Arithmetic Of Elliptic Curves Research Articles

An Efficient Audio Encryption Scheme Based on Elliptic Curve over Finite Fields

Elliptic curve (EC) based cryptographic systems are more trustworthy than the currently used cryptographic approaches since they require less computational work while providing good security. This paper shows how to use an EC to make a good cryptosystem for encrypting digital audio. As a preliminary step, the system uses an EC of a particular type over a binary extension field to distort the digital audio pixel position. It reduces the inter-correlation between pixels in the original audio, making the system resistant to statistical attacks. In creating confusion in the data, an EC over a binary extension field is used to make a different number of substitution boxes (S-boxes). The suggested design employs a unique curve that relies on efficient EC arithmetic operations in the diffusion module. As a result, it generates high-quality pseudo-random numbers (PRNs) and achieves optimal diffusion in encrypted audio files with less processing work. Audio files of various sizes and kinds can all be encrypted using the provided algorithm. Moreover, the results show that this method effectively protects many kinds of audio recordings and is more resistant to statistical and differential attacks.

Design a cryptosystem using elliptic curves cryptography and Vigenère symmetry key

In this paper describes the basic idea of elliptic curve cryptography (ECC) as well as Vigenère symmetry key. Elliptic curve arithmetic can be used to develop elliptic curve coding schemes, including key exchange, encryption, and digital signature. The main attraction of elliptic curve cryptography compared to Rivest, Shamir, Adleman (RSA) is that it provides equivalent security for a smaller key size, which reduces processing costs. From the theorical basic, we proposed a cryptosystem using elliptic curves and Vigenère cryptography. We proposed and implemented our encryption algorithm in an integrated development environment named visual studio 2019 to design a safe, secure, and effective cryptosystem.

Quantum Binary Field Multiplication with Optimized Toffoli Depth and Extension to Quantum Inversion

The Shor's algorithm can find solutions to the discrete logarithm problem on binary elliptic curves in polynomial time. A major challenge in implementing Shor's algorithm is the overhead of representing and performing arithmetic on binary elliptic curves using quantum circuits. Multiplication of binary fields is one of the critical operations in the context of elliptic curve arithmetic, and it is especially costly in the quantum setting. Our goal in this paper is to optimize quantum multiplication in the binary field. In the past, efforts to optimize quantum multiplication have centred on reducing the Toffoli gate count or qubits required. However, despite the fact that circuit depth is an important metric for indicating the performance of a quantum circuit, previous studies have lacked sufficient consideration for reducing circuit depth. Our approach to optimizing quantum multiplication differs from previous work in that we aim at reducing the Toffoli depth and full depth. To optimize quantum multiplication, we adopt the Karatsuba multiplication method which is based on the divide-and-conquer approach. In summary, we present an optimized quantum multiplication that has a Toffoli depth of one. Additionally, the full depth of the quantum circuit is also reduced thanks to our Toffoli depth optimization strategy. To demonstrate the effectiveness of our proposed method, we evaluate its performance using various metrics such as the qubit count, quantum gates, and circuit depth, as well as the qubits-depth product. These metrics provide insight into the resource requirements and complexity of the method. Our work achieves the lowest Toffoli depth, full depth, and the best trade-off performance for quantum multiplication. Further, our multiplication is more effective when not used in stand-alone cases. We show this effectiveness by using our multiplication to the Itoh-Tsujii algorithm-based inversion of F(x8+x4+x3+x+1).

Torsion divisors of plane curves with maximal flexes and Zariski pairs

AbstractThere is a close relationship between the embedded topology of complex plane curves and the (group‐theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves that include a special smooth component, via the torsion properties induced by the divisors in the special curve associated to the remaining components, which is an arithmetic property. When this special curve has maximal flexes, there is a natural isomorphism between its Jacobian variety and the degree zero part of its Picard group. In this paper, we consider curve arrangements that contain a special smooth component with a maximal flex and exploit these properties to obtain Zariski tuples, which show the interplay between topology, geometry, and arithmetic.

Efficient Elliptic Curve Arithmetic for Lightweight Cryptographic Schemes for IoT Applications

The Internet of Things’ (IoT) market is expected to grow exponentially at the global level in the coming years, due to the proliferation of more reliable and faster networks resulting from the extensive rollout of 5 to 10 G mobile networks. By 2025, it is expected that worldwide projection of IoT connected devices will be pegged at 30.9 billion units. Despite the potential benefits of the new technology, security in IoT is a major threat. According to HP, 70% of IoT devices are vulnerable to sniffing attacks and reliable solution is yet to be found. The standard cryptographic algorithms such as RSA and AES provide good security but their utilization in IoT is questionably due to hardware and energy constraints for computationally expensive encryption schemes. However, elliptic curve- based cryptography, a recent paradigm in public key cryptography, achieves the same level of security with smaller key sizes. On the other hand, the total score of performance of an elliptic curve-based cryptosystem depends largely on the efficiency of the arithmetic operations performed in it. It is against this background that this paper proposes efficient elliptic curve arithmetic for implementing ECC based schemes suitable for IoT systems implementations. Elliptic curve point arithmetic implementations in projective coordinate systems over binary extension fields introduce higher efficiencies in software. In this regard, this paper has proposed an improved López-Dahab point arithmetic methods on non-supersingular elliptic curves over . The results show 69.20% improvement in Point Doubling, 44.68% in Point Addition and the scalar point multiplication execution time is decreased by 48.80%.

What Do Networks and Elliptic Curves Have in Common?

Graph theory used in networking and the arithmetic of elliptic curves have more in common than meets the eye. We present how their zeta functions fuse them seamlessly.

Pseudorandom vector generation using elliptic curves and applications to Wiener processes

In this paper we present, using the arithmetic of elliptic curves over finite fields, an algorithm for the efficient generation of a sequence of uniform pseudorandom vectors in high dimensions, that simulates a sample of a sequence of i.i.d. random variables, with values in the hypercube [0,1]d with uniform distribution. As an application, we obtain, in the discrete time simulation, an efficient algorithm to simulate, uniformly distributed sample path sequence of a sequence of independent standard Wiener processes. This could be employed for use, in the full history recursive multi-level Picard approximation method, for numerically solving the class of semilinear parabolic partial differential equations of the Kolmogorov type.

On The Batch Outsourcing Of Pairing Computations

Abstract Pairing-based cryptography is utilized in a wide range of devices, such as servers, mobile devices, smart cards and sensors. Pairing computation would be a burden for power and/or computation-restricted devices. Protocols for outsourcing pairing computations from limited devices to more resourceful devices are already proposed. These protocols naturally require verification of the computation and secrecy of the inputs and/or outputs. Similarly, batch pairing outsourcing protocols aim to improve efficiency over multiple runs of the state-of-the-art single pairing delegation protocols. Here, we will cover efficient, privacy preserving, secure batch pairing outsource protocols for each type based on secrecy of inputs and outputs. We propose the first generic outsourcing protocol where inputs and outputs of the pairing function are secret. In addition to this, we give some methods to avoid certain type of attacks, increase efficiency and get rid of pairing arithmetic. The proposed protocols enable limited devices to outsource pairing computations with only elliptic curve arithmetic.

Dependable and Provable Secure Two-Factor Mutual Authentication Scheme Using ECC for IoT-Based Telecare Medical Information System.

With the recent tremendous growth in technology, the Internet of Things (IoT) Telecare Medicine Information System (TMIS) is the most widely used medical information system with prominent achievements. Authentication schemes, which use Smart cards, offer the best solution for TMIS applications that in turn provide efficiency and security. Furthermore, authentication schemes that combine passwords and smart cards are considered to be an effective solution for the two-factor authentication scheme. Such schemes contribute to high security along with the public-key cryptosystem. In this research work, a two-factor authentication technique that is both efficient and secure, which makes use of Elliptic Curve Cryptography (ECC) with smart cards, has been proposed. Here, we have used the fundamental assumptions of strong and collision free cryptographic Hash function and Elliptic Curve arithmetic. The proposed authentication technique protects user privacy by allowing registered users to change their passwords without revealing their identity to the server. The proposed authentication scheme has been subjected to formal and informal security investigations. In terms of efficiency and performance, the proposed two-factor authentication technique was compared with the existing relevant two-factor authentication schemes based on ECC. This scheme satisfies the two-factor authentication scheme's basic security standards.

Simplified isogeny formulas on twisted Jacobi quartic curves

Isogenies between elliptic curves play a very important role in elliptic curve related cryptosystems and cryptanalysis. It is widely known that different models of elliptic curves would induce different computational costs of elliptic curve arithmetic, and several works have been devoted to accelerate the isogeny computation on various curve models. This paper studies the case of the Jacobi quartic model, which is a classic form of elliptic curves. A new w-coordinate system on extended Jacobi quartic curves is introduced for Montgomery-like group arithmetic. Explicit formulas for 2-isogenies and odd ℓ-isogenies on the specific curves are presented, and based on the w-coordinate system, the computation of such isogenies could be further simplified.

Tiling number problem and $\boldsymbol{\theta}$-congruent number problem

The tiling number problem is closely related to the arithmetic of elliptic curves. In this paper, we construct a family of tiling numbers with multiple prime factors by using Birch lemma and the induction method developed by one of the authors(Ye Tian) to prove the non-torsioness of Heegner points. In addition, we prove the $2$-part BSD conjecture of the related elliptic curves. We deal with the quadratic twist family of elliptic curves without complex multiplication and the distribution of $2$-Selmer groups of this family is not predicted by the general conjecture or the known results.

P r -Selmer companion modular forms

The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $\chi$ of $K$, the $n$-Selmer groups of $E_1^\chi $ and $E_2^\chi$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogue of this for Bloch-Kato $p^r$-Selmer group of modular forms. We compare the Bloch-Kato Selmer groups of a modular form respectively with the Greenberg Selmer group when the modular form is $p$-ordinary and with the signed Selmer group of Lei-Loeffler-Zerbes when the modular form is non-ordinary at $p$. We also indicate the corresponding results over $\Q_\cyc$ and its relation with the well known congruence results of the special values of the corresponding $L$-functions due to Vatsal.

Non-monogenic division fields of elliptic curves

For various positive integers n, we show the existence of infinite families of elliptic curves over Q with n-division fields that are not monogenic, i.e., such that the ring of integers does not admit a power integral basis. We parametrize some of these families explicitly. Moreover, we show that every E/Q without CM has infinitely many non-monogenic division fields. Our main technique combines a global description of the Frobenius obtained by Duke and Tóth with an algorithm based on ideas of Dedekind. As a counterpoint, we are able to use different aspects of the arithmetic of elliptic curves to exhibit a family of monogenic 2-division fields.

Derived Beilinson–Flach elements and the arithmetic of the adjoint of a modular form

Kings, Lei, Loeffler and Zerbes constructed in \[LLZ], \[KLZ1] a three-variable Euler system $\kappa(\textbf{g},\textbf{h})$ of Beilinson–Flach elements associated to a pair of Hida families $(\textbf{g},\textbf{h})$ and exploited it to obtain applications to the arithmetic of elliptic curves, extending the earlier work \[BDR]. The aim of this article is to show that this Euler system also encodes arithmetic information concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when $\textbf{g}$ and $\textbf{h}$ specialize in weight 1 to $p$-stabilizations of eigenforms such that one is dual to the other. We encounter an exceptional zero phenomenon which forces the specialization of $\kappa(\textbf{g},\textbf{h})$ to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of the main conjecture of \[DLR4] on iterated integrals and the main conjecture of \[DR1] for Beilinson–Flach elements in the adjoint setting. The main point of this paper is that the methods of \[DLR1], \[DLR4] and \[CH], where the above conjectures are proved when the weight 1 eigenforms have CM, do not apply to our setting and new ideas are required. In the previous works, a crucial ingredient is a factorization of $p$-adic $L$-functions, which in our scenario is not available due to the lack of critical points. Instead we resort to the principle of improved Euler systems and $p$-adic $L$-functions to reduce our problems to questions which can be resolved using Galois deformation theory. We expect this approach may be adapted to prove other cases of the Elliptic Stark Conjecture and of its generalizations that appear in the literature.

Mazur\u2019s rational torsion result for pointless genus one curves: examples

This note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over mathbb {Q} in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over mathbb {Q} corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

Elliptic Curve Cryptography Point Multiplication Core for Hardware Security Module

In today's technology, a sheer number of Internet of Things applications use hardware security modules for secure communications. The widely used algorithms in security modules, for example, digital signatures and key agreement, are based upon elliptic curve cryptography (ECC). A core operation used in ECC is the point multiplication, which is computationally expensive for many Internet of things applications. In many IoT applications, such as intelligent transportation systems and distributed control systems, thousands of safety messages need to be signed and verified within a very short time-frame. Considerable research has been conducted in the design of a fast elliptic curve arithmetic on finite fields using residue number systems (RNS). In this article, we propose an RNS-based ECC core hardware for the two families of elliptic curves that are short Weierstras and twisted Edwards curves. Specifically, we present RNS implementations for SECP256K1 and ED25519 standard curves. We propose an RNS hardware architecture supporting fast elliptic curve point-addition (ECPA), point-doubling (ECPD), and point-tripling (ECPT). We implemented different ECC point multiplication algorithms on the Xilinx FPGA platform. The test results confirm that the performance of our fully RNS ECC point multiplication is better than the fastest ECC point multiplication cores in the literature.

The Integral Points On A Class Of Elliptic Curve

It is very important to study the integer points of various elliptic curves in elementary number theory. On the one hand, as an important research object in elementary number theory, elliptic curve plays an indispensable role in the development of mathematics. On the other hand, as an important research object in elementary number theory, elliptic curve has practical application in many aspects. So far, there are some conclusions about elliptic curve y 2 = x 3 + a x + b, a, b ∈ Z. In 1987, D. Zagier proposed the integer points problem on y 2 = x 3 - 27x + 62, which is important for studying the arithmetic and geometric properties of elliptic curves. In 2009, Zhu H L and Chen J H used the methods of P-adic analysis and algebraic number theory to solve the problem of the integer points on y 2 = x 3 - 27x + 62. In 2010, Wu H M used the elementary methods to find all the integral points of elliptic curves y 2 = x 3 - 27x - 62. In 2015, Li Y Z and Cui B J used the elementary methods to solve the problem of the integer points on y 2 = x 3 - 21x - 90. In 2016, Guo J used the elementary methods to solve the problem of the integer points on y 2 = x 3 + 27x + 62. In 2017, Guo J used the elementary method proves that y 2 = x 3 - 21x + 90 has no integer points. Up to now, there is no relevant conclusions on the integral points of elliptic curves y 2 = x 3 + 19x - 46, which is the subject of this paper. By using congruence and Legendre Symbol, it can be proved that elliptic curve y 2 = x 3 + 19x - 46 has only one integer point: (x, y) = (2,0).

Optimized CSIDH Implementation Using a 2-Torsion Point

The implementation of isogeny-based cryptography mainly use Montgomery curves, as they offer fast elliptic curve arithmetic and isogeny computation. However, although Montgomery curves have efficient 3- and 4-isogeny formula, it becomes inefficient when recovering the coefficient of the image curve for large degree isogenies. Because the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) requires odd-degree isogenies up to at least 587, this inefficiency is the main bottleneck of using a Montgomery curve for CSIDH. In this paper, we present a new optimization method for faster CSIDH protocols entirely on Montgomery curves. To this end, we present a new parameter for CSIDH, in which the three rational two-torsion points exist. By using the proposed parameters, the CSIDH moves around the surface. The curve coefficient of the image curve can be recovered by a two-torsion point. We also proved that the CSIDH while using the proposed parameter guarantees a free and transitive group action. Additionally, we present the implementation result using our method. We demonstrated that our method is 6.4% faster than the original CSIDH. Our works show that quite higher performance of CSIDH is achieved while only using Montgomery curves.

An Application of the Arithmetic of Elliptic Curves to the Class Number Problem for Quadratic Fields

Let $l$ be the prime $3$, $5$ or $7$, and let $m_{1}$,~$m_{2}$, $n_{1}$ and $n_{2}$ be non-zero rational numbers. We construct an infinite family of pairs of distinct quadratic fields $\mathbb{Q}(\sqrt{m_{1}D+n_{1}})$ and $\mathbb{Q}(\sqrt{m_{2}D+n_{2}})$ with $D\in\mathbb{Q}$ such that both class numbers are divisible by $l$, using rational points on an elliptic curve with positive Mordell-Weil rank to parametrize such quadratic fields.

Exterior powers of the adjoint representation and the Weyl ring of E8

I derive explicitly all polynomial relations in the character ring of E8 of the form χ∧ke8−pk(χ1,…,χ8)=0, where ∧ke8 is an arbitrary exterior power of the adjoint representation and χi is the ith fundamental character. This has simultaneous implications for the theory of relativistic integrable systems, Seiberg–Witten theory, quantum topology, orbifold Gromov–Witten theory, and the arithmetic of elliptic curves. The solution is obtained by reducing the problem to a (large, but finite) dimensional linear problem, which is amenable to an efficient solution via distributed computation.