Munesh Kumari

Kalika Prasad, Hrishikesh Mahato, Munesh Kumari

In this article, we have proposed a generalized Lucas matrix (recursive matrix of higher order) having relation with generalized Fibonacci sequences and established many special properties in addition to that usual matrix algebra. Further, we have proposed a modified public key cryptography using these matrices as keys in Affine cipher and key agreement for encryption-decryption with the combination of terms of generalized Lucas sequences under residue operations. In this scheme, instead of exchanging the whole key matrix, only a pair of numbers(parameters) need to be exchanged, which reduces the time complexity as well as space complexity of the key transmission and has a large key-space.

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.

Kalika Prasad, Munesh Kumari

In this review article, we discussed the Mathematics and mechanics behind the Enigma machine with an analysis of security strength. The German army used the Enigma machine during the second world war to encrypt communications. Due to its complexity, the encryption done by the Enigma Machine was assumed to be almost unbreakable. However, the Polish believed that people with good background and deep knowledge of science and mathematics would have a better chance to break the encryption done by Enigma. They appointed twenty mathematicians from Poznan University to work on this problem at the Polish Cipher Bureau. Three of those, Marian Rejewski, Jerzy Rozycki and Henryk Zygalski were able to exploit certain flaws in the encryption, and by using permutation group theory finally managed to decipher the Enigma messages. The mathematics discovered by them is presented here.

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).