chapter, the authors have defined a new ElGamal cryptosystem by using the power Fibonacci sequence module m. Then they have defined a new sequence module m and the other ElGamal cryptosystem by using the new sequence. In addition, they have compared that the new ElGamal cryptosystems and ElGamal cryptosystem in terms of cryptography. Then the authors have defined the third ElGamal cryptosystem. They have, particularly, called the new system as composite ElGamal cryptosystem. The authors made an application of composite ElGamal cryptosystem. Finally, the authors have compared that composite ElGamal cryptosystem and ElGamal cryptosystem in terms of cryptography and they have obtained that composite ElGamal cryptosystem is more advantageous than ElGamal cryptosystem.

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How Would Quantum Computing Impact the Security of Bitcoin by Enhancing Our Ability to Solve the Elliptic Curve Discrete Logarithm Problem?

SSRN Electronic Journal ◽

10.2139/ssrn.3232101 ◽

2018 ◽

Author(s):

Neeraj Samtani

Keyword(s):

Quantum Computing ◽

Elliptic Curve ◽

Discrete Logarithm ◽

Discrete Logarithm Problem

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On the first fall degree of summation polynomials

Journal of Mathematical Cryptology ◽

10.1515/jmc-2017-0022 ◽

2019 ◽

Vol 13 (3-4) ◽

pp. 229-237

Author(s):

Stavros Kousidis ◽

Andreas Wiemers

Keyword(s):

Elliptic Curve ◽

Gröbner Basis ◽

Discrete Logarithm ◽

Polynomial Systems ◽

Discrete Logarithm Problem ◽

Grobner Basis ◽

Weil Descent ◽

Degree Bound

Abstract We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev’s summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gröbner basis algorithms.

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Improved cryptanalysis of a ElGamal Cryptosystem Based on Matrices Over Group Rings

Journal of Mathematical Cryptology ◽

10.1515/jmc-2019-0054 ◽

2020 ◽

Vol 15 (1) ◽

pp. 266-279

Author(s):

Atul Pandey ◽

Indivar Gupta ◽

Dhiraj Kumar Singh

Keyword(s):

Discrete Logarithm ◽

Public Key Cryptography ◽

Public Key ◽

Quantum Computers ◽

Group Rings ◽

Diffie Hellman ◽

Algebra Approach ◽

Elgamal Cryptosystem ◽

Diffie Hellman Key Exchange ◽

Equivalent Keys

AbstractElGamal cryptosystem has emerged as one of the most important construction in Public Key Cryptography (PKC) since Diffie-Hellman key exchange protocol was proposed. However, public key schemes which are based on number theoretic problems such as discrete logarithm problem (DLP) are at risk because of the evolution of quantum computers. As a result, other non-number theoretic alternatives are a dire need of entire cryptographic community.In 2016, Saba Inam and Rashid Ali proposed a ElGamal-like cryptosystem based on matrices over group rings in ‘Neural Computing & Applications’. Using linear algebra approach, Jia et al. provided a cryptanalysis for the cryptosystem in 2019 and claimed that their attack could recover all the equivalent keys. However, this is not the case and we have improved their cryptanalysis approach and derived all equivalent key pairs that can be used to totally break the ElGamal-like cryptosystem proposed by Saba and Rashid. Using the decomposition of matrices over group rings to larger size matrices over rings, we have made the cryptanalysing algorithm more practical and efficient. We have also proved that the ElGamal cryptosystem proposed by Saba and Rashid does not achieve the security of IND-CPA and IND-CCA.

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