Elliptic curve based key generation for symmetric encryption

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 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

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.

Background: Today’s developing era data and information security plays an important role in unsecured communication between Internet of Things (IoT) elements. As the technologies are developing huge amount of information are exchanged with each other which increases the demand of a secure encryption standards. Objectives: To protect the information transmission between nodes in the perceptual layer of IoT, security is to be enhanced such that the Men in Middle (MIM) attack of information by intruders in the communication link is not possible. Methods/Statistical analysis: The security method is implemented using the concept of symmetric, asymmetric and hash algorithms between the sender and receiver. Elliptic Curve Cryptography (ECC) is performed for encrypting and sharing the Advanced Encryption Standard (AES) secret key between the sender and receiver. Elliptic Curve digital signature are added with the encrypted data for validation in transmission. Secured Hash Algorithm (SHA) is considered for generating Elliptical Curve Digital Signature Algorithm (ECDSA). All the algorithms are combined to enhance the security. Findings: Elliptic Curve Cryptography is generally applied for encrypting and sharing the secret key between the sender and receiver when Advanced Encryption Standard is used for encryption/decryption of data. A highly efficient architectures for asymmetric key encryption is applied with symmetric key encryption for encrypting the data transmitted between the devices present in the perceptual layers in the IoT. Due to security issues NIST selected SHA-3 Keccak-f[1600] algorithm. An authentication process based on ECC is applied to perceptual layer involving initialization and authentication phase apart from encryption and signature generation. Sender combines the encrypted data and key with signature together before transmitting to the receiver. The receiver verifies the signature and decrypts the key and obtains original information. Hence it is very difficult for the man in middle to access the information exchanged between the Perceptual Layer devices. Application/Improvements: This generally doesn’t require an innovative technique but an approach to develop the method implemented successfully in the perceptual layer of Internet of Things. The key is encrypted and decrypted using a novel ECC algorithm with improved ECDSA.

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.

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.

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> .

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

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.

Due to reduced key size, Elliptic Curve Cryptography (ECC) was utilized in this study for tenant authentication, data encryption, and decryption. The proposed ECC-based authentication technique efficiently defends against many related threats while allowing authorized users access to confidential data. The authors choose the exact and ideal values of elliptic curves to provide more secure data encryption by combining nature-inspired optimizations like the Moth Search Algorithm (MSA) with ECC. The DNS encryption and the ECC encryption technique are combined in the proposed encryption and decryption method. Analysis of the proposed method's security shows that it is effective against certain types of attacks, including denial-of-service attacks, spoofing attacks, reaction attacks, and simple both the selected ciphertext attack and text attacks.

An addition law for an elliptic curve is complete if it is defined for all possible pairs of input points on the elliptic curve. In Elliptic Curve Cryptography (ECC), a complete addition law provides a natural protection against side-channel attacks which are based on Simple Power Analysis (SPA). Montgomery curves are a specific family of elliptic curves that play a crucial role in ECC because of its well-known Montgomery ladder, particularly in the Elliptic Curve Diffie-Hellman Key Exchange (ECDHKE) protocol and the Elliptic Curve factorization Method (ECM). However, the complete addition law for Montgomery curves, as stated in the literature, has a computational cost of 14M+ 2D, where M,D denote the costs of a field multiplication and a field multiplication by a constant, respectively. The lack of a competitive complete addition law has led implementers towards twisted Edwards curves, which offer a complete addition law at a lower cost of 8M+ 1D for appropriately chosen curve constants.In this paper, we introduce extended Montgomery coordinates as a novel representation for points on Montgomery curves. This coordinate system enables us to define birational multiplication-free maps between the extended twisted Edwards coordinates and extended Montgomery coordinates. Using this map, we can transfer the complete addition laws from twisted Edwards curves to Montgomery curves without incurring additional multiplications or squarings. In addition, we employ a technique known as scaling to refine the addition laws for twisted Edwards curves, which results in having i) Complete addition laws with the costs varying between 8M+1D and 9M+1D for a broader range of twisted Edwards curves, ii) Incomplete addition laws for twisted Edwards curves with the cost of 8M. Consequently, by leveraging our birational multiplication-free maps, we present complete addition laws for Montgomery curves with the cost of 8M+1D. This shows a significant improvement for complete addition law for Montgomery curves by reducing the computational cost by 6M+ 1D. This improvement makes Montgomery curves a more attractive option for applications where an efficient complete addition law is essential.

The Residue Number System (RNS) is a numeral representation enabling for more efficient addition and multiplication implementations. However, due its non-positional nature, modular reductions, required for example by Elliptic Curve (EC) Cryptography (ECC), become costlier. Traditional approaches to RNS modular reduction resort to the Montgomery algorithm, underpinned by large basis extensions. Recently, Hybrid-Positional Residue Number Systems (HPRs) have been proposed, providing a trade-off between the efficiency of RNS and the flexibility of positional number representations. Numbers are represented in a positional representation with the coefficients represented in RNS. By crafting primes of a special form, the complexity of reductions modulo those primes is mitigated, relying on extensions of smaller bases. Due to the need of crafting special primes, this approach is not directly extensible to group operations over currently standardised elliptic curves. In this paper, the Hybrid-Polynomial Residue Number System (HyPoRes) is proposed, enabling for improved modular reductions for any prime. Experimental results show that the modular reduction of HyPoRes, although at most 1.4 times slower than HPR for HPR-crafted primes, is up to 1.4 times faster than a generic RNS approach for primes of ECC standards.

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.

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.

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.