Jacobian Coordinates Research Articles

Complete group law for genus 2 Jacobians on Jacobian coordinates

AbstractThis manuscript provides complete, inversion-free, and explicit group law formulas in Jacobian coordinates for the genus 2 hyperelliptic curves of the form $$y^2 = x^5+a_3 x^3+a_2 x^2+a_1 x+a_0$$ y 2 = x 5 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 over a field K with $$\textrm{char}(K) \ne 2$$ char ( K ) ≠ 2 . The formulas do not require the use of polynomial arithmetic operations such as resultant, mod, or gcd computations but only operations in K.

Modified elliptic curve cryptography for efficient data protection in wireless sensor network

Wireless Sensor Networks (WSN) have a crucial function in gathering and transmitting data in different applications, making data protection a top priority. Elliptic Curve Cryptography (ECC) is known for its strong security features, but it requires substantial computational resources in resource-limited WSN environments. To tackle this challenge, this study presents a new alteration to ECC by employing a Point Compression Algorithm, specifically utilizing the Modified Jacobian coordinates. This study assesses the effectiveness of the proposed Modified ECC (MECC) by analyzing key performance metrics such as efficiency, scalability, and communication overhead. These metrics are crucial for ensuring the successful implementation of data protection in WSN. The analysis provides a thorough examination of performance, specifically highlighting encryption time, decryption time, key generation time, and key size. It offers a comprehensive comparison with the conventional ECC approach. The results demonstrate that the MECC, incorporating the Point Compression Algorithm in Modified Jacobian coordinates, presents a favorable approach for improving data security in WSN. The enhanced efficacy, diminished communication burden, and expandability of the suggested methodology are apparent, showcasing its capability to safeguard data transmission while conserving valuable network resources. This research aims to enhance the security of WSN making them more robust and flexible in meeting the changing demands of modern data protection needs.

Hardware Implementations of Elliptic Curve Cryptography Using Shift-Sub Based Modular Multiplication Algorithms

Elliptic curve cryptography (ECC) over prime fields relies on scalar point multiplication realized by point addition and point doubling. Point addition and point doubling operations consist of many modular multiplications of large operands (256 bits for example), especially in projective and Jacobian coordinates which eliminate the modular inversion required in affine coordinates for every point addition or point doubling operation. Accelerating modular multiplication is therefore important for high-performance ECC. This paper presents the hardware implementations of modular multiplication algorithms, including (1) interleaved modular multiplication (IMM), (2) Montgomery modular multiplication (MMM), (3) shift-sub modular multiplication (SSMM), (4) SSMM with advance preparation (SSMMPRE), and (5) SSMM with CSAs and sign detection (SSMMCSA) algorithms, and evaluates their execution time (the number of clock cycles and clock frequency) and required hardware resources (ALMs and registers). Experimental results show that SSMM is 1.80 times faster than IMM, and SSMMCSA is 3.27 times faster than IMM. We also present the ECC hardware implementations based on the Secp256k1 protocol in affine, projective, and Jacobian coordinates using the IMM, SSMM, SSMMPRE, and SSMMCSA algorithms, and investigate their cost and performance. Our ECC implementations can be applied to the design of hardware security module systems.

Efficient Elliptic Curve Operators for Jacobian Coordinates

The speed up of group operations on elliptic curves is proposed using a new type of projective coordinate representation. These operations are the most common computations in key exchange and encryption for both current and postquantum technology. The boost this improvement brings to computational efficiency impacts not only encryption efforts but also attacks. For maintaining security, the community needs to take note of this development as it may need to operate changes in the key size of various algorithms. Our proposed projective representation can be viewed as a warp on the Jacobian projective coordinates, or as a new operation replacing the addition in a Jacobian projective representation, basically yielding a new group with the same algebra elements and homomorphic to it. Efficient algorithms are introduced for computing the expression Pk+Q where P and Q are points on the curve and k is an integer. They exploit optimized versions for particular k values. Measurements of the numbers of basic computer instructions needed for operations based on the new representation show clear improvements. The experiments are based on benchmarks selected using standard NIST elliptic curves.

Lightweight Architecture for Elliptic Curve Scalar Multiplication over Prime Field

In this paper, we present a novel lightweight elliptic curve scalar multiplication architecture for random Weierstrass curves over prime field Fp. The elliptic curve scalar multiplication is executed in Jacobian coordinates based on the Montgomery ladder algorithm with (X,Y)-only common Z coordinate arithmetic. At the finite field operation level, the adder-based modular multiplier and modular divider are optimized by the pre-calculation method to reduce the critical path while maintaining low resource consumption. At the group operation level, the point addition and point doubling methods in (X,Y)-only common Z coordinate arithmetic are modified to improve computation parallelism. A compact scheduling method is presented to improve the architecture’s performance, which includes appropriate scheduling of finite field operations and specific register connections. Compared with existing works, our design is implemented on the FPGA platform without using DSPs or BRAMs for higher portability. It utilizes 6.4~6.5k slices in Kintex-7, Virtex-7, and ZYNQ FPGA and executes an elliptic curve scalar multiplication for a field size of 256-bit in 1.73 ms, 1.70 ms, and 1.80 ms, respectively. Additionally, our design is resistant to timing attacks, simple power analysis attacks, and safe-error attacks. This architecture outperforms most state-of-the-art lightweight designs in terms of area-time products.

Fast scalar multiplication of degenerate divisors for hyperelliptic curve cryptosystems

Scalar multiplication is the main operation in the implementation of hyperelliptic curve cryptosystems, where the basic arithmetic of reduced divisor classes is required. In this work, some explicit formulas are proposed for the arithmetic of reduced divisor classes by exploiting Jacobian coordinates when the degenerate divisor is involved. Moreover, an efficiency analysis shows that the degenerate divisor as a base element can be a valid alternative in hyperelliptic curve cryptosystems as well.

An efficient hardware implementation of the elliptic curve cryptographic processor over prime field,

SummaryElliptic curve cryptography (ECC) schemes are widely adopted for the digital signature applications due to their key sizes, hardware resources, and higher security per bit than Rivest‐Shamir‐Adleman (RSA). In this work, we proposed a new hardware architecture for elliptic curve scalar multiplication (ECSM) in Jacobian coordinates over prime field, . This is a combination of point doubling and point addition architecture, implemented using resource sharing concept to achieve high speed and low hardware resources, which is synthesized both in field‐programmable gate array (FPGA) and application‐specific integrated circuit (ASIC). The proposed ECSM takes 1.76 and 2.44 ms on Virtex‐7 FPGA platform over 224‐bit and 256‐bit prime field, respectively. Similarly, ASIC (GF 40 nm complementary metal‐oxide semiconductor [CMOS]) technology implementation provides energy efficient with a latency of 0.46 and 0.6 ms over prime field and , respectively. This design provides better area‐delay product and high throughput value in both FPGA and ASIC when compared with other designs.

A Fully RNS based ECC Processor

This work proposes a residue number system (RNS) based hardware architecture of an elliptic curve cryptography (ECC) processor over prime field. The processor computes point multiplication in Jacobian coordinates by a combined architecture of point doubling and point addition. An optimized modular reduction architecture is also presented that achieves low area by dividing the RNS moduli in small groups and processes the groups one by one. The proposed ECC processor is generic and supports any random curve over Fp256. The implementation on Virtex-7 and Virtex-6 FPGAs shows comparable performance to the state-of-the-art binary and RNS based ECC processors.

Fuzzy Control for Walking Balance of the Biped Robot Using ZMP Criterion

The dynamic walking stability of biped robot is a challenging research topic. In this paper, the fuzzy controller on zero movement point (ZMP) criterion is implemented to maintain the balance of dynamic walking. We designed a force sensitive resistor (FSR) sensor circuits to collect information from the sensors for the ZMP trajectory tracking. The robot walking has an imbalance when the ZMP shows the risk of moving out the footprint. The fuzzy controller is provided to drive the ZMP inside the ZMP reference area, by adjusting a suitable angle for the ankle and hip joint. In order to determine the ZMP reference, we compute the inverse kinematic and Jacobian coordinates from the biped robot model. In this research, we also introduce the biped robot configuration for control dynamic walking. A user interface is designed to conventionally observe the ZMP trajectory generation. The experimental results for stability of dynamic walking are presented to appreciate the proposed method.

High‐performance elliptic curve cryptography processor over NIST prime fields

This study presents a description of an efficient hardware implementation of an elliptic curve cryptography processor (ECP) for modern security applications. A high-performance elliptic curve scalar multiplication (ECSM), which is the key operation of an ECP, is developed both in affine and Jacobian coordinates over a prime field of size p using the National Institute of Standards and Technology standard. A novel combined point doubling and point addition architecture is proposed using efficient modular arithmetic to achieve high speed and low hardware utilisation of the ECP in Jacobian coordinates. This new architecture has been synthesised both in application-specific integrated circuit (ASIC) and field-programmable gate array (FPGA). A 65 nm CMOS ASIC implementation of the proposed ECP in Jacobian coordinates takes between 0.56 and 0.73 ms for 224-bit and 256-bit elliptic curve cryptography, respectively. The ECSM is also implemented in an FPGA and provides a better delay performance than previous designs. The implemented design is area-efficient and this means that it requires not many resources, without any digital signal processing (DSP) slices, on an FPGA. Moreover, the area–delay product of this design is very low compared with similar designs. To the best of the authors’ knowledge, the ECP proposed in this study over F p performs better than available hardware in terms of area and timing.

Jacobian Coordinates on Genus 2 Curves

This paper presents a new projective coordinate system and new explicit algorithms which together boost the speed of arithmetic in the divisor class group of genus 2 curves. The proposed formulas generalize the use of Jacobian coordinates on elliptic curves, and their application improves the speed of performing cryptographic scalar multiplications in Jacobians of genus 2 curves over prime fields by an approximate factor of 1.25x. For example, on a single core of an Intel Core i7-3770 (Ivy Bridge), we show that replacing the previous best formulas with our new set improves the cost of generic scalar multiplications from 239,000 to 192,000 cycles and drops the cost of specialized GLV-style scalar multiplications from 155,000 to 123,000 cycles.

New adaptive time step symplectic integrator: an application to the elliptic restricted three-body problem

The time-transformed leapfrog scheme of Mikkola & Aarseth was specifically designed for a second-order differential equation with two individually separable forms of positions and velocities. It can have good numerical accuracy for extremely close two-body encounters in gravitating few-body systems with large mass ratios, but the non-time-transformed one does not work well. Following this idea, we develop a new explicit symplectic integrator with an adaptive time step that can be applied to a time-dependent Hamiltonian. Our method relies on a time step function having two distinct but equivalent forms and on the inclusion of two pairs of new canonical conjugate variables in the extended phase space. In addition, the Hamiltonian must be modified to be a new time-transformed Hamiltonian with three integrable parts. When this method is applied to the elliptic restricted three-body problem, its numerical precision is explicitly higher by several orders of magnitude than the nonadaptive one's, and its numerical stability is also better. In particular, it can eliminate the overestimation of Lyapunov exponents and suppress the spurious rapid growth of fast Lyapunov indicators for high-eccentricity orbits of a massless third body. The present technique will be useful for conservative systems including N-body problems in the Jacobian coordinates in the the field of solar system dynamics, and nonconservative systems such as a time-dependent barred galaxy model in a rotating coordinate system.

A generalized Zakharov–Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter

In this paper, a generalized Zakharov–Shabat equation ( g-ZS equation), which is an isospectral problem, is introduced by using a loop algebra G ∼ . From the stationary zero curvature equation we define the Lenard gradients { g j } and the corresponding generalized AKNS ( g-AKNS) vector fields { X j } and X k flows. Employing the nonlinearization method, we obtain the generalized Zhakharov–Shabat Bargmann ( g-ZS-B) system and prove that it is Liouville integrable by introducing elliptic coordinates and evolution equations. The explicit relations of the X k flows and the polynomial integrals { H k } are established. Finally, we obtain the finite-band solutions of the g-ZS equation via the Abel–Jacobian coordinates. In addition, a soliton hierarchy and its Hamiltonian structure with an arbitrary parameter k are derived.

Using 5-isogenies to quintuple points on elliptic curves

Finding multiples of points on elliptic curves is the most important computation in elliptic curve cryptography. Extending the work of C. Doche, T. Icart, and D. Kohel (Efficient scalar multiplication by isogeny decomposition, in: M. Yung, Y. Dodis, A. Kiayias, T.G. Malkin (Eds.), Public Key Cryptography 2006, in: Lecture Notes in Comput. Sci., vol. 3958, Springer, Heidelberg, 2006, pp. 191–206) we use 5-isogenies to compute multiples of a point on an elliptic curve. Specifically, we find explicit formulas for quintupling a point. We compare the results with other published formulas for quintupling. We find that when the point is represented in Jacobian coordinates with z = 1 , our method is potentially among the fastest on specially chosen elliptic curves. We also see that using l-isogenies to compute the multiplication by l map (for l larger than five) is unlikely to be more efficient than other techniques.

Faster computation of the Tate pairing

Text This paper proposes new explicit formulas for the doubling and addition steps in Miller's algorithm to compute the Tate pairing on elliptic curves in Weierstrass and in Edwards form. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in addition and doubling. The Tate pairing on Edwards curves can be computed by using these functions in Miller's algorithm. Computing the sum of two points or the double of a point and the coefficients of the corresponding functions is faster with our formulas than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also improve the formulas for Tate pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairing-friendly Edwards curves. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=nideQo-K9ME/.

Efficient scalar multiplication for elliptic curves over binary fields

Scalar multiplication [n]P is the kernel and the most time-consuming operation in elliptic curve cryptosystems. In order to improve scalar multiplication, in this paper, we propose a tripling algorithm using Lopez and Dahab projective coordinates, in which there are 3 field multiplications and 3 field squarings less than that in the Jacobian projective tripling algorithm. Furthermore, we map P to ϕe−1(P), and compute [n]ϕe−1 (P) on elliptic curve Ee, which is faster than computing [n]P on E, where ϕe is an isomorphism. Finally we calculate ϕe([n]ϕe−1(P)) = [n]P. Combined with our efficient point tripling formula, this method leads scalar multiplication using double bases to achieve about 23% improvement, compared with Jacobian projective coordinates.

Correlation diagrams in collisions of three identical particles

We discuss collision of three identical particles and derive scattering selection rules from initial to final states of the particles. We use either hyperspherical or Jacobian coordinates depending on which one is best suited to describe three different configurations of the particles: (1) three free particles, (2) a quasi-bound trimer or (3) a dimer and a free particle. We summarize quantum numbers conserved during the collision as well as quantum numbers that are appropriate for a given configuration but may change during the scattering process. The total symmetry of the system depends on these quantum numbers. Based on the selection rules, we construct correlation diagrams between different configurations before and after a collision. In particular, we describe a possible fragmentation of the system into one free particle and a dimer, which can be used, for example, to identify possible decay products of quasi-stationary three-body states or three-body recombination.

Wave packet study of the UV photodissociation of the Ar2HBr complex

The photodissociation dynamics of the Ar2HBr van der Waals molecule is studied using the multiconfiguration time-dependent Hartree method. Standard Jacobian coordinates are used to describe the molecule. Two four-dimensional calculations are carried out where the rotation of the Ar2 molecule and, in addition, either the vibration of Ar2-Br or that of Ar2 are frozen. The time-evolution of the probability density in the different modes and the calculation of the dissociative flux show that the dissociating hydrogen atom preferentially moves out of the plane defined by Ar2 and Br. A comprehensive study of the cage effect in the process is presented.

Zero-Value Register Attack on Elliptic Curve Cryptosystem

Differential power analysis (DPA) might break implementations of elliptic curve cryptosystem (ECC) on memory constraint devices. Goubin proposed a variant of DPA using a point (0, y), which is not randomized in Jacobian coordinates or in an isomorphic class. This point often exists in standardized elliptic curves, and we have to care this attack. In this paper, we propose zero-value register attack as an extension of Goubin's attack. Note that even if a point has no zero-value coordinate, auxiliary registers might take zero value. We investigate these zero-value registers that cannot be randomized by the above randomization. Indeed, we have found several points P = (x, y) which cause the zero-value registers, e.g., (1) 3x 2 +a = 0, (2) 5x 4 + 2ax 2 -4bx + a 2 = 0, (3) P is y-coordinate self-collision point, etc. We demonstrate the elliptic curves recommended in SECG that have these points. Interestingly, some conditions required for zero-value register attack depend on explicit implementation of addition formulae - in order to resist this type of attacks, we have to care how to implement the addition formulae. Finally, we note that Goubin's attack and the proposed attack assume that a base point P can be chosen by attackers and a secret scalar d is fixed, so that they are not applicable to ECDSA.

Behaviour of a two-planetary system on a cosmogonic time-scale

The orbital evolution of planetary systems similar to our Solar one represents one of the most important problems of Celestial Mechanics. In the present work we use Jacobian coordinates, introduce two systems of osculating elements, construct the Hamiltonian expansions in Poisson series for all the elements for the planetary three-body problem (including the problem Sun–Jupiter–Saturn). Further we construct the averaged Hamiltonian by the Hori–Deprit method with accuracy up to second order with respect to the small parameter, the generating function, the change of variables formulae, and the right-hand sides of the averaged equations. The averaged equations for the Sun–Jupiter–Saturn system are integrated numerically over a time span of 10 Gyr. The Liapunov Time turns out to be 14 Myr (Jupiter) and 10 Myr (Saturn).To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html