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.

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.

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.

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.

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.

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.

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.

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.

A Generalized MSST Algorithm for Counting Points of Elliptic Curves over F_(p)^(n)

Elliptic curve cryptography is an important part of nowaday's public key cryptosystem.Counting points of elliptic curves over finite fields is of great significance to the selection of safety curves.At present,there are many p-adic algorithms,such as SST algorithm,generalized AGM algorithm,Kedlaya algorithm,etc.,which can deal with the situation of finite fields of small characteristics.In this paper,the authors generalize the MSST algorithm of characteristic 2 to general fields of odd characteristic,and propose the generalized MSST algorithm.The generalized MSST algorithm is achieved by combining the advantages of the SST algorithm and the generalized AGM algorithm.If the time complexity of the multiplication of two n-bit numbers is denoted as O((n)^(μ)),then the time complexity of the generalized MSST algorithm is O(n^(2μ+1/1+μ)),which is the same as the improved SST algorithm.In practical experiments,the running time of the generalized MSST algorithm is less than that of the improved SST algorithm.

Message Verification Protocol Based on Bilinear Pairings and Elliptic Curves for Enhanced Security in Vehicular Ad Hoc Networks

Vehicular ad hoc networks(VANETs)provide intelligent navigation and efficient route management,resulting in time savings and cost reductions in the transportation sector.However,the exchange of beacons and messages over public channels among vehicles and roadside units renders these networks vulnerable to numerous attacks and privacy violations.To address these challenges,several privacy and security preservation protocols based on blockchain and public key cryptography have been proposed recently.However,most of these schemes are limited by a long execution time and massive communication costs,which make them inefficient for on-board units(OBUs).Additionally,some of them are still susceptible to many attacks.As such,this study presents a novel protocol based on the fusion of elliptic curve cryptography(ECC)and bilinear pairing(BP)operations.The formal security analysis is accomplished using the Burrows–Abadi–Needham(BAN)logic,demonstrating that our scheme is verifiably secure.The proposed scheme’s informal security assessment also shows that it provides salient security features,such as non-repudiation,anonymity,and unlinkability.Moreover,the scheme is shown to be resilient against attacks,such as packet replays,forgeries,message falsifications,and impersonations.From the performance perspective,this protocol yields a 37.88%reduction in communication overheads and a 44.44%improvement in the supported security features.Therefore,the proposed scheme can be deployed in VANETs to provide robust security at low overheads.

Extreme central L-values of almost prime quadratic twists of elliptic curves

In this paper,we prove the extreme values of L-functions at the central point for almost prime quadratic twists of an elliptic curve.As an application,we get the extreme values for the Tate-Shafarevich groups in the quadratic twist family of an elliptic curve under the Birch and Swinnerton-Dyer conjecture.

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.

Cryptography on elliptic curves over p-adic number fields

In this paper we introduce a cryptosystem based on the quotient groups of the group of rational points of an elliptic curve defined over p-adic number field. Some additional parameters are taken in this system, which have an advantage in performing point multiplication while keeping the security of ECC over finite fields. We give a method to select generators of the cryptographic groups, and give a way to represent the elements of the quotient groups with finitely bounded storage by establishing a bijection between these elements and their approximate coordinates. The addition formula under this representation is also presented.

Elliptic curves and positive definite ternary forms

For two given ternary quadratic forms f( x, y, z) and g( x, y, z), let r( f, n) and r( g,n) be the numbers of representations of n represented by f( x, y, z) and g( x, y, z) respectively. In this paper we study the following problem: when will we have r( f, n) = r( g, n) or r( f, n) ≠ r( g, n).Our method is to use elliptic curves and the corresponding new forms.

A Low-Cost and High-Performance Cryptosystem Using Tripling-Oriented Elliptic Curve

Developing a high-performance public key cryptosystem is crucial for numerous modern security applications.The Elliptic Curve Cryptosystem(ECC)has performance and resource-saving advantages compared to other types of asymmetric ciphers.However,the sequential design implementation for ECC does not satisfy the current applications’performance requirements.Therefore,several factors should be considered to boost the cryptosystem performance,including the coordinate system,the scalar multiplication algo-rithm,and the elliptic curve form.The tripling-oriented(3DIK)form is imple-mented in this work due to its minimal computational complexity compared to other elliptic curves forms.This experimental study explores the factors playing an important role in ECC performance to determine the best combi-nation that leads to developing high-speed ECC.The proposed cryptosystem uses parallel software implementation to speed up ECC performance.To our knowledge,previous studies have no similar software implementation for 3DIK ECC.Supported by using parallel design,projective coordinates,and a fast scalar multiplication algorithm,the proposed 3DIK ECC improved the speed of the encryption process compared with other counterparts and the usual sequential implementation.The highest performance level for 3DIK ECC was achieved when it was implemented using the Non-Adjacent Form algorithm and homogenous projection.Compared to the costly hardware implementations,the proposed software implementation is cost effective and can be easily adapted to other environments.In addition,the power con-sumption of the proposed ECC is analyzed and compared with other known cryptosystems.thus,the current study presents a detailed overview of the design and implementation of 3DIK ECC.

On the Birch-Swinnerton-Dyer Conjecture of Elliptic Curves E_D:y^2=x^3-D^2x

We prove in this paper that the BSD conjecture holds for a certain kind of elliptic curves.

Congruent elliptic curves with non-trivial Shafarevich-Tate groups

We study 2-primary parts Ш（E（n）/Q）[2 ∞] of Shafarevich-Tate groups of congruent elliptic curves E（n） ： y2 = x3-n2x, n ∈ Q×/Q×2. Previous results focused on finding sufficient conditions for Ш（E（n）/Q）[2∞] trivial or isomorphic to （Z/2Z）2. Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E（n） is isomorphic to （Z/2Z）2 and the Mordell-Weil rank of E（n） is zero, provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for Ш（E（n）/Q）[2∞] （Z/2Z）2k, where k ≥ 2.

Congruent elliptic curves with non-trivial Shafarevich-Tate groups： Distribution part

Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.

On the isomorphism classes of Legendre elliptic curves over finite fields

In this paper,the number of isomorphism classes of Legendre elliptic curves over finite field is enumerated.

A Criterion for Elliptic Curves with Second Lowest 2-Power in L(1)(Ⅱ)

Let D = p1p2 …pm, where p1,p2, ……,pm are distinct rational primes with p1 ≡p2 ≡3（mod 8）, pi =1（mod 8）（3 ≤ i ≤ m）, and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the complex L-function of the congruent elliptic curve ED2 ： y^2 = x^3- D^2x at s = 1, divided by the period ω defined below, to be exactly divisible by 2^2m-2, the second lowest 2-power with respect to the number of the Gaussian prime factors of D. As a corollary, we obtain a new series of non-congruent numbers whose prime factors can be arbitrarily many. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer.