Extended Explicit Relations Between Trace, Definition Field, and Embedding Degree

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

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> .

Elliptic curve cryptosystem (ECC) was proposed by Miller [10] and Koblitz [9] which relies on the difficulty of elliptic curve discrete logarithmic problem (ECDLP). It is gaining wide acceptance as an alternative to the conventional public key cryptosystem such as RSA [24], DSA [25], DH [26]. Also it is noted that the wireless sensor networks (WSN) based on the rapid progress of wireless communications and embedded micro electro mechanical systems technologies are becoming important part in our daily life. The security of the WSN becomes one of the major concerns in its applications. Even ECC prominently offers great potential benefits for WSN security there is still a lot of work needs to be done due to WSN has very restraint running conditions such as limited energy source, capability of computing, etc. It is well known that scalar multiplication operation in ECC accounts for about 80% of key calculation time on wireless sensor network motes. In this paper we present an optimized dynamic window based on our previous research works. The whole quality of service (QoS) has been improved under this algorism in particularly the power consuming is more efficiently. The simulation results showed that the average calculation time, due to fuzzy conditions decreased from previous 26 to current 9 as a whole the calculation time, decreased by approximately 18% in comparison to our previous algorithms in an ECC wireless sensor network [23].

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

Session Initiation Protocol(SIP) provides authentication and session key agreement to ensure the security of the successive session.In 2010,Yoon et al.(YOON E-J,YOO K-Y.A three-factor authenticated key agreement scheme for SIP on elliptic curves.NSS '10: 4th International Conference on Network and System Security.Piscataway: IEEE,2010: 334-339.) proposed a three-factor authenticated key agreement scheme named TAKA_SIP for SIP.However,the scheme is vulnerable to insider attack,server-spoofing attack,off-line password attack,and losing token attack.Moreover,it does not provide mutual authentication.To overcome these flaws of TAKA_SIP,a new three-factor authentication scheme named ETAKA_SIP based on Elliptic Curve Cryptosystem(ECC) was proposed.ETAKA_SIP,on the basis of elliptic curve discrete logarithm problem,provides higher security than TAKA_SIP.It needs 7 elliptic curve scalar multiplication operations,1 additional operation and up to 6 Hash operations,and of high efficiency.

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.

The concept of public key cryptography was first introduced by Diffie and Hellman in 1976 using discrete logarithm problem as base of difficulty. In 1985, T. ElGamal proposed public key cryptosystem scheme based on discrete logarithm problem. Elliptic curve cryptosystems were first proposed in 1985 independently by Neil Koblitz and Victor Miller. Elliptic curve cryptosystems are unique in using elliptic curve groups for arithmetic. This cryptosystem is based on discrete logarithm problem in the group of points of an elliptic curve defined over a finite field. The discrete logarithm problem in an elliptic curve group appears to be much harder than the discrete logarithm problem in other groups. Hence elliptic curves cryptosystem can match the security of other cryptosystems while using smaller key. In this paper we will discuss a VLSl implementation of elliptic curves cryptosystem for ElGamal encryption scheme.

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.

There have been many digital signature schemes were developed based on the discrete logarithm problem on a finite field. In this study, we use the elliptic curve discrete logarithm problem to build new collective signature schemes. The cryptosystem on elliptic curve allows to generate digital signatures with the same level of security as other cryptosystems but with smaller keys. To extend practical applicability and enhance the security level of the group signature protocols, we propose two new types of collective digital signature schemes based on the discrete logarithm problem on the elliptic curve: i) the collective digital signature scheme shared by several signing groups and ii) the collective digital signature scheme shared by several signing groups and several individual signers. These two new types of collective signatures have combined the advantages of group digital signatures and collective digital signatures. These signatures have a fixed size and do not depend on the number of members participating in the creation of the final collective signature. One of the advantages of the proposed collective signature protocols is that they can be deployed on top of the available public key infrastructures.

In this paper we presents an Elliptic Curve Discrete Logarithm Problem (ECDLP) based blind signature scheme. Elliptic Curve Cryptosystem is the best known public-key cryptosystem till date, and is suitable for resource constrained environment because of smaller key size. The proposed scheme is untraceable and proved to be resistant against active attacks. The scheme is also universally verifiable. We compared the performance with some existing scheme and found that, the computational overhead of the proposed scheme is very low. Hence it can be applicable to real life scenarios, such as, e-voting, e-cash and e-commerce applications.

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.

In this digital world, communication systems have witnessed abundant usage of media over the platforms. Among these, providing security in transmission of images is highly important, and attained a lot of research interest due to its high consideration in both the industry and the academic community. This paper proposes a hybrid asymmetric image encryption algorithm using Elliptic curve cryptosystem (ECC), Hadamard transform and Hill cipher algorithms. Based on the Diffie–Hellman public key exchange method a point on the elliptic curve is selected and agreed between both the sender and receiver. The key relies upon the ECC and it is difficult to resolve the ECDLP to get it. The proposed algorithm involves two stages of encryption, primarily, the XOR function is applied on the Elliptic curve Diffie-Hellman (ECDH) shared secret key and the hadamard image. In the subsequent stage, ECC is combined with the hill cipher algorithm. Encryption and decryption uses self-invertible key matrix, hence the process of finding inverse key becomes redundant during decryption which improves the speed of execution. It also enhances the security and efficiency compared to original hill cipher method. The results are compared with other ECC methods proves that the current cryptosystem attains large key space, highly key sensitive, low correlation and can resist against differential and statistical attacks.

With the rapid development of the Internet, many users start to take action putting personal or company information on it, and share with everyone. The Internet is public as it were, if we do not control its limit of authority to assure security, it's possible that those attackers do illegal access of important information and destroy them. Not only personal privacy is invaded, but the mass property damage. Therefore, effective access control system has been more and more emphasized these days. To fight against these network attacks, it is necessary to establish an effective and safe access control system; here we proposed a scheme, a key management which called Lagrange interpolation mainly takes access control model as framework, and use Elliptic Curve Cryptography system to enhance security. The fact we choose Lagrange interpolation is the key we use is randomized, no relationship between each key, so is relatively hard. As Elliptic Curve Cryptography system, we want attackers to encounter Elliptic Curve Discrete Logarithm Problem. Once the prime number is big enough, attackers will have trouble deciphering the key. Access control is so comprehensive, such as electronic documents, online television system sand wireless networks and so on, from previous literature; mobile agent technology applied in access control and key management would waste space and do exist some flaws in security, also we still have a lot of works to do on medical application. Hence, we propose these schemes in mobile agent in order to reach improvement, and then analyze of security and try to simulate what attackers will do. We conclude four common attacks: External Collective Attack, Internal Attack, Collusion Attacks and Equation Breaking Attack. As results, attackers are hard to decipher the key because of no relationship between each key and will face Elliptic Curve Discrete Logarithm Problem. We confirm that the proposed schemes can be more efficiently and safety to protect mobile agent.

This paper examines the cryptographic security of fixed versus random elliptic curves over the field GF(p). Its basic assumption is that a large precomputation to aid in breaking the elliptic curve discrete logarithm problem (ECDLP) can be made for a fixed curve. We take this into account when examining curve security as well as considering a variation of Pollard's rho method where computations from solutions of previous ECDLPs can be used to solve subsequent ECDLPs on the same curve. We present a lower bound on the expected time to solve such ECDLPs using this method, as well as an approximation of the expected time remaining to solve an ECDLP when a given size of precomputation is available. We conclude that adding 5 bits to the size of a fixed curve to avoid general software attacks and an extra 6 bits to avoid attacks on special moduli and a parameters provides an equivalent level of security.

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.