Sensitive Information Security Based on Elliptic Curves

The elliptic curve cryptography algorithm represents a major advancement in the field of computer security. This innovative algorithm uses elliptic curves to encrypt and secure data, providing an exceptional level of security while optimizing the efficiency of computer resources. This study focuses on how elliptic curves cryptography helps to protect sensitive data. Text is encrypted using the elliptic curve technique because it provides great security with a smaller key on devices with limited resources, such as mobile phones. The elliptic curves cryptography of this study is better than using a 256-bit RSA key. To achieve equivalent protection by using the elliptic curves cryptography, several Python libraries such as cryptography, pycryptodome, pyQt5, secp256k1, etc. were used. These technologies are used to develop a software based on elliptic curves. If built, the software helps to encrypt and decrypt data such as a text messages and it offers the authentication for the communication.

Remote sensing image encryption algorithm based on novel hyperchaos and an elliptic curve cryptosystem

Remote sensing images carry crucial ground information,often involving the spatial distribution and spatiotemporal changes of surface elements.To safeguard this sensitive data,image encryption technology is essential.In this paper,a novel Fibonacci sine exponential map is designed,the hyperchaotic performance of which is particularly suitable for image encryption algorithms.An encryption algorithm tailored for handling the multi-band attributes of remote sensing images is proposed.The algorithm combines a three-dimensional synchronized scrambled diffusion operation with chaos to efficiently encrypt multiple images.Moreover,the keys are processed using an elliptic curve cryptosystem,eliminating the need for an additional channel to transmit the keys,thus enhancing security.Experimental results and algorithm analysis demonstrate that the algorithm offers strong security and high efficiency,making it suitable for remote sensing image encryption tasks.

Cryptanalysis and Improvement of Signcryption Schemes on Elliptic Curves

In this paper, we analyze two signcryption schemes on elliptic curves proposed by Zheng Yu-liang and Hideki Imai. We point out a serious problem with the schemes that the elliptic curve based signcryption schemes lose confidentiality to gain non-repudiation. We also propose two improvement versions that not only overcome the security leak inherent in the schemes but also provide public verifiability or forward security. Our improvement versions require smaller computing cost than that required by signature-then-encryption methods.

Text Encryption Using Pell Sequence and Elliptic Curves with Provable Security

The demand for data security schemes has increased with the significant advancement in the field of computation and communication networks.We propose a novel three-step text encryption scheme that has provable security against computation attacks such as key attack and statistical attack.The proposed scheme is based on the Pell sequence and elliptic curves,where at the first step the plain text is diffused to get a meaningless plain text by applying a cyclic shift on the symbol set.In the second step,we hide the elements of the diffused plain text from the attackers.For this purpose,we use the Pell sequence,a weight function,and a binary sequence to encode each element of the diffused plain text into real numbers.The encoded diffused plain text is then confused by generating permutations over elliptic curves in the third step.We show that the proposed scheme has provable security against key sensitivity attack and statistical attacks.Furthermore,the proposed scheme is secure against key spacing attack,ciphertext only attack,and known-plaintext attack.Compared to some of the existing text encryption schemes,the proposed scheme is highly secure against modern cryptanalysis.

Binary Sequences from a Pair of Elliptic Curves

A family of binary sequences were constructed by using an elliptic curve and its twisted curves over finite fields. It was shown that these sequences possess ＂good＂ cryptographie properties of 0-1 distribution, long period and large linear complexity. The results indicate that such se quences provide strong potential applications in cryptography.

Computation of Complex Primes Using Elliptic Curves: Application for Cryptosystem Design

This paper provides several generalizations of Gauss theorem that counts points on special elliptic curves. It is demonstrated how to implement these generalizations for computation of complex primes, which are applicable in several protocols providing security in communication networks. Numerical examples illustrate the ideas discussed in this paper.

Constructing quasi-random subsets of Z_N by using elliptic curves

Let ε ： y^2 = x3 ＋ Ax ＋ B be an elliptic curve defined over the finite field Zp（p 〉 3） and G be a rational point of prime order N on ε. Define a subset of ZN, the residue class ring modulo N, asS：={n：n∈ZN,n≠0,（X（nG）/p）=1} where X（nG） denotes the x-axis of the rational points nC and （＊/P） is the Legendre symbol. Some explicit results on quasi-randomness of S are investigated. The construction depends on the intrinsic group structures of elliptic curves and character sums along elliptic curves play an important role in the proofs.

An Algebraic Proof of the Associative Law of Elliptic Curves

In this paper we revisit the addition of elliptic curves and give an algebraic proof to the associative law by use of MATHEMATICA. The existing proofs of the associative law are rather complicated and hard to understand for beginners. An ‘‘elementary” proof to it based on algebra has not been given as far as we know. Undergraduates or non-experts can master the addition of elliptic curves through this paper. After mastering it they should challenge the elliptic curve cryptography.

On the Torsion Subgroups of Certain Elliptic Curves over Q

Let E be an elliptic curve over a given number field . By Mordell’s Theorem, the torsion subgroup of E defined over Q is a finite group. Using Lutz-Nagell Theorem, we explicitly calculate the torsion subgroup E(Q)tors for certain elliptic curves depending on their coefficients.

On Two Problems About Isogenies of Elliptic Curves Over Finite Fields

Isogenies occur throughout the theory of elliptic curves.Recently,the cryptographic protocols based on isogenies are considered as candidates of quantum-resistant cryptographic protocols.Given two elliptic curves E1,E2 defined over a finite field k with the same trace,there is a nonconstant isogeny b from E2 to E1 defined over k.This study gives out the index of Homk(E1,E2)b as a nonzero left ideal in Endk(E2)and figures out the correspondence between isogenies and kernel ideals.In addition,some results about the non-trivial minimal degree of isogenies between two elliptic curves are also provided.

Cryptographic Schemes Based on Elliptic Curves over the Ring Zp[i]

Elliptic Curve Cryptography recently gained a lot of attention in industry. The principal attraction of ECC compared to RSA is that it offers equal security for a smaller key size. The present paper includes the study of two elliptic curve and defined over the ring where . After showing isomorphism between and , we define a composition operation (in the form of a mapping) on their union set. Then we have discussed our proposed cryptographic schemes based on the elliptic curve . We also illustrate the coding of points over E, secret key exchange and encryption/decryption methods based on above said elliptic curve. Since our proposed schemes are based on elliptic curve of the particular type, therefore the proposed schemes provides a highest strength-per-bit of any cryptosystem known today with smaller key size resulting in faster computations, lower power assumption and memory. Another advantage is that authentication protocols based on ECC are secure enough even if a small key size is used.

SPEED UP RATIONAL POINT SCALAR MULTIPLICATIONS ON ELLIPTIC CURVES BY FROBENIUS EQUATIONS

Let q be a power of a prime and φ be the Frobenius endomorphism on E（Fqk）, then q = tφ - φ^2. Applying this equation, a new algorithm to compute rational point scalar multiplications on elliptic curves by finding a suitable small positive integer s such that q^s can be represented as some very sparse φ-polynomial is proposed. If a Normal Basis （NB） or Optimal Normal Basis （ONB） is applied and the precomputations are considered free, our algorithm will cost, on average, about 55% to 80% less than binary method, and about 42% to 74% less than φ-ary method. For some elliptic curves, our algorithm is also taster than Mǖller＇s algorithm. In addition, an effective algorithm is provided for finding such integer s.

On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields

We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quadratic fields Q(m) for m≤200. These results of computations give best-possible data including structures of Mordell-Weil groups over some real quadratic fields via two-descent. We also prove similar results for the case of certain cubic fields. Especially, we give the first example of elliptic curve having everywhere good reduction over a pure cubic field using our method.

Secure and efficient elliptic curve cryptography resists side-channel attacks

An embedded cryptosystem needs higher reconfiguration capability and security. After analyzing the newly emerging side-channel attacks on elliptic curve cryptosystem （ECC）, an efficient fractional width-w NAF （FWNAF） algorithm is proposed to secure ECC scalar multiplication from these attacks. This algorithm adopts the fractional window method and probabilistic SPA scheme to reconfigure the pre-computed table, and it allows designers to make a dynamic configuration on pre-computed table. And then, it is enhanced to resist SPA, DPA, RPA and ZPA attacks by using the random masking method. Compared with the WBRIP and EBRIP methods, our proposals has the lowest total computation cost and reduce the shake phenomenon due to sharp fluctuation on computation performance.

Addition Sequence Method of Scalar Multiplication of Elliptic Curve over OEF

A new elliptic curve scalar multiplication algorithm is proposed. Thealgorithm uses the Frobenius map on optimal extension field (OEF) and addition sequence We introducea new algorithm on generating addition sequence efficiently and also give some analysis about it.Based on this algorithm, a new method of computing scalar multiplication of elliptic curve over anOEF is presented. The new method is more efficient than the traditional scalar multiplicationalgorithms of elliptic curve over OEF. Thecomparisons of traditional method and the new method arealso given.

Integral Points on a Class of Elliptic Curve

We prove all integral points of the elliptic curve y^2=x^2-30x＋133 are （x,y） = （-7,0）,（-3,±14）,（2, ±9）,（6,±13）, （5143326,±11664498677）, by using the method of algebraic number theory and p-adic analysis. Furthermore, we develop a computation method to find all integral points on a class of elliptic curve y^2= （x＋α）（x^2-α）（x^2-αx＋b） ,α ,b∈Z,α^2〈4b and find all integer solutions of hyperelliptic Diophantine equation Dy^2=Ax^4 ＋ Bx^2 ＋C,B^2〈4AC.

Verifiable (t, n) Threshold Signature Scheme Based on Elliptic Curve

Based on tht difficulty of solving the ECDLP (elliptic curve discretelogarithm problem) on the finite field, we present a (t, n) threshold signature scheme and averifiable key agreement scheme without trusted party. Applying a modified elliptic curve signatureequation, we gel a more efficient signature scheme than the existing ECDSA (elliptic curve digitalsignature algorithm) from the computability and security view. Our scheme has a shorter key, fastercomputation, and better security.

RGB Based Multiple Share Creation in Visual Cryptography with Aid of Elliptic Curve Cryptography

In visual cryptography, many shares are generated which are illogical containing certain message within themselves. When all shares are piled jointly, they tend to expose the secret of the image. The notion of visual secret sharing scheme is to encrypt a secret image into n illogical share images. It is unable to reveal any data on the original image if at least one of the shares is not achieved. The original image, in fact, is realized by overlapping the entire shares directly, in order that the human visual system is competent to identify the collective secret image without employing any complicated computational tools. Therefore, they are communicated steadily as number of shares. The elliptic curve cryptography approach, in turn, is employed to augment the privacy and safety of the image. The new.fangled technique is utilized to generate the multiple shares which are subjected to encryption and decryption by means of the elliptic curve cryptography technique. The test outcomes have revealed the fact that the peak signal to noise ratio is 58.0025, Mean square error value is 0.1164 and the correlation coefficient is 1 for the decrypted image without any sort of distortion of the original image.

Elliptic Curve Scalar Multiplication with x-Coordinate

This paper gives a comprehensive method to do Elliptic Curve Scalar Multiplication with only x-coordinate. Explicit point operation formulae for all types of defining equations of the curves are derived. For each type of curve, the performance is analyzed. The formulae are applied in Montgomery Ladder to get scalar multiplication algorithm operated with only x-coordinate. The new scalar multiplication has the same security level and computation amount with protected binary scalar multiplication （PBSM） against side channel attack, and has the advantages of higher security and little memory needed.