Elliptic Curve Point Research Articles

Random Point Blinding Methods for Koblitz Curve Cryptosystem

While the elliptic curve cryptosystem (ECC) is getting more popular in securing numerous systems, implementations without consideration for side-channel attacks are susceptible to critical information leakage. This paper proposes new power attack countermeasures for ECC over Koblitz curves. Based on some special properties of Koblitz curves, the proposed methods randomize the involved elliptic curve points in a highly regular manner so the resulting scalar multiplication algorithms can defeat the simple power analysis attack and the differential power analysis attack simultaneously. Compared with the previous countermeasures, the new methods are also noticeable in terms of computational cost.

Endomorphisms for Faster Elliptic Curve Cryptography on a Large Class of Curves

Efficiently computable homomorphisms allow elliptic curve point multiplication to be accelerated using the Gallant–Lambert–Vanstone (GLV) method. Iijima, Matsuo, Chao and Tsujii gave such homomorphisms for a large class of elliptic curves by working over ${\mathbb{F}}_{p^{2}}$. We extend their results and demonstrate that they can be applied to the GLV method. In general we expect our method to require about 0.75 the time of previous best methods (except for subfield curves, for which Frobenius expansions can be used). We give detailed implementation results which show that the method runs in between 0.70 and 0.83 the time of the previous best methods for elliptic curve point multiplication on general curves.

A high performance ECC hardware implementation with instruction-level parallelism over GF(2 163)

In this work we propose a high performance elliptic curve cryptographic processor over GF(2 163) for the applications that require high performance. It has three finite field (FF) RISC cores and a main controller to achieve instruction-level parallelism (ILP) for elliptic curve point multiplication. Customized instructions are proposed to decrease clock cycles. The interconnection among three FF cores and the main controller is obtained based on the analysis of both data dependency and critical path. The proposed design can reach 185 MHz with 20,807 slices when implemented on Xilinx XC4VLX80 FPGA device and 263 MHz with 217,904 gates when synthesized with TSMC .18 μm CMOS technology.

Random Binary and Non-Binary Sequences Derived from Random Sequence of Points on Cyclic Elliptic Curve Over Finite Field GF(2 m ) and Their Properties

In this paper, Cyclic Elliptic Curves of the form y 2 + xy = x 3 + ax 2 + b,a,b ∈ GF(2 m ) with order M is considered. A finite field GF(p) (p ≥ N, where N is the order of point P) is considered. Random sequence {k i } of integers is generated using Linear Feedback Shift Register (LFSR) over GF(p) for maximum period. Every element in sequence {k i } is mapped to k i P which is a point on Cyclic Elliptic Curve with co-ordinates say (x i , y i ). The sequence {k i P} is a random sequence of elliptic curve points. From the sequence (x i , y i ) several binary and non-binary sequences are derived and their randomness properties are investigated. The results are discussed. It is found that these sequences pass FIPS-140, NIST tests and exhibit good Hamming Correlation properties. These sequences find applications in Stream Cipher Systems. Here, Cyclic Elliptic Curve over GF(28) is chosen for analysis.

A SINGLE FORMULA AND ITS IMPLEMENTATION IN FPGA FOR ELLIPTIC CURVE POINT ADDITION USING AFFINE REPRESENTATION

A formula for point addition in elliptic curves using affine representation and its implementation in FPGA is presented. The use of this new formula in hardware implementations of scalar multiplications for elliptic curve cryptography has the main advantages of: (i) reducing area for the implementations of elliptic curve point addition, and (ii) increasing the resistance to side channel attacks of the hardware implementation itself. Hardware implementation of scalar multiplication for elliptic curve cryptography using this new formulation requires low area resources while keeping high performance compared to implementations using projective coordinates, which are usually considered faster than the affine coordinates.

Elliptic Curves over the Perfect Closure of a Function Field

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

Elliptic Gauss sums and applications to point counting

We define a class of algebras over finite fields, called polynomially cyclic algebras, which extend the class of abelian field extensions. We study the structure of these algebras; furthermore, we define and investigate properties of Lagrange resolvents and Gauss and Jacobi sums. Natural examples of polynomially cyclic algebras are for instance algebras of the form F p [ X ] / ( F q ( X ) ) where p , q are distinct odd primes and F q is the q - th cyclotomic polynomial. Further examples occur similarly on replacing the cyclotomic polynomials with factors of division polynomials of elliptic curves. Finally, Gauss and Jacobi sums over polynomially cyclic algebras are applied for improving current algorithms for counting the number of points of elliptic curves over finite fields.

Large Family of Sequences from Elliptic Curves over Residue Class Rings

An upper bound is established for certain exponential sums on the rational points of an elliptic curve over a residue class ring $\mathbb{Z}_N$, N=pq for two distinct odd primes p and q. The result is a generalization of an estimate of exponential sums on rational point groups of elliptic curves over finite fields. The bound is applied to showing the pseudorandomness of a large family of binary sequences constructed by using elliptic curves over $\mathbb{Z}_N$.

An O(m 2 )-depth quantum algorithm for the elliptic curve discrete logarithm problem over GF(2 m ) a

We consider a quantum polynomial-time algorithm which solves the discrete logarithmproblem for points on elliptic curves over GF(2m). We improve over earlier algorithmsby constructing an efficient circuit for multiplying elements of binary finite fields andby representing elliptic curve points using a technique based on projective coordinates.The depth of our proposed implementation, executable in the Linear Nearest Neighbor(LNN) architecture, is O(m2), which is an improvement over the previous bound ofO(m3) derived assuming no architectural restrictions.

An O(m^2)-depth quantum algorithm for the elliptic curve discrete logarithm problem over GF(2^m)

We consider a quantum polynomial-time algorithm which solves the discrete logarithm problem for points on elliptic curves over $GF(2^m)$. We improve over earlier algorithms by constructing an efficient circuit for multiplying elements of binary finite fields and by representing elliptic curve points using a technique based on projective coordinates. The depth of our proposed implementation, executable in the Linear Nearest Neighbor (LNN) architecture, is $O(m^2)$, which is an improvement over the previous bound of $O(m^3)$ derived assuming no architectural restrictions.

Efficient FPGA Implementation of a Programmable Architecture for GF(p) Elliptic Curve Crypto Computations

This paper presents a processor architecture for elliptic curve cryptography computations over GF(p). The speed to compute the Elliptic-curve point multiplication over the prime fields GF(p) is increased by using the maximum degree of parallelism, and by carefully selecting the most appropriate coordinates system. The proposed Elliptic Curve processor is implemented using FPGAs. The time, area and throughput results are obtained, analyzed, and compared with previously proposed designs showing interesting performance and features.

An RNS Implementation of an $F_{p}$ Elliptic Curve Point Multiplier

Elliptic curve point multiplication is considered to be the most significant operation in all elliptic curve cryptography systems, as it forms the basis of the elliptic curve discrete logarithm problem. Designs for elliptic curve cryptography point multiplication are area demanding and time consuming. Thus, the efficient realization of point multiplication is of fundamental importance for the performance of an elliptic curve system. In this paper, a hardware architecture of an elliptic curve point multiplier is proposed that exploits the intrinsic parallelism of the residue number system (RNS), in order to speed up the elliptic curve point calculations and minimize the area complexity of the elliptic curve point multiplier. The architecture proves to be the fastest among all known design approaches, while complexity is less than half of that of previous efforts. This architecture also supports the required input (binary-to-RNS) and output (RNS-to-binary) conversions. Through a graph-oriented approach, the area of the elliptic curve point multiplier is minimized, by optimizing the point addition and doubling algorithms. Also, through this approach, the number of execution steps for point addition is matched to the number of execution steps for point doubling. Additionally, the impact of various RNS bases, in terms of number of moduli and their bit lengths, on the area and speed of the proposed implementation is analyzed, in an effort to define the potential for using RNS in elliptic curve cryptography.

ENHANCED MENEZES-VANESTONE ELLIPTIC CURVES CRYPTOSYSTEM

The group of the elliptic curve points forms an Abelian group, which is a suitable choice for constructing a good problem similar to Discrete Logarithm Problem. This paper introduces a brief overview of the Menezes-Vanestone Elliptic Curves Cryptosystem (MVECC), and attempts to improve it by suggesting a new variation on MVECC to make encryption more efficient than the original elliptic curves cryptosystems.

Quasi-quadratic elliptic curve point counting using rigid cohomology

Let E be a nonsupersingular elliptic curve over the finite field with p n elements. We present a deterministic algorithm that computes the zeta function and hence the number of points of such a curve E in time quasi-quadratic in n . An older algorithm having the same time complexity uses the canonical lift of E , whereas our algorithm uses rigid cohomology combined with a deformation approach. An implementation in small odd characteristic turns out to give very good results.

On the Chow-forms of elliptic normal curves of degree 4

In this note, we shall aim, using explicit theta constants,to describe the Chow-form of an elliptic normal curve of degree 4 embedded in $P^{3}$ by using theta functions, at any point of the upper half plane. In this case, the embedded elliptic curve is a complete intersection which is defined by two quadratic forms. In order to calculate its Chow-form, we use the elimination theory. Our main result is Th. 1' (§2). In our case, Chow-forms are divisors of the Grassmann variety of lines in $P^{3}$. From the theorem, we see that the Chow-forms of elliptic normal curves of degree 4 lie on a linear subspace $M$ (cf. 2.3) of dimension 23. §2 is the main part of this note.In §3, using the theory of elliptic modular forms of level 4, we consider the geometric meaning of Th. 1'. Then our theorem shows that the Chow point is given by modular forms of weight 2 (of weight 1 in the traditional sense) and of level 4. The compactification of the moduli space of level 4 is isomorphic to a plane conic. Thus, we see that the Chow points of the projective elliptic normal curves of degree 4 determined by the points of the upper half plane form a rational curve of degree 4 in $M$ which is essentially the image of the 2-uple embedding of the above plane conic (Th. 2, §3. cf. Comments after the proof of Th. 2). Furthermore, we see that the correspondence which assigns the Chow point to a point of the upper half plane can be extended to the inequivalent cusps of the principal congruence subgroup of level 4 (Cor. 1, §3). Finally, we give a remark about the structure of the corresponding component of the Chow variety parametrizing l-cycles of degree 4 in $P^{3}$ (Cor. 2, §3).

Torsion points of elliptic curves with good reduction

We consider the torsion points of elliptc curves over certain number fields with good reduction everywhere.

New stream ciphers based on elliptic curve point multiplication

Three new algorithms for the development of stream cipher based on elliptic curve point multiplication are presented in this paper. Extensive analysis of the properties of keystreams produced by various algorithms available in literature for the generation of Pseudorandom Bit Sequence (PRBS) based on the elliptic curve is carried out. New algorithms are proposed as modifications over these schemes in such a way that the weakness of these schemes in terms of reduced cryptographic strength and periodicity are overcome. Since elliptic curve cryptography is replacing RSA in most of the cryptosystems as the standard for key exchange, an EC based stream cipher provides a good option for encryption using the available infrastructure for elliptic curve point multiplication.

On Parallelization of High-Speed Processors for Elliptic Curve Cryptography

This paper discusses parallelization of elliptic curve cryptography hardware accelerators using elliptic curves over binary fields F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sub> . Elliptic curve point multiplication, which is the operation used in every elliptic curve cryptosystem, is hierarchical in nature, and parallelism can be utilized in different hierarchy levels as shown in many publications. However, a comprehensive analysis on the effects of parallelization has not been previously presented. This paper provides tools for evaluating the use of parallelism and shows where it should be used in order to maximize efficiency. Special attention is given for a family of curves called Koblitz curves because they offer very efficient point multiplication. A new method where the latency of point multiplication is reduced with parallel field arithmetic processors is introduced. It is shown to outperform the previously presented multiple field multiplier techniques in the cases of Koblitz curves and generic curves with fixed base points. A highly efficient general elliptic curve cryptography processor architecture is presented and analyzed. Based on this architecture and analysis on the effects of parallelization, a few designs are implemented on an Altera Stratix II field-programmable gate array (FPGA).

Singular values of some modular functions and their applications to class fields

Since the modular curve \(X(5)=\Gamma(5)\backslash\frak H^{*}\) has genus zero, we have a field isomorphism \(\mathcal{K}(X(5))\approx \mathbb{C}(X_{2}(z))\) where X2(z) is a product of Klein forms. We apply it to construct explicit class fields over an imaginary quadratic field K from the modular function jΔ,25(z):=X2(5z). And, for every integer N≥7 we further generate ray class fields K(N) over K with modulus N just from the two generators X2(z) and X3(z) of the function field \(\mathcal{K}(X_{1}(N))\) , which are also the product of Klein forms without using torsion points of elliptic curves.

Torsion points on elliptic curves over function fields and a theorem of Igusa

If F is a global function field of characteristic p > 3 , we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a fixed finite set of places of F. We end the paper with an application to torsion points rational over abelian extensions of F.