Jagmohan Tanti

Munesh Kumari, Jagmohan Tanti

In this paper, we have proposed a modified cryptographic scheme based on the application of recursive matrices as key in ECC and ElGamal. For encryption, we consider mapping analogous to affine Hill cipher in which a plaintext matrix has been constructed by points corresponding to letters on elliptic curves. In the formation of key-space, the generalized Fibonacci matrices have been taken into account, which is the sequence of matrices. The beauty of considering Fibonacci matrices is their construction where we need only two parameters(integers) in place of $n^2$ elements. The use of a recursive matrix makes a large keyspace for our proposed scheme and increases its efficiency. Thus, it reduces time as well space complexity, and its security \& strength is based on EC-DLP which is a hard problem in number theory.

Munesh Kumari, Jagmohan Tanti

In this paper, we have proposed a public key cryptography using recursive block matrices involving generalized Fibonacci numbers over a finite field Fp. For this, we define multinacci block matrices, a type of upper triangular matrix involving multinacci matrices at diagonal places and obtained some of its algebraic properties. Moreover, we have set up a method for key element agreement at end users, which makes the cryptography more efficient. The proposed cryptography comes with a large keyspace and its security relies on the Discrete Logarithm Problem(DLP).

Md. Helal Ahmed, Jagmohan Tanti, Sumant Pushp

In this paper, we present the fast computational algorithms for the Jacobi sums of orders $l^2$ and $2l^{2}$ with odd prime $l$ by formulating them in terms of the minimum number of cyclotomic numbers of the corresponding orders. We also implement two additional algorithms to validate these formulae, which are also useful for the demonstration of the minimality of cyclotomic numbers required.

Md. Helal Ahmed, Jagmohan Tanti, Sumant Pushp

Confidentiality and Integrity are two paramount objectives in the evaluation of information and communication technology. In this paper, we propose an arithmetic approach for designing asymmetric key cryptography. Our method is based on the formulation of cyclotomic matrices correspond to the diophantine system. The proposed cyclotomic asymmetric cryptosystem (CAC) utilizes the cyclotomic matrices, whose entries are cyclotomic numbers of order $2l^{2}$, $l$ be prime over a finite field $\mathbb{F}_{p}$ of $p$ elements. The method utilize cyclotomic matrices to design a one-way function. The outcome of a one-way function that is efficient to compute however difficult to compute its inverse unless if secret data about the trapdoor is known. We demonstrate that the encryption and decryption can be efficiently performed with asymptotic complexity of $\mathcal{O}(e^{2.373})$. Besides, we study the computational complexity of the CAC.