Elliptic Prime Curves Based New Symmetric Key Cryptosystem

A public-key encryption algorithm and a practical hybrid scheme are proposed in this paper. This algorithm uses the advanced transform techniques, i.e. the Number Theoretic Transforms (NTTs) to diffuse data, and uses a non-linear technique based on the Elliptic Curve Discrete Logarithm Problem (ECDLP) to confuse data. The main advantage of the proposed algorithm is its high efficiency and capability. The plaintext can be encrypted in blocks in parallel, thus saves computational time. The proposed algorithm also can be easily applied in a hybrid scheme for practical applications.

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

There are many threats from virus, hacker, electronic eavesdropper, and electronic deceiver in the network. The growth of the computer system and network makes the organizations and personal users to depend on the information that circulate in these systems more and more. This makes us be aware of that protecting the data and resources from leak, insuring the reliability of data and information, and protecting the system from suffering the hit from network are very important.Submit a cryptography that is affirmative by people to reach the demand on the safety of SET system. The algorithm submitted is public, and everyone can use and to verify its safety, using Elliptic Curve Cryptography to execute the key exchange and encryption decryption, and making sure its performance of SET is better while the safety of SET is same as before, and the overload of compute and transmission is yet lower.

Cloud storage is a useful service of cloud computing which allows users to upload their data in the cloud and share it with others. Although, this new service is affordable and practical, but it is associated with several privacy and security challenges. Nowadays, Attribute Based Encryption (ABE) is widely used to provide secure data sharing in the distributed environment such as cloud computing. Unfortunately, most of the existing ABE schemes are not suitable for resource- constraint cloud systems, because they use expensive bilinear pairing operation and thus they cause a very high encryption and decryption computation overhead. In this paper, we propose an efficient no-pairing and revocable ABE data sharing scheme based on Elliptic Curve Cryptography (ECC) for cloud storage systems. Moreover, a comprehensive security and performance analysis shows that our scheme is both secure and efficient.

We propose to represent a scalar in the form of d (1/2)(superscript a)3(superscript b), where a and b are integers and d is an odd integer belonging to a given set. This representation is a combination of the extended double-base number system (DBNS) and the double-base chain representation using powers of 1/2 and 3. Experimental results show that our approach leads to a shorter DBNS expansion and a lower complexity in elliptic curve scalar multiplication, with the cost of only a few pre-computations and storages. This contributes to the efficient implementation of elliptic curve cryptography (ECC).

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

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 reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sh

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.

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.

Elliptic curves are beautiful mathematical objects that again and again appear in the most surprising places. Their history certainly originates at least in ancient Greece, whereas the study of arithmetic properties of elliptic curves as objects in algebra, geometry, and number theory traces back to the nineteenth century. Curiously, the earliest use of the term “elliptic curve” in the literature seems to have been by James Thomson in 1727 in “A Poem sacred to the Memory of Sir Isaac Newton”: “He, first of Men, with awful Wing pursu’d the Comet tro’ the long Elliptic Curve.” In 1985, Koblitz and Miller independently proposed to use elliptic curves in cryptography which can only be described as a magnificent and practical application of elliptic curves. This paper intends to mostly present a low-brow introduction of elliptic curves and their use in real-world applications of public-key cryptography.

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 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.

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.

This document updates RFC 5480 to specify semantics for the keyEncipherment and dataEncipherment key usage bits when used in certificates that support Elliptic Curve Cryptography.