An implementation of ElGamal elliptic curves cryptosystems

Наведено порівняльний аналіз відомих протоколів узгодження таємного ключа через відкритий канал звʼязку. Запропоновано узагальнення відомого протоколу з використанням некомутативного множення матриць над простим скінченним полем на випадок довільного скінченного поля. A symmetric cryptosystem requires a secret key agreed by both parties. The Diffie-Hellman protocol was originally proposed for its exchange via an open communication channel. It is based on the computational complexity of the discrete logarithm problem in certain finite groups (the multiplicative group of a finite field, the group of points of an elliptic curve over a finite field). The availability of a powerful quantum computer will allow solving the discrete logarithm problem in these groups. Therefore, the issue of construction of secret key exchange protocols that will be resistant to attacks using a quantum computer has become urgent. The paper provides a comparative analysis of known protocols for a secret key exchange via an open communication channel. A generalization of the known protocol using non-commutative matrix multiplication over a prime finite field for the case of an arbitrary finite field is proposed. In this protocol, two high order elements from the general linear group over a finite field should be used, which satisfy an additional condition. It consists in the fact that each of the matrices cannot be reduced to a diagonal form by conjugation transformation. This condition follows from the considerations of avoiding an attack on the protocol. It is shown how to construct such elements. For the proposed generalization of the protocol, the sizes of public and private keys, secret key, the size of the finite field, which ensure the appropriate level of security, are calculated. The dimensions of matrices (elements of the general linear group) that we use are also given. The generalization of the protocol to the case of an arbitrary finite field increases the flexibility of choosing protocol parameters to ensure the desired level of security.

We show that finding an efficiently computable injective homomorphism from the XTR subgroup into the group of points over GF(p2) of a particular type of supersingular elliptic curve is at least as hard as solving the Diffie–Hellman problem in the XTR subgroup. This provides strong evidence for a negative answer to the question posed by Vanstone and Menezes at the Crypto 2000 Rump Session on the possibility of efficiently inverting the MOV embedding into the XTR subgroup. As a side result we show that the Decision Diffie–Hellman problem in the group of points on this type of supersingular elliptic curves is efficiently computable, which provides an example of a group where the Decision Diffie–Hellman problem is simple, while the Diffie–Hellman and discrete logarithm problems are presumably not. So-called distortion maps on groups of points on elliptic curves that play an important role in our cryptanalysis also lead to cryptographic applications of independent interest. These applications are an improvement of Joux’s one round protocol for tripartite Diffie–Hellman key exchange and a non-refutable digital signature scheme that supports escrowable encryption. We also discuss the applicability of our methods to general elliptic curves defined over finite fields which includes a classification of elliptic curve groups where distortion maps exist.

We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over ${\text {GF}}({2^n})$. We discuss the question of primitive points on an elliptic curve modulo p, and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point.

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 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.

The new signature scheme presented by the authors in [13] is the first signature scheme based on the discrete logarithm problem that gives message recovery. The purpose of this paper is to show that the message recovery feature is independent of the choice of the signature equation and that all ElGamal-type schemes have variants giving message recovery. For each of the six basic ElGamal-type signature equations five variants are presented with different properties regarding message recovery, length of commitment and strong equivalence. Moreover, the six basic signature schemes have different properties regarding security and implementation. It turns out that the scheme proposed in [13] is the only inversionless scheme whereas the message recovery variant of the DSA requires computing of inverses in both generation and verification of signatures. In general, message recovery variants can be given for ElGamal-type signature schemes over any group with large cyclic subgroup as the multiplicative group of GF(2n) or elliptic curve over a finite field. The present paper also shows how to integrate the DLP-based message recovery schemes with secret session key establishment and ElGamal encryption. In particular, it is shown that with DLP-based schemes the same functionality as with RSA can be obtained. However, the schemes are not as elegant as RSA in the sense that the signature (verification) function cannot at the same time be used as the decipherment (encipherment) function.

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 subexponential algorithm for discrete logarithms over hyperelliptic curves of large genus over GF( q)

An elliptic curve cryptosystem (ECC) is one of public key cryptosystem, whose security is based on elliptic curve discrete logarithm problem (ECDLP). An elliptic curve is uniquely determined by mathematical parameters such as j-invariant of an elliptic curve. By giving trace of elliptic curve, t, a definition field \(\mathbb F_{p}\), and discriminant D, an elliptic curve with order \(\sharp {E(\mathbb F_{p})} = n\) is determined. Therefore it is an open problem to determine explicit relations between the mathematical parameters and the embedding degrees k. Hirasawa and Miyaji presented concrete relations between the mathematical parameters and the embedding degrees. In this research, a new explicit relation between elliptic-curve parameters and embedding degrees is investigated by generalizing their research.

A nonlinear elliptic curve cryptosystem based on matrices

The importance of data encryption has grown dramatically, especially in terms of personal data. The elliptic curve cryptosystem is the major solution for data security because it has become more prevalent. Security and privacy are required to ensure the data has recently generated much concern within the research community. This paper's objective is to obtain a complicated and secure ciphertext and make cryptanalysis difficult. In this paper, we modified the El-Gamal Elliptic Curve Cryptosystem (ECC) by producing new secret keys for encrypting data and embedding messages by using Discrete Logarithm Problem (DLP) behavior. This modification is to offer enhanced encryption standards and improve the security. The experiential results show that the proposed algorithm is more complex than the original method.

A simple new technique of parallelizing methods for solving search problems which seek collisions in pseudorandom walks is presented. This technique can be adapted to a wide range of cryptanalytic problems which can be reduced to finding collisions. General constructions are given showing how to adapt the technique to finding discrete logarithms in cyclic groups, finding meaningful collisions in hash functions, and performing meet-in-the-middle attacks such as a known-plaintext attack on double encryption. The new technique greatly extends the reach of practical attacks, providing the most cost-effective means known to date for defeating: the small subgroup used in certain schemes based on discrete logarithms such as Schnorr, DSA, and elliptic curve cryptosystems; hash functions such as MD5, RIPEMD, SHA-1, MDC-2, and MDC-4; and double encryption and three-key triple encryption. The practical significance of the technique is illustrated by giving the design for three $10 million custom machines which could be built with current technology: one finds elliptic curve logarithms in GF(2155) thereby defeating a proposed elliptic curve cryptosystem in expected time 32 days, the second finds MD5 collisions in expected time 21 days, and the last recovers a double-DES key from two known plaintexts in expected time 4 years, which is four orders of magnitude faster than the conventional meet-in-the-middle attack on double-DES. Based on this attack, double-DES offers only 17 more bits of security than single-DES.

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

In this paper, we introduce a new method to solve the elliptic curve discrete logarithm probelm (ECDLP) over a finite field by using the elliptic curve lifting problem. Moreover, we propose to find a non-trivial point in E1(Q) in order to get a lifted elliptic curve with rank smaller than the number of lifted points. By this method, we conclude that finding a non-trivial point in E1(Q) implies solving the ECDLP, the discrete logarithm problem (DLP) and the integer factorization problem (IFP). Finally, we find that the minimum of canonical height of a point in E1(Q) is almost O(|Ẽ(Fp)|), which means that it is too large to be found by the brute force search.

The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (DLOG) has the same difficulty for all curves over a given finite field with the same order. We prove that this is essentially true by showing polynomial time random reducibility of DLOG among such curves, assuming the Generalized Riemann Hypothesis (GRH). We do so by constructing certain expander graphs, similar to Ramanujan graphs, with elliptic curves as nodes and low degree isogenies as edges. The result is obtained from the rapid mixing of random walks on this graph. Our proof works only for curves with (nearly) the same endomorphism rings. Without this technical restriction such a DLOG equivalence might be false; however, in practice the restriction may be moot, because all known polynomial time techniques for constructing equal order curves produce only curves with nearly equal endomorphism rings.