Cryptosystems based on Elliptic Curves and Recurrent Sequences

Abstract

People have been using the internet to transmit and store data in recent years. The latest advancements in computing and communication technology have increased the necessity for data security strategies. In this thesis, we generated random numbers and permutations using the points on elliptic curves to build one of the most e cient and unbreakable cryptographic techniques. Thetwofundamental categories of data security are cryptography and steganography. Steganog raphy is the study of data security schemes where condential information is incorporated into host data so that the attackers cannot detect the existence of secret data [1]. Cryptography is the study of data security schemes where secret data is transformed into an unreadable data [2]. Encryption is the key element within cryptography for the provision of security. In encryption, keys of the same or sometimes smaller length are used for encoding messages or data. Mostly ciphering is used for the accomplishment of encryption and decryption at both transmission ends. Cipher is an appropriate way of encrypting or decrypting messages, where ciphering is mostly dependent on the encryption key. A single key is utilized In symmetric cryptography, for both encryption and decryption. Asymmetric cryptography uses two dis tinct keys, rst one encrypts the data and the other for decryption. Recently, elliptic curves (ECs) are used to design strong cryptosystems which creates smaller, quicker, and more ef fective keys with high security. In the second and third century A.D. the elliptic curve rst appeared in the Diophantuss work. In fact, a polynomial equation with integer or rational solutions is known as a Diophantine equation. An elliptic curve, E over a eld K is a set of points that satis es a plane cubic curve. In the eld of cryptography, we use elliptic curve over a nite eld. A non-singular ii cubic curve E which can be represented as follows Y2 =X3+ X+ , is an elliptic curve over eld K with a point , where the point exists on every vertical line and belong to eld K. For an elliptic curve, E over a eld K, the chord-and-tangent rule exists to add two points of the curve. The set of all points of the elliptic curve along with this addition operation generates an abelian group. The number of points lying on an elliptic curve over the nite eld K are also nite, so they construct a nite abelian group. Torsion point is a point of nite order and if it has order n then it is called n-torsion point. Hasses theorem provides bounds to compute number of points on an EC. In general group law makes it di cult to calculate all curve points [3]. The readers are referred to [2] and [3] for additional information regarding cryptography and elliptic curves, respectively. Based on various mathematical structures, several data security methods have been presented. Schneier [4] developed a new secret-key block cipher, Blowsh. Rahouma [5] suggested a block cipher data scheme for computer network security. Gupta et al. [6] proposed a data security algorithm depending on logical and shifting operations. Pattanayak et al. [7] used extended Euclidean algorithm and linear congruences to design a text encryption scheme. Abdullah et al. [8] proposed a cryptosystem based on fuzzy logic where triangular fuzzy numbers are used to represent plaintext and ciphertext. In the same way, elliptic curves gained consider able interest in the eld of cryptography due to their comparative protection against modern cryptanalysis with low key size. Miller developed an encryption scheme based on elliptic curve cryptography similar but about 20 percent quicker than the Di e-Hellman key exchange pro tocol. The concept of a discrete logarithmic problem used in Di e-public Hellmans key cryptography was applied by Koblitz et al. [9] to the EC group. Amara et al. [10] describe elliptic curve cryptographys network security task with a smaller key size and contrast RSA to elliptic curve cryptography, and conclude that it is a better encryption alternative. An EC iii cryptography scheme using the microcontroller with fuzzy modular arithmetic was designed by Ganapathy et al. [11]. Balamurugan et al. [12] used a non-singular matrix to construct a rapid mapping scheme by mapping the plaintext to points of elliptic curve and the ElGamal encryption approach is used to decrypt points using a non-singular matrix. Hayat et al. [13] used elliptic curves to generate pseudo-random numbers as well as S-boxes, then they applied these S-boxes and random numbers in image encryption. Several authors who have applied text encryption and decryption using elliptic curve cryp tography have used an accepted table consisting of mapping characters and elliptic curve coordinates, or the ASCII values of the characters are used to derive a ne elliptic curve coordinates by performing point multiplication on the generator G and the corresponding character ASCII value. For cryptographic and other applications, researchers are still using ECs. But from literature review, it follows that mathematically structured elliptic curves particularly as ordered elliptic curves, have not yet been deployed for data security as per our knowledge. As a result, this fact drives us to develop new mathematical structures based on elliptic curves to enhance data security. We focus on Mordell elliptic curves, a special elliptic curve, to achieve the following objectives. i. To construct the sequences of random numbers by using elliptic curves. ii. Use the above structures to create confusion and di usion in the text encryption algo rithm. iii. Utilizing the existing elliptic curve structures to construct new structures. iv. Next, create new schemes employing the above novel structures to construct non-linear cryptosystem components. v. An image encryption algorithm using the aforementioned constructed non-linear cryp tosystem components to evaluate their e ectiveness. iv vi. To get an encryption algorithm well suited for large size data that can be used for any script with speci ed ASII values, not just for English scripts. vii. This work also aims at getting e cient encryption algorithm, accuracy, and safety fea tures to maintain the security of data during decryption process. viii. Stating and proving theoretical results. The outline of this thesis is structured as follows: In chapter 1, basic concepts related to cryptography, pell sequences and elliptic curves over nite elds/rings are brie y discussed. In chapter 2, We suggest a brand-new, three-step text encryption method that can be shown to be secure against computation-based attacks like key and statistical attack. The rst stage of the suggested system, which is based on elliptic curves and the pell sequence, is applying a cyclic shift to the symbol set to transform the plaintext into an abstract plaintext. The second phase involves hiding the elements of the di used plaintext from the intruders. In the third stage, permutations over elliptic curves are generated in order to confuse the dis tributed plaintext that has been encoded. We demonstrate the provable security of the sug gested approach against some important attacks. The suggested system is also resistant tokey spacing attacks,known-plaintext attacks, andciphertext-only attacks. The suggested scheme is extremely secure against present cryptanalysis than some of the existing text encryption algorithms. In chapter 3, a novel method of new substitution box construction based on ECs over nite rings is proposed. In addition, the newly developed method is thoroughly evaluated and com pared with a few other methods that already exist. Experimental results show that the newly designed substitution box is signi cantly more resistant to linear attacks and has a higher capacity for confusion than some of the existing substitution boxes. In chapter 4, we introduced elliptic curves over nite rings based image encryption algorithm. Our scheme consists of three main steps, the rst of which is to use points from an EC over v a nite ring to mask the plain image. In the second phase, we generate di usion by mapping the EC over the nite ring to the EC over the nite eld. A substitution box (S-box) is used to permute the pixels of the di used image to create a cipher image, which will signi cantly confuse the plaintext. We demonstrated that the suggested scheme is more secure against di erential, statical, and linear attacks than existing cryptosystems using computational ex periments. Additionally, color image encryption takes less time on average than other existing methods. In chapter 5, the work provided in this thesis is summarized and some prospective avenues for further research are discussed