Elliptic Curve arithmetic over extension field to intensify security and privacy

The elliptic curve discrete logarithm problem is one of the most important problems in cryptography. In recent years, several index calculus algorithms have been introduced for elliptic curves defined over extension fields, but the most important curves in practice, defined over prime fields, have so far appeared immune to these attacks. In this paper we formally generalize previous attacks from binary curves to prime curves. We study the efficiency of our algorithms with computer experiments and we discuss their current and potential impact on elliptic curve standards. Our algorithms are only practical for small parameters at the moment and their asymptotic analysis is limited by our understanding of Grobner basis algorithms. Nevertheless, they highlight a potential vulnerability on prime curves which our community needs to explore further.

Elliptic curves cryptography was introduced independently by Victor Miller (Miller, 1986) and Neal Koblitz (Koblitz, 1987(Koblitz, ) in 1985. .At that time elliptic curve cryptography was not actually seen as a promising cryptographic technique.As time progress and further research and intensive development done especially on the implementation side, elliptic curve cryptography is now being implemented widely.Elliptic curves cryptography offers smaller key size, bandwidth savings and faster in implementations when compared to the RSA (Rivest-Shamir-Adleman) cryptography which based its security on the integer factorization problem.The most interesting feature of the elliptic curves is the group structure of the points generated by the curves, where points on the elliptic curves form a group.The security of elliptic curves cryptography relies on the elliptic curves discrete logarithm problem.The elliptic curve discrete logarithm problem is analogous to the ordinary algebraic discrete logarithm problem, l = g x , where given the l and g, it is infeasible to compute the x.Elliptic curve discrete logarithm problem deals with solving for n the relation P = nG.Given the point P and the point G, then it is very hard to find the integer n.To implement the discrete logarithm problem in elliptic curve cryptography, the main task is to compute the order of group of the curves or in other words the number of points on the curve.Computation to find the number of points on a curve, has given rise to several point counting algorithms.The Schoof and the SEA (Schoof-Elkies-Atkin) point counting algorithms will be part of the discussion in this chapter.This chapter is organized as follows: Section 2, gives some preliminaries on elliptic curves, and in section 3, elliptic curve discrete logarithm problem is discussed.Some relevant issues on elliptic curve cryptography is discussed in section 4, in which the Diffie-Hellman key exchange scheme, ElGamal elliptic curve cryptosystem and elliptic curve digital signature scheme are discussed here accompanied with some examples.Section 5 discussed the two point counting algorithms, Schoof algorithm and the SEA (Schoof-Elkies-Atkin) algorithm.Following the discussion in section 5, section 6 summaries some similarities and the differences between these two algorithms.Section 7 gives some brief literature on these two point counting algorithms.Finally, section 8 is the concluding remarks for this chapter. Elliptic curvesElliptic curves obtained their name from their relation to elliptic integrals that arise from the computation of the arc length of ellipses (Lawrence & Wade, 2006).Elliptic curves are www.intechopen.com Cryptography and Security in Computing1 1 2 3 4 2 2 4 Fig. 1.1. 2(1 ) (1 ) yx x x   www.intechopen.com

The use of public-key cryptosystems (PKC) received considerable attention. They are beneficial in encryption as well as signing which play an essential role in electronic banking and financial transactions. Elliptic Curve Cryptography (ECC) is one of best public key techniques because of its small key size and high security. Secure applications in smart cards present implementation challenges particular to the platform's memory, bandwidth, and computation constraints. ECC's unique properties make it especially well suited to smart card applications. ECC systems provide the highest strength per bit of any cryptosystem known today. The paper presents a new method for smart card implementation of elliptic curves, explaining how ECC can not only significantly reduce the cost, but also accelerate the deployment of smart cards in new applications. ECC permits reductions in key and certificate size that translate to smaller memory requirements (especially for EEPROM), which represent significant cost savings. This added functionality can play an effective role in electronic payment and online banking technologies.

The elliptic curve cryptography is almost used in the literature to develop the public key cryptosystems. A new general-purpose symmetric key cryptosystem using elliptic prime curves is proposed in this paper. For a given prime number P, there are many elliptic prime curves on which each point in the quadrant (0,0) to (P-1,P-1) lays. The set of minimum number of eligible elliptic prime curves for a given prime number P constitutes a lookup table that is the secret key of the proposed cryptosystem. The sender can pick up one eligible elliptic curve for each 2n-bit plaintext block. The sender sends two integers cipher for each 2n-bit plaintext block with encryption rate = 0.8. Moreover, a small prime number can be used efficiently with high security and good robustness against brute force attack.

This paper investigates how to reduce the elliptic curve discrete logarithm problem over prime fields to the quadratic unconstrained binary optimization (QUBO) problem in order to obtain as few logical qubits as possible. In the best case scenario, if n is the bitlength of a characteristic of prime field Fp, approximately 3n³ logical qubits are required for such a reduction in the Edwards curve case. We present a practical attack on an elliptic curve discrete logarithm problem over the 3-bit prime field F7 for an elliptic curve with the subgroup of order 8. We solved this problem using the D-Wave Advantage QPU. To the best of the authors' knowledge, no one has made, so far, a practical attack on the elliptic curve discrete logarithm over a prime field using the direct quantum method.

A promising direction for constructing cryptographically stable pseudo-random sequence generators is an application of transformations in a group of points of elliptic and hypereliptic curves. This will allow building evidence-stable crypto algorithms, the problem of finding the private key in which is associated with solving a theoretically complex elliptic curve discrete logarithm problem. This paper proposes a method for generating pseudo-random sequences of the maximal period using transformations on elliptic curves. This method consists in the application of recurrent transformations with sequential formation of elements of points group of elliptic curves. This allows providing the maximum period of pseudo-random sequences with the reduction of the problem of finding the private key to the solution of the theoretically complex elliptic curve discrete logarithm problem. The block diagram of the device for generating pseudo-random sequences and the scheme for generating the internal states of the generator are given. We also present the results of statistical testing of some generators, which show that the generated sequences are indistinguishable in a statistical sense from truly random ones.KeywordsPseudo-random sequencesElliptic curvesMaximum period

This work presents an evaluation of different software implementations of algorithms to compute the most demanding operation in cryptographic schemes based on Elliptic Curve Cryptography (ECC), the scalar multiplication. Five different methods were studied, including traditional and more sophisticated methods for elliptic curve cryptography defined over prime and binary fields. For evaluation, the key generation algorithm in ECC consisting on a single scalar multiplication was implemented on a P500h LG smarthphone, which includes an ARM processor running at 600MHz. Both execution time and memory usage was evaluated. It was found that in general, scalar multiplication runs 8 times faster for ECC defined over prime fields, being the NAF the best performer method. For ECC over binary fields, the best performer method was wNAF. The results presented in this work could help a designer to select the most appropriate method when implementing ECC-based cryptographic schemes such as encryption or digital signatures on mobile devices like smartphones, meeting implementation requirements in terms of execution time and memory usage.

Elliptic curve cryptography is a public key cryptosystem based on the elliptic curve discrete logarithm problem. The reason for the attractiveness of ECC is that there is no sub-exponential algorithm known to solve the elliptic curve discrete logarithm problem. This leads to lower memory requirements, lower computation requirements and higher security than other public key cryptosystems. This project describes an implementation of a micro coded elliptic curve processor using FPGA technology. This processor implements optimal normal basis field operations. The design is generated by a parameterized module generator which can accommodate arbitrary n and also produce field multipliers with different speed/area tradeoffs. The control part of the processor is micro coded, enabling curve operations to be incorporated into the processor hence reducing the chip's I/O requirements. The micro coded approach also facilitates rapid development and algorithmic optimization. for example, projective and affine coordinates were supported using different sets of micro codes. The design was successfully tested on a Xilinx Spartan -II kit device. Elliptic curve processor is particularly suitable for recourse limited devices because of its lower requirements of memory and computation. It is applicable to be implanted into smart cards, cellular phones and other hand-held devices so that secure communication can be provided. For based on implement the FPGA Using Verilog HDL.

Many Cryptanalysis trials performed on the Elliptic Curve Cryptography (ECC). Some of these trials succeeded to crack the privacy of data using ECC. A proposed method by Singh et al. has been executed successfully and the cipher image is decrypted revealing the original image. It took advantage that the order of elliptic curve is not large enough to protect from attacks A robust encoding method for elliptic curve cryptosystem is proposed in this paper to provide high security level, more efficient and complexity if compared with the traditional encoding method. The proposed method encodes data and key selection over prime field. Compared with hacked ECC algorithm, the proposed encoding scheme proved its robustness based on some statistical analyzes. Correlation Coefficient (CC), Entropy, Peak Signal to Noise Ratio (PSNR), Number of changing pixel rate (NPCR), and Unified Average Changing Intensity (UACI) are used to quantify ciphertext strength and securing performance on gray and colored image to measure how much robustness is the proposed encoding technique. Together with the encryption scheme, a Least Significant Bit (LSB) steganography technique is subjoined after encryption process to hide the ciphertext. Overall results of the previews metrics were satisfied and its recommended to use the proposed encoding method with the ECC.

A pairing over an elliptic curve E(F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pm</sub> ) to an extension field of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pmk</sub> has begun to be attractive in cryptosystems, where k is called the embedding degree. The cryptosystems using a pairing are called the pairing-based cryptosystems. The embedding degree k is also an indicator of the relationship between the elliptic curve discrete logarithm problem (ECDLP) and the discrete logarithm problem (DLP), where ECDLP over E(F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pm</sub> ) is reduced to DLP over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pmk</sub> . An elliptic curve is determined by j-invarient or order, however the explicit condition between these parameters and an embedding degree has been described only in some degrees. In this paper, we investigate a new condition of the existence of elliptic curves with pre-determined embedding degrees, and present some examples of the elliptic curves over 160-bit, 192-bit and 224-bit F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pm</sub> .

We give precise quantum resource estimates for Shor's algorithm to compute discrete logarithms on elliptic curves over prime fields. The estimates are derived from a simulation of a Toffoli gate network for controlled elliptic curve point addition, implemented within the framework of the quantum computing software tool suite LIQ$Ui|\rangle$. We determine circuit implementations for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. We conclude that elliptic curve discrete logarithms on an elliptic curve defined over an $n$-bit prime field can be computed on a quantum computer with at most $9n + 2\lceil\log_2(n)\rceil+10$ qubits using a quantum circuit of at most $448 n^3 \log_2(n) + 4090 n^3$ Toffoli gates. We are able to classically simulate the Toffoli networks corresponding to the controlled elliptic curve point addition as the core piece of Shor's algorithm for the NIST standard curves P-192, P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to recent resource estimates for Shor's factoring algorithm. The results also support estimates given earlier by Proos and Zalka and indicate that, for current parameters at comparable classical security levels, the number of qubits required to tackle elliptic curves is less than for attacking RSA, suggesting that indeed ECC is an easier target than RSA.

Embedded systems are increasingly providing secure functionalities, which often rely on some dedicated hardware for symmetric and public-key cryptography. When resources are limited, elliptic curve cryptography (ECC) may be chosen instead of the more widely known RSA, which needs much longer keys for the same security level. However, ECC may be vulnerable, as any other cryptographic implementation, to side channel analysis, which may reveal secret information by analyzing collateral sources of information, such as power consumption. Countermeasures must be thus adopted at the design level, in order to ensure robust and secure operation of the device. We propose here a new scalar multiplication algorithm on an elliptic curve, based on a novel randomized window method. This design is protected against side channel attacks (Timing, Simple and Differential Power Analysis) and it is implemented over prime fields, but it can be applied to binary fields as well. In order to evaluate this countermeasure, we provide its costs, and an estimation of the additional entropy added to the computation against side channels attacks.

Elliptic curve cryptography (ECC) is based in one of the hardest arithmetic problems, the elliptic curve discrete logarithm problem, making ECC a reliable cryptographic technique. In addition to this advantage, ECC requires shorter keys compared to other public key algorithms, which suggests its use in low-end systems such as smart cards because of its efficiency and limited hardware requirements. Currently the leading smart card technology is Java card. Though both Java 2 standard edition (J2SE) 5.0 and Java card (JC) 2.2.1 include classes and interfaces making ECC available to Java programs, they are not complete, making it necessary to use additional, non-standard packages if the programmer wants to use ECC for different operations (digital signature, data encryption and decryption, etc.). In the present work, we provide a complete review of the state of the art of ECC in J2SE and Java card, with a complete list of operations available in both standard implementations and proprietary packages.

With the explosion of networks and the huge amount of data transmitted along, securing data content is becoming more and more important. Data encryption is widely used to ensure security in open networks such as the internet. This paper presents the implementation of stream cipher, where the key stream is generated based on the properties of Linear Feedback Shift Register and cyclic Elliptic Curve over a finite prime field. In this paper we illustrate the process of encryption/decryption of an image in spatial domain and also encrypt key file parameters needed for generating the key stream to other parties using Elliptic Curve Cryptography. Therefore the encrypted key file parameters are only transmitted and not the entire full-length key. Since Elliptic Curve Cryptography is replacing RSA for key exchange and Elliptic Curve based stream cipher offers a good choice for encryption in real time application. The strength of the proposed cipher lies in the generation of random sequence using Linear Feedback Shift Register over GF(p), difficulty of Elliptic Curve Discrete Logarithmic Problem and the entire key need not be transmitted in the encryption process. This paper also discusses the security aspects of the proposed cipher which is secure against all kinds of attacks.

Objectives: The role of digital images are vast in todays trends and it is exposed to illicit access, modifications that must needs confidentiality during the stages of data archival and communication1, systems, wireless devices. In order to facilitate secret communication, image encryption have bring into being a significant place in both public and private services like as military supervision, health systems, financial services like online trading and video conferencing and so on. Some methods are proposed on the image encryption which they include the Paper2 have developed a method for A Chaotic Key Based Algorithm (CKBA) to change the pixel values of the plain-image. In 3 have developed a protocol for random combinational image encryption approach with bit, pixel and block permutations Paper4 proposed a technique to encrypt an image for secure transmission using the digital signature of the image. Methods: The proposed work is developed that provides secured authentication to integrate the image encryption with Elliptic Curve Cryptography to provide security. In this work, transformation operation of the matrix operations are performed on the original image matrix and the transformed image is further encrypted with the help of a key sequence which is generated from the elliptic Curve. The encrypted image will be authenticated using Elliptic Curve Diffie Helman (ECDH) key exchange algorithm5 and finally decrypted by using the key generated from the chosen elliptic curve and then we perform the reverse transformations on the matrix obtained . So after decryption process to acquire the original image. Findings: In this work we can get better security to image ecryption by application of ECC with matrix operations performed on image data. Because ECC has the well attention of researchers and robust mathematical development and top security among the former existing algorithms. As a result Elliptic Curve Cryptography for image data is developed to meet the current security needs. Application: To transmit securely image data over the wireless channels and systems and so on. Currently this work was currently developed in an ad-hoc fashion and it can be further improved with ECC binary field and various measures of security that includes different attacks.