Elliptic Curve Point Research Articles

Chaotic Map-Based Anonymous User Authentication Scheme With User Biometrics and Fuzzy Extractor for Crowdsourcing Internet of Things

The recent proliferation of mobile devices, such as smartphones and wearable devices has given rise to crowdsourcing Internet of Things (IoT) applications. E-healthcare service is one of the important services for the crowdsourcing IoT applications that facilitates remote access or storage of medical server data to the authorized users (for example, doctors, patients, and nurses) via wireless communication. As wireless communication is susceptible to various kinds of threats and attacks, remote user authentication is highly essential for a hazard-free use of these services. In this paper, we aim to propose a new secure three-factor user remote user authentication protocol based on the extended chaotic maps. The three factors involved in the proposed scheme are: 1) smart card; 2) password; and 3) personal biometrics. As the proposed scheme avoids computationally expensive elliptic curve point multiplication or modular exponentiation operation, it is lightweight and efficient. The formal security verification using the widely-accepted verification tool, called the ProVerif 1.93, shows that the presented scheme is secure. In addition, we present the formal security analysis using the both widely accepted real-or-random model and Burrows–Abadi–Needham logic. With the combination of high security and appreciably low communication and computational overheads, our scheme is very much practical for battery limited devices for the healthcare applications as compared to other existing related schemes.

Reconfigurable Hardware Architecture for Authenticated Key Agreement Protocol Over Binary Edwards Curve

In this article, we present a high-performance hardware architecture for Elliptic curve based (authenticated) key agreement protocol “Elliptic Curve Menezes, Qu and Vanstone” (ECMQV) over Binary Edwards Curve (BEC). We begin by analyzing inversion module on a 251-bit binary field. Subsequently, we present Field Programmable Gate Array (FPGA) implementations of the unified formula for computing elliptic curve point addition on BEC in affine and projective coordinates and investigate the relative performance of these two coordinates. Then, we implement the w -coordinate based differential addition formulae suitable for usage in Montgomery ladder. Next, we present a novel hardware architecture of BEC point multiplication using mixed w -coordinates of the Montgomery laddering algorithm and analyze it in terms of resistance to Simple Power Analysis (SPA) attack. In order to improve the performance, the architecture utilizes registers efficiently and uses efficient scheduling mechanisms for the BEC arithmetic implementations. Our implementation results show that the proposed architecture is resistant against SPA attack and yields a better performance when compared to the existing state-of-the-art BEC designs for computing point multiplication (PM). Finally, we present an FPGA design of ECMQV key agreement protocol using BEC defined over GF(2 251 ). The execution of ECMQV protocol takes 66.47μs using 32,479 slices on Virtex-4 FPGA and 52.34μs using 15,988 slices on Virtex-5 FPGA. To the best of our knowledge, this is the first FPGA design of the ECMQV protocol using BEC.

Constant-time hardware computation of elliptic curve scalar multiplication around the 128 bit security level

In this paper we present two classes of scalar multiplication hardware architectures that compute a constant-time variable-base point multiplication over the Galbraith–Lin–Scott (GLS) family of binary elliptic curves. Our first architecture is speed-optimized and utilizes the available hardware resources to achieve the fastest possible elliptic curve point multiplication operation. The second architecture, on the other hand, targets a more effective resource utilization by optimizing time-area product. Our hardware designs are especially tailored for the quadratic extension field F22n, with n=127, which allows us to attain a security level close to 128 bits. We explore extensively the usage of digit-based and Karatsuba multipliers for performing the quadratic field arithmetic associated to GLS elliptic curves and report the area and time performance obtained by these two types of multipliers. Targeting a Xilinx Kintex-7 FPGA device, we report on real hardware implementations of our designs, the fastest of which achieves a delay of just 7.97 µs for computing one scalar multiplication on a 1-core design.

Fast elliptic curve point multiplication based on window Non-Adjacent Form method

This article presents window Non-Adjacent Form (wNAF) of scalar representation method for developing algorithms of computing scalar multiplication in groups of points on an elliptic curve over finite fields. A new efficient wNAF-based algorithm has been presented. The algorithm was developed based on simple and composite operations with a point and also based on affine and Jacobi coordinates systems taking into account the latest achievements in computing cost reduction. The theorem concerning its computational complexity is formulated and proved for this new algorithm. In the end of this article, an efficiency analysis of the proposed algorithm depending on various parameters is presented, namely, the characteristic of the field over which the curve is defined, the scalar length and the window width of the precomputations and the coordinates of the precomputed points. A comparative analysis of the new algorithm and previously known efficient algorithm is also presented.

Euclidean addition chains scalar multiplication on curves with efficient endomorphism

Random Euclidean addition chain generation has proven to be an efficient low memory and SPA secure alternative to standard ECC scalar multiplication methods in the context of fixed base point [21]. In this work, we show how to generalize this method to random point scalar multiplication on elliptic curves with an efficiently computable endomorphism. In order to do so we generalize results from [21] on the relation of random Euclidean chains generation and elliptic curve point distribution obtained from those chains. We propose a software implementation of our method on various platforms to illustrate the impact of our approach. For that matter, we provide a comprehensive study of the practical computational cost of the modular multiplication when using Java and C standard libraries developed for the arithmetic over large integers.

Elliptic Curve Integral Points on y2 = x3 + 3x − 14

The positive integer points and integral points of elliptic curves are very important in the theory of number and arithmetic algebra, it has a wide range of applications in cryptography and other fields. There are some results of positive integer points of elliptic curve y2 = x3 + ax + b, a, b ∈ Z In 1987, D. Zagier submit the question of the integer points on y2 = x3 − 27x + 62, it count a great deal to the study of the arithmetic properties of elliptic curves.In 2009, Zhu H L and Chen J H solved the problem of the integer points on y2 = x3 − 27x + 62 by using algebraic number theory and P-adic analysis method.In 2010, By using the elementary method, Wu H M obtain all the integral points of elliptic curves y2 = x3 − 27x − 62.In 2015, Li Y Z and Cui B J solved the problem of the integer points on y2 = x3 − 21x − 90 By using the elementary method.In 2016, Guo J solved the problem of the integer points on y2 = x3 + 27x + 62 by using the elementary method.In 2017, Guo J proved that y2 = x3 − 21x + 90 has no integer points by using the elementary method.Up to now, there is no relevant conclusions on the integral points of elliptic curves y2 = x3 + 3x − 14, which is the subject of this paper.By using congruence and Legendre Symbol, it can be proved that elliptic curve y2 = x3 + 3x − 14 has only one integer point: (x, y) = (2, 0).

Cartan images and $$\ell $$ ℓ -torsion points of elliptic curves with rational j-invariant

Let $$\ell $$ be an odd prime and d a positive integer. We determine when there exists a degree-d number field K and an elliptic curve E / K with $$j(E)\in \mathbb {Q}\setminus \{0,1728\}$$ for which $$E(K)_\mathrm {tors}$$ contains a point of order $$\ell $$ , conditionally on a conjecture of Sutherland. We likewise determine when there exists such a pair (K, E) for which the image of the associated mod- $$\ell $$ Galois representation is contained in a Cartan subgroup or its normalizer. We do the same under the stronger assumption that E is defined over $$\mathbb {Q}$$ .

On the class numbers of the fields of the p^n-torsion points of elliptic curves over \protect \mathbb{Q}

Let $E$ be an elliptic curve over $\mathbb{Q}$ which has multiplicative reduction at a fixed prime $p$. For each positive integer $n$ we put $K_n:=\mathbb{Q}(E[p^n])$. The aim of this paper is to extend the author's previous our results concerning with the order of the $p$-Sylow group of the ideal class group of $K_n$ to more general setting. We also modify the previous lower bound of the order and describe the new lower bound in terms of the Mordell-Weil rank of $E(\mathbb{Q})$ and the ramification related to $E$.

Digital Image Encryption Algorithm Based on Elliptic Curve Public Cryptosystem

With the increasing use of media in communications, the content security of digital images attracts much attention in both the academia and the industry. Meanwhile, for the symmetric cryptosystem, the key transmission and management is burden on users. This paper proposes an asymmetric image encryption algorithm based on an elliptic curve cryptosystem (ECC). The sender and the recipient agree on an elliptic curve point based on the Diffie–Hellman public key sharing technique. First, to reduce the encryption times, the sender groups pixel values together and converts them into big integers. Second, the sender encrypts big integers with ECC and the chaotic system. Finally, the encrypted image is obtained from encrypted big integers. The proposed algorithm makes the key transmission and management relatively simple and secure. Simulation data show that the proposed algorithm exhibits both the strong security and the high efficiency.

Fast elliptic curve point multiplication based on binary and binary non-adjacent scalar form methods

This article presents two methods for developing algorithms of computing scalar multiplication in groups of points on an elliptic curve over finite fields. Two new effective algorithms have been presented: one of them is based on a binary Non-Adjacent Form of scalar representation and another one on a binary of scalar representation method. All algorithms were developed based on simple and composite operations with point and also based on affine and Jacobi coordinates systems taking into account the latest achievements in computing cost reduction. Theorems concerning their computational complexity are formulated and proved for these new algorithms. In the end of this article comparative analysis of both new algorithms among themselves and previously known algorithms are represented.

65‐nm CMOS low‐energy RNS modular multiplier for elliptic‐curve cryptography

Modular multiplication (MM) is the main operation in cryptography algorithms such as elliptic-curve cryptography (ECC) and Rivest-Shamir-Adleman, where repeated MM is used to perform elliptic curve point multiplication and modular exponentiation, respectively. The algorithm for the proposed architecture is derived from the Chinese remainder theorem and performs MM completely within a residue number system (RNS). Moreover, a 40-channel RNS moduli-set is proposed for this architecture to benefit from the short-channel width of the RNS moduli-set. The throughput of the architecture is enhanced by pipelining and pre-computations. The proposed architecture is fabricated as an ASIC using 65-nm CMOS technology. The measurement results are obtained for energy dissipation at different voltage levels from 0.43 to 1.25 V. The maximum throughput of the proposed design is 1037 Mbps while operating at a frequency of 162 MHz with an energy dissipation of 48 nJ. The proposed architecture enables the construction of low-voltage and energy-efficient ECCs.

On parametric spaces of bicentric quadrilaterals

Abstract In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as tangents. Consider a bicentric quadrilateral with rational sides. We discuss the problem of finding such quadrilaterals where the ratio of the radii of the circumcircle and incircle is rational. We show that this problem can be formulated in terms of a family of elliptic curves given by Ea : y 2 = x 3 + (a 4 − 4a 3 − 2a 2 − 4a + 1)x 2 + 16a 4 x which have, in general, (ℤ/8ℤ), and in rare cases (ℤ/2ℤ × ℤ/8ℤ) as torsion subgroups. We show the existence of infinitely many elliptic curves Ea of rank at least two with torsion subgroup ℤ/8ℤ, parametrised by the points of an elliptic curve of rank at least one, and give five particular examples of rank 5. We, also, show the existence of a subfamily of Ea whose torsion subgroup is ℤ/2ℤ × ℤ/8ℤ.

A robust image encryption scheme based on chaotic system and elliptic curve over finite field

The paper proposes a robust image encryption scheme based on chaotic system and elliptic curve over a finite field. The sender and receiver agree on an elliptic curve point based on Diffie-Hellman public key sharing technique. The logistic map is used to generate a chaotic sequence with initial conditions derived from the shared elliptic curve point. The chaotic sequence is converted to integers and the point multiplication is performed with the shared elliptic curve point. The resulting elliptic curve points are converted to byte values to generate a random sequence. The image to be encrypted is scrambled using Arnold’s transform where the number of scrambling rounds is derived from the shared elliptic curve point. The scrambled image pixels value is XOR with the random sequence to generate the cipher image. Statistical, performance, security and robustness analyses show that the proposed scheme is a robust encryption scheme with the ability to resist from different types of attacks.

FPGA Based Crypto Processor for Elliptic Curve Point Multiplication (ECPM) Over GF(2233)

FPGA Based Crypto Processor for Elliptic Curve Point Multiplication (ECPM) Over GF(2233)

Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm

The negation map can be used to speed up the computation of elliptic curve discrete logarithms using either the baby-step giant-step algorithm (BSGS) or Pollard rho. Montgomery's simultaneous modular inversion can also be used to speed up Pollard rho when running many walks in parallel. We generalize these ideas and exploit the fact that for any two elliptic curve points X and Y, we can efficiently get X-Y when we compute X+Y. We apply these ideas to speed up the baby-step giant-step algorithm. Compared to the previous methods, the new methods can achieve a significant speedup for computing elliptic curve discrete logarithms in small groups or small intervals.Another contribution of our paper is to give an analysis of the average-case running time of Bernstein and Lange's 'grumpy giants and a baby' algorithm, and also to consider this algorithm in the case of groups with efficient inversion.Our conclusion is that, in the fully-optimised context, both the interleaved BSGS and grumpy-giants algorithms have superior average-case running time compared with Pollard rho. Furthermore, for the discrete logarithm problem in an interval, the interleaved BSGS algorithm is considerably faster than the Pollard kangaroo or Gaudry-Schost methods.

High performance hardware support for elliptic curve cryptography over general prime field

Secure information exchange in resource constrained devices can be accomplished efficiently through elliptic curve cryptography (ECC). Due to the high computational complexity of ECC arithmetic, a high performance dedicated hardware architecture is essential to provide sufficient performance in a computation of elliptic curve scalar multiplication. This paper presents a high performance hardware support for elliptic curve cryptography over a prime field GF(p). It exploited a best available possible parallelism of elliptic curve points in projective representation. The proposed hardware for ECC is implemented on Xilinx Virtex-4, Virtex-5 and Virtex-6 FPGAs. A 256-bit scalar multiplication is completed in 2.01 ms, 2.62 ms and 3.91 ms on Virtex-6, Virtex-5 and Virtex-4 FPGA platforms, respectively. The results show that the proposed design is 1.96 times faster with insignificant increase in area consumption as compared to the other reported designs. Therefore, it is a good choice to be used in many ECC based schemes.

A 5.1μJ per point‐multiplication elliptic curve cryptographic processor

SummarySecurity features such as privacy and device authentication are required in wireless sensor networks, electronic IDs, Radio Frequency Identification tags, and many other applications. These features are provided using cryptography. Symmetric key cryptography, where the key is distributed between the communication parties prior to communication, does not provide adequate solution for large scalable systems such as sensor networks. In these cases, public‐key cryptography should be used. However, public‐key algorithms are typically more computationally intensive than their symmetric key counterparts, which creates difficulties in meeting the strict area, power, and energy requirements. Elliptic curve cryptography, because of relatively small operand sizes, can be used to answer the imposed challenges. In this paper, we present a processor for elliptic curve cryptography over GF(2163). This processor can perform elliptic curve point multiplication as well as general modular operations. The processor is flexible enough to support multiple cryptographic protocols. The chip is fabricated using UMC .13 μm 1P8M process, resulting in a core area of 0.54 mm2. The energy consumption to perform one elliptic curve point multiplication is 5.1 μJ. The design features lightweight countermeasures against side‐channel attacks. A security evaluation shows the effectiveness of such countermeasures. Copyright © 2016 John Wiley & Sons, Ltd.

Improved elliptic curve hashing and point representation

For a large class of functions $$f:\mathbb {F}_q\rightarrow E(\mathbb {F}_q)$$f:FqźE(Fq) to the group of points of an elliptic curve $$E/\mathbb {F}_q$$E/Fq (typically obtained from certain algebraic correspondences between E and $$\mathbb {P}^1$$P1), Farashahi et al. (Math Comput 82(281):491---512, 2013) established that the map $$(u,v)\mapsto f(u)+f(v)$$(u,v)źf(u)+f(v) is regular, in the sense that for a uniformly random choice of $$(u,v)\in \mathbb {F}_q^2$$(u,v)źFq2, the elliptic curve point $$f(u)+f(v)$$f(u)+f(v) is close to uniformly distributed in $$E(\mathbb {F}_q)$$E(Fq). This result has several applications in cryptography, mainly to the construction of elliptic curve-valued hash functions and to the Elligator Squared technique by Tibouchi (in: Christin and Safavi-Naini (eds) Financial cryptography. LNCS, vol 8437, pp 139---156. Springer, Heidelberg, 2014) for representating uniform points on elliptic curves as close to uniform bitstrings. In this paper, we improve upon Farashahi et al.'s character sum estimates in two ways: we show that regularity can also be obtained for a function of the form $$(u,v)\mapsto f(u)+g(v)$$(u,v)źf(u)+g(v) where g has a much smaller domain than $$\mathbb {F}_q$$Fq, and we prove that the functions f considered by Farashahi et al. also satisfy requisite bounds when restricted to large intervals inside $$\mathbb {F}_q$$Fq. These improved estimates can be used to obtain more efficient hash function constructions, as well as much shorter Elligator Squared bitstring representations.

Fields generated by torsion points of elliptic curves

Let K be a field of characteristic char(K)≠2,3 and let E be an elliptic curve defined over K. Let m be a positive integer, prime with char(K) if char(K)≠0; we denote by E[m] the m-torsion subgroup of E and by Km:=K(E[m]) the field obtained by adding to K the coordinates of the points of E[m]. Let Pi:=(xi,yi) (i=1,2) be a Z-basis for E[m]; then Km=K(x1,y1,x2,y2). We look for small sets of generators for Km inside {x1,y1,x2,y2,ζm} trying to emphasize the role of ζm (a primitive m-th root of unity). In particular, we prove that Km=K(x1,ζm,y2), for any odd m⩾5. When m=p is prime and K is a number field we prove that the generating set {x1,ζp,y2} is often minimal, while when the classical Galois representation Gal(Kp/K)→GL2(Z/pZ) is not surjective we are sometimes able to further reduce the set of generators. We also describe explicit generators, degree and Galois groups of the extensions Km/K for m=3 and m=4.

Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves

In the present paper, we prove the {\it finiteness} of the set of {\it moderate} rational points of a once-punctured elliptic curve over a number field. This {\it finiteness} may be regarded as an analogue for a once-punctured elliptic curve of the well-known {\it finiteness} of the set of torsion rational points of an abelian variety over a number field. In order to obtain the {\it finiteness}, we discuss the {\it center} of the image of the pro-$l$ outer Galois action associated to a hyperbolic curve. In particular, we give, under the assumption that $l$ is {\it odd}, a {\it necessary and sufficient condition} for a certain hyperbolic curve over a generalized sub-$l$-adic field to have {\it trivial center}.