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

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.

scholarly journals Modification of Traditional RSA into Symmetric-RSA Cryptosystems

Cryptography ◽  
2020 ◽  
pp. 120-128
Author(s):  
Prerna Mohit ◽  
G. P. Biswas
Keyword(s):  
Digital Signature ◽  
Discrete Logarithm ◽  
Prime Numbers ◽  
Public Key ◽  
Key Exchange Protocol ◽  
Factorization Problem ◽  
Rsa Algorithm ◽  
Diffie Hellman ◽  
Diffie Hellman Key Exchange ◽  
Rsa Cryptography

This paper addresses the modification of RSA cryptography namely Symmetric-RSA, which seem to be equally useful for different cryptographic applications such as encryption, digital signature, etc. In order to design Symmetric-RSA, two prime numbers are negotiated using Diffie-Hellman key exchange protocol followed by RSA algorithm. As the new scheme uses Diffie-Hellman and RSA algorithm, the security of the overall system depends on discrete logarithm as well as factorization problem and thus, its security is more than public-key RSA. Finally, some new cryptographic applications of the proposed modifications are described that certainly extend the applications of the existing RSA.

Modification of Traditional RSA into Symmetric-RSA Cryptosystems

2017 ◽  
Vol 13 (1) ◽  
pp. 66-73
Author(s):  
Prerna Mohit ◽  
G. P. Biswas
Keyword(s):  
Digital Signature ◽  
Discrete Logarithm ◽  
Prime Numbers ◽  
Public Key ◽  
Key Exchange Protocol ◽  
Factorization Problem ◽  
Rsa Algorithm ◽  
Diffie Hellman ◽  
Diffie Hellman Key Exchange ◽  
Rsa Cryptography

This paper addresses the modification of RSA cryptography namely Symmetric-RSA, which seem to be equally useful for different cryptographic applications such as encryption, digital signature, etc. In order to design Symmetric-RSA, two prime numbers are negotiated using Diffie-Hellman key exchange protocol followed by RSA algorithm. As the new scheme uses Diffie-Hellman and RSA algorithm, the security of the overall system depends on discrete logarithm as well as factorization problem and thus, its security is more than public-key RSA. Finally, some new cryptographic applications of the proposed modifications are described that certainly extend the applications of the existing RSA.

scholarly journals Algebraic generalization of Diffie–Hellman key exchange

2018 ◽  
Vol 12 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Juha Partala
Keyword(s):  
Cyclic Group ◽  
Discrete Logarithm ◽  
Key Exchange ◽  
Public Key ◽  
Algebraic Operation ◽  
Average Case ◽  
Diffie Hellman ◽  
Average Case Complexity ◽  
Algebraic Generalization ◽  
Diffie Hellman Key Exchange

AbstractThe Diffie–Hellman key exchange scheme is one of the earliest and most widely used public-key primitives. Its underlying algebraic structure is a cyclic group and its security is based on the discrete logarithm problem (DLP). The DLP can be solved in polynomial time for any cyclic group in the quantum computation model. Therefore, new key exchange schemes have been sought to prepare for the time when quantum computing becomes a reality. Algebraically, these schemes need to provide some sort of commutativity to enable Alice and Bob to derive a common key on a public channel while keeping it computationally difficult for the adversary to deduce the derived key. We suggest an algebraically generalized Diffie–Hellman scheme (AGDH) that, in general, enables the application of any algebra as the platform for key exchange. We formulate the underlying computational problems in the framework of average-case complexity and show that the scheme is secure if the problem of computing images under an unknown homomorphism is infeasible. We also show that a symmetric encryption scheme possessing homomorphic properties over some algebraic operation can be turned into a public-key primitive with the AGDH, provided that the operation is complex enough. In addition, we present a brief survey on the algebraic properties of existing key exchange schemes and identify the source of commutativity and the family of underlying algebraic structures for each scheme.

scholarly journals Survey on Asymmetric Key Cryptographic Algorithms

2020 ◽  
pp. 404-408
Author(s):  
Sabitha S ◽  
Binitha V Nair
Keyword(s):  
Public Key Cryptography ◽  
Public Key ◽  
The Public ◽  
Private Key ◽  
Design And Implementation ◽  
Diffie Hellman ◽  
Encryption And Decryption ◽  
Cryptographic Algorithms ◽  
Symmetric Key ◽  
Symmetric Key Cryptography

Cryptography is an essential and effective method for securing information’s and data. Several symmetric and asymmetric key cryptographic algorithms are used for securing the data. Symmetric key cryptography uses the same key for both encryption and decryption. Asymmetric Key Cryptography also known as public key cryptography uses two different keys – a public key and a private key. The public key is used for encryption and the private key is used for decryption. In this paper, certain asymmetric key algorithms such as RSA, Rabin, Diffie-Hellman, ElGamal and Elliptical curve cryptosystem, their security aspects and the processes involved in design and implementation of these algorithms are examined.

Problems in Cryptography and Cryptanalysis

2018 ◽  
pp. 23-39
Author(s):  
Kannan Balasubramanian ◽  
Rajakani M.
Keyword(s):  
Elliptic Curve ◽  
Optimization Problems ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Random Number Generators ◽  
Factorization Problem ◽  
Diffie Hellman ◽  
Integer Factorization Problem ◽  
Diffie Hellman Key Exchange ◽  
Membership Problems

The integer factorization problem used in the RSA cryptosystem, the discrete logarithm problem used in Diffie-Hellman Key Exchange protocol and the Elliptic Curve Discrete Logarithm problem used in Elliptic Curve Cryptography are traditionally considered the difficult problems and used extensively in the design of cryptographic algorithms. We provide a number of other computationally difficult problems in the areas of Cryptography and Cryptanalysis. A class of problems called the Search problems, Group membership problems, and the Discrete Optimization problems are examples of such problems. A number of computationally difficult problems in Cryptanalysis have also been identified including the Cryptanalysis of Block ciphers, Pseudo-Random Number Generators and Hash functions.

Sustainability of Public Key Cryptosystem in Quantum Computing Paradigm

2018 ◽  
pp. 563-588
Author(s):  
Krishna Asawa ◽  
Akanksha Bhardwaj
Keyword(s):  
Quantum Computing ◽  
Secure Communication ◽  
Prime Numbers ◽  
Cryptographic Protocols ◽  
Public Key ◽  
Quantum Computers ◽  
New Paradigm ◽  
Public Key Cryptosystems ◽  
Computing Paradigm ◽  
Diffie Hellman

With the emergence of technological revolution to host services over Internet, secure communication over World Wide Web becomes critical. Cryptographic protocols are being in practice to secure the data transmission over network. Researchers use complex mathematical problem, number theory, prime numbers etc. to develop such cryptographic protocols. RSA and Diffie Hellman public key crypto systems have proven to be secure due to the difficulty of factoring the product of two large primes or computing discrete logarithms respectively. With the advent of quantum computers a new paradigm shift on public key cryptography may be on horizon. Since superposition of the qubits and entanglement behavior exhibited by quantum computers could hold the potential to render most modern encryption useless. The aim of this chapter is to analyze the implications of quantum computing power on current public key cryptosystems and to show how these cryptosystems can be restructured to sustain in the new computing paradigm.

Sustainability of Public Key Cryptosystem in Quantum Computing Paradigm

2016 ◽  
pp. 664-688
Author(s):  
Krishna Asawa ◽  
Akanksha Bhardwaj
Keyword(s):  
Quantum Computing ◽  
Secure Communication ◽  
Prime Numbers ◽  
Cryptographic Protocols ◽  
Public Key ◽  
Quantum Computers ◽  
New Paradigm ◽  
Public Key Cryptosystems ◽  
Computing Paradigm ◽  
Diffie Hellman

With the emergence of technological revolution to host services over Internet, secure communication over World Wide Web becomes critical. Cryptographic protocols are being in practice to secure the data transmission over network. Researchers use complex mathematical problem, number theory, prime numbers etc. to develop such cryptographic protocols. RSA and Diffie Hellman public key crypto systems have proven to be secure due to the difficulty of factoring the product of two large primes or computing discrete logarithms respectively. With the advent of quantum computers a new paradigm shift on public key cryptography may be on horizon. Since superposition of the qubits and entanglement behavior exhibited by quantum computers could hold the potential to render most modern encryption useless. The aim of this chapter is to analyze the implications of quantum computing power on current public key cryptosystems and to show how these cryptosystems can be restructured to sustain in the new computing paradigm.

scholarly journals A Group Law on the Projective Plane with Applications in Public Key Cryptography

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 734
Author(s):  
Raúl Durán Díaz ◽  
Luis Hernández Encinas ◽  
Jaime Muñoz Masqué
Keyword(s):  
Projective Plane ◽  
Multiplicative Group ◽  
Mathematical Problem ◽  
Discrete Logarithm ◽  
Public Key Cryptography ◽  
Public Key ◽  
Arbitrary Field ◽  
Base Field ◽  
Computational Difficulty ◽  
Group Law

In the context of new threats to Public Key Cryptography arising from a growing computational power both in classic and in quantum worlds, we present a new group law defined on a subset of the projective plane F P 2 over an arbitrary field F , which lends itself to applications in Public Key Cryptography and turns out to be more efficient in terms of computational resources. In particular, we give explicitly the number of base field operations needed to perform the mentioned group law. Based on it, we present a Diffie-Hellman-like key agreement protocol. We analyze the computational difficulty of solving the mathematical problem underlying the proposed Abelian group law and we prove that the security of our proposal is equivalent to the discrete logarithm problem in the multiplicative group of the cubic extension of the finite field considered. We present an experimental setup in order to show real computation times along a comparison with the group operation in the group of points of an elliptic curve. Based on current state-of-the-art algorithms, we provide parameter ranges suitable for real world applications. Finally, we present a promising variant of the proposed group law, by moving from the base field F to the ring Z / p q Z , and we explain how the security becomes enhanced, though at the cost of a longer key length.

A key exchange protocol using matrices over group ring

2019 ◽  
Vol 12 (05) ◽  
pp. 1950075
Author(s):  
Indivar Gupta ◽  
Atul Pandey ◽  
Manish Kant Dubey
Keyword(s):  
Group Ring ◽  
Complexity Analysis ◽  
Discrete Logarithm ◽  
Key Exchange ◽  
Group Rings ◽  
Secret Key ◽  
Key Exchange Protocol ◽  
Diffie Hellman ◽  
Shared Secret ◽  
Invertible Matrices

The first published solution to key distribution problem is due to Diffie–Hellman, which allows two parties that have never communicated earlier, to jointly establish a shared secret key over an insecure channel. In this paper, we propose a new key exchange protocol in a non-commutative semigroup over group ring whose security relies on the hardness of Factorization with Discrete Logarithm Problem (FDLP). We have also provided its security and complexity analysis. We then propose a ElGamal cryptosystem based on FDLP using the group of invertible matrices over group rings.

Explicit constructions of extremal graphs and new multivariate cryptosystems

2015 ◽  
Vol 52 (2) ◽  
pp. 185-204 ◽  
Author(s):  
Vasyl Ustimenko
Keyword(s):  
Discrete Logarithm ◽  
Public Key Infrastructure ◽  
Public Key ◽  
Polynomial Degree ◽  
Affine Spaces ◽  
Private Key ◽  
Explicit Constructions ◽  
Diffie Hellman ◽  
Polynomial Transformations ◽  
Finite Commutative Ring

New multivariate cryptosystems are introduced. Sequences f(n) of bijective polynomial transformations of bijective multivariate transformations of affine spaces Kn, n = 2, 3, ... , where K is a finite commutative ring with special properties, are used for the constructions of cryptosystems. On axiomatic level, the concept of a family of multivariate maps with invertible decomposition is proposed. Such decomposition is used as private key in a public key infrastructure. Requirements of polynomiality of degree and density allow to estimate the complexity of encryption procedure for a public user. The concepts of stable family and family of increasing order are motivated by studies of discrete logarithm problem in Cremona group. Statement on the existence of families of multivariate maps of polynomial degree and polynomial density with the invertible decomposition is formulated. We observe known explicit constructions of special families of multivariate maps. They correspond to explicit constructions of families of nonlinear algebraic graphs of increasing girth which appeared in Extremal Graph Theory. The families are generated by pseudorandom walks on graphs. This fact ensures the existence of invertible decomposition; a certain girth property guarantees the increase of order for the family of multivariate maps, good expansion properties of families of graphs lead to good mixing properties of graph based private key algorithms. We describe the general schemes of cryptographic applications of such families (public key infrastructure, symbolic Diffie—Hellman protocol, functional versions of El Gamal algorithm).

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