MODIFICATION OF ELGAMAL ELLIPTIC CURVE CRYPTOSYSTEM ALGORITHM

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.

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

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.

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].

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.

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.

The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$G=?P? of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$P,?P,?2P,?,?dP?G, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$??Zp? in the case of $$d|(p\pm 1)$$d|(p±1) with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$O((p±1)/d+di) group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$O(max{(p±1)/d,d}) storage ($$i=\frac{1}{2}$$i=12 or 1 for $$d|(p-1)$$d|(p-1) case or $$d|(p+1)$$d|(p+1) case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$P,?P,?kP,?k2P,?k3P,?,?k?(d)-1P?G and multiplicative cyclic group $$K=\langle k \rangle $$K=?k? are given, where d is a prime, $$\varphi (d)$$?(d) is the order of K and $$\varphi $$? is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$??Zp? can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$O((p-1)/d+d) group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$O((p-1)/d) storage.

For a finite group G to be used in the MOR public key cryptosystem, it is necessary that the discrete logarithm problem(DLP) over the inner automorphism group Inn (G) of G must be computationally hard to solve. In this paper, under the assumption that the special conjugacy problem of G is easy, we show that the complexity of the MOR system over G is about log |G| times larger than that of DLP over G in a generic sense. We also introduce a group-theoretic method, called the group extension, to analyze the MOR cryptosystem. When G is considered as a group extension of H by a simple abelian group, we show that DLP over Inn (G) can be ‘reduced’ to DLP over Inn (H). On the other hand, we show that the reduction from DLP over Inn (G) to DLP over G is also possible for some groups. For example, when G is a nilpotent group, we obtain such a reduction by the central commutator attack.

In 1976 Diffie and Hellman [152] described the framework for public-key cryptography. It was not until 1978 that three designs for public-key cryptosystems were published. Rivest, Shamir, and Adleman [431] showed how the discrete logarithm and factorization problems could be used to construct a public-key cryptosystem. This is the well-known RSA cryptosystem. Merkle and Hellman [339] used the knapsack problem in their construction. McEliece [329] built a system based on error correcting codes. Later in 1985 ElGamal [163] designed a public-key cryptosystem using the discrete logarithm problem. Koblitz [283] and Miller [346] suggested the use of elliptic curves in the design of public-key cryptosystems. Nowadays, there are quite a few more suggestions as to how to design public-key cryptosystems, but none so popular as the RSA and ElGamal cryptosystems.

Elliptic curve cryptosystem — The answer to strong, fast public-key cryptography for securing constrained environments

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 acceleration of cryptographic applications on embedded devices is a topic of increasingly importance, due to their massive use. In this paper, the efficiency of these devices when operating as cryptographic accelerators is assessed, using a two-tiered parallelism model, where not only multi-core, but also Single Instruction Multiple Data (SIMD) parallelism is exploited to increase the throughput of modular multiplications. In addition, the applicability of the developed parallel modular multiplications are tested for the Rivest-Shavir-Adleman (RSA) and Elliptic Curve (EC) cryptosystems. The relative efficiency of the two-tiered model for implementing cryptosystems on general purpose processors and embedded devices is analysed and experimentally evaluated. It can be concluded that it is possible to achieve the same levels of execution enhancement on both platforms. Moreover, speedups of up to 7.3 and 3.9 are obtained for the RSA and EC cryptosystems on the ARM A15 quad-core processor, respectively.

Since their invention in the mid 1980s, Elliptic Curve Cryptosystems (ECC) have become an alternative to common Public-Key (PK) cryptosystems such as, e.g., RSA. The utilization of Elliptic Curves (EC) in cryptography is very promising because of their resistance against powerful index-calculus attacks. Providing a similar level of security as RSA, ECC allows for efficient implementation due to a significantly smaller bit size of the operands. It is widely accepted that the only feasible way to attack actual cryptosystems, if at all, is the application of dedicated hardware. In times of continuous technological improvements and increasing computing power, the question of the security of ECC against attacks based on special-purpose hardware and, in particular based on recently emerged low-cost FPGAs, arises.This work presents the first architecture with a corresponding FPGA implementation of an attack against ECC over prime fields. We describe an FPGA-based multi-processing hardware architecture for the Pollard-Rho method which is, to our knowledge, currently the most efficient attack against ECC. The implementation is running on a contemporary low-cost FPGA which allows for a much better cost-performance ratio than conventional CPUs. With the implementation at hand, a fairly accurate estimate about the cost of an FPGA-based attack can be given. We will extrapolate the results on actual ECC key lengths (128 bits and above) and estimate the expected runtimes for a successful attack. Since FPGA-based attacks are out of reach for key lengths exceeding 128 bits, we provide estimates for an ASIC design.Based on our results, currently used elliptic curve cryptosystems (160 bit and above) are infeasible to break with available computational and financial resources. However, some of the security standards proposed by the SECG in [2, 3] become subject to attacks based on low-cost FPGAs.

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.