From Golden to Unimodular Cryptography

2019

A.P. Stakhov in [6] proposed the concepts golden matrices and new kind of cryptography. In this paper, we proposed a digital signature scheme using Golden matrices based on Matrix extension of RSA Cryptosystem. This proposed work is very rapid fast and simple for technical recognition and can be used for signature protection of digital signals like telecommunication and measurement system.

Boletim da Sociedade Paranaense de Matemática, 2017

In this work we introduce a new method of cryptography based on the matrices over a finite field Fq, were q is a power of a prime number p. The first time we construct the matrix M = A 1 A 2 0 A 3 were A 1 , A 2 and A 3 are matrices of order n with coefficients in Fq and 0 is the zero matrix of order n. We prove that M l = A l 1

Electronic Journal of Mathematical Analysis and Applications

We are aware that a major cryptosystem element plays a crucial part in maintaining the security and robustness of cryptography. Various researchers are focusing on creating new forms of cryptography and improving those that already exist using the principles of number theory and linear algebra. In this article, we have proposed an Extended generalized Fibonacci matrix (recursive matrix of higher order) having a relation with Extended generalized Fibonacci sequences and established some properties in addition to that usual matrix algebra. Further, we proposed a modified public key cryptography using these matrices as keys in Affine-Hill Cipher and key agreement for encryption-decryption with the combination of terms of Extended generalized Fibonacci sequences under prime modulo. This system has a large key space and reduces the time complexity as well as space complexity of the key transmission by only requiring the exchange of pair of numbers(parameters) as opposed to the entire key matrix.

Cryptography - Recent Advances and Future Developments, 2021

Quantum computers are distinguished by their enormous storage capacity and relatively high computing speed. Among the cryptosystems of the future, the best known and most studied which will resist when using this kind of computer are cryptosystems based on error-correcting codes. The use of problems inspired by the theory of error-correcting codes in the design of cryptographic systems adds an alternative to cryptosystems based on number theory, as well as solutions to their vulnerabilities. Their security is based on the problem of decoding a random code that is NP-complete. In this chapter, we will discuss the cryptographic properties of error-correcting codes, as well as the security of cryptosystems based on code theory.

arXiv: Number Theory, 2020

In this article, we propose a method to construct self orthogonal matrix, orthogonal matrix and anti orthogonal matrix over the finite field. Orthogonal matrices has numerous applications in cryptography, so here we demonstrate the application of weighted orthogonal matrix into cryptography. Using the proposed method of construction we see that it is very easy to transmit the private key and can easily convert the encrypted message into original message and at the same time it will be difficult to get the key matrix for intruder.

Tatra Mountains Mathematical Publications, 2019

In this paper we introduce a new cryptographic system which is based on the idea of encryption due to [McEliece, R. J. A public-key cryptosystem based on algebraic coding theory, DSN Progress Report. 44, 1978, 114–116]. We use the McEliece encryption system with a new linear error-correcting code, which was constructed in [Hannusch, C.—Lakatos, P.: Construction of self-dual binary 22 k, 22 k−1, 2 k-codes, Algebra and Discrete Math. 21 (2016), no. 1, 59–68]. We show how encryption and decryption work within this cryptosystem and we give the parameters for key generation. Further, we explain why this cryptosystem is a promising post-quantum candidate.

EURASIP Journal on Wireless Communications and Networking, 2006

Securing transmission over a wireless network is especially challenging, not only because of the inherently insecure nature of the medium, but also because of the highly error-prone nature of the wireless environment. In this paper, we take a joint encryptionerror correction approach to ensure secure and robust communication over the wireless link. In particular, we design an errorcorrecting cipher (called the high diffusion cipher) and prove bounds on its error-correcting capacity as well as its security. Towards this end, we propose a new class of error-correcting codes (HD-codes) with built-in security features that we use in the diffusion layer of the proposed cipher. We construct an example, 128-bit cipher using the HD-codes, and compare it experimentally with two traditional concatenated systems: (a) AES (Rijndael) followed by Reed-Solomon codes, (b) Rijndael followed by convolutional codes. We show that the HD-cipher is as resistant to linear and differential cryptanalysis as the Rijndael. We also show that any chosen plaintext attack that can be performed on the HD cipher can be transformed into a chosen plaintext attack on the Rijndael cipher. In terms of error correction capacity, the traditional systems using Reed-Solomon codes are comparable to the proposed joint error-correcting cipher and those that use convolutional codes require 10% more data expansion in order to achieve similar error correction as the HD-cipher. The original contributions of this work are (1) design of a new joint error-correction-encryption system, (2) design of a new class of algebraic codes with built-in security criteria, called the high diffusion codes (HD-codes) for use in the HD-cipher, (3) mathematical properties of these codes, (4) methods for construction of the codes, (5) bounds on the error-correcting capacity of the HD-cipher, (6) mathematical derivation of the bound on resistance of HD cipher to linear and differential cryptanalysis, (7) experimental comparison of the HD-cipher with the traditional systems.

Advances in Science, Technology and Engineering Systems Journal, 2017

Journal of the Nigerian Society of Physical Sciences

Goldreich-Goldwasser-Halevi (GGH) encryption scheme is lattice-based cryptography with its security based on the shortest vector problem (SVP) and closest vector problem (CVP) with immunity to almost all attacks, including Shor's quantum algorithm and Nguyen's attack of higher lattice dimension. To improve the efficiency and security of the GGH Scheme by reducing the size of the public basis to be transmitted, we use an hourglass matrix obtained from quadrant interlocking factorization as a public key. The technique of quadrant interlocking factorization to yield a nonsingular hourglass matrix compensates the encryption scheme with better efficiency and security.

International Journal of Computer and Information Technology(2279-0764)

Achieving security is the most important goal for any digital signature scheme. The security of RSA, the most widely used signature is based on the difficulty of factoring of large integers. The minimum key size required for RSA according to current technology is 1024 bits which can be increased with the advancement in technology. Representation of message in the form of matrix can reduce the key size and use of Tribonacci matrices can double the security of RSA. Recently M.Basu et.al introduced a new coding theorycalled Tribonacci coding theory based onTribonacci numbers, that are the generalization ofthe Fibonacci numbers. In this paper we present anew and efficient digital signature scheme usingTribonacci matrices and factoring.