Implementation of Embedded Mobile Device Elliptic Curve Algorithm Fast Generating Algorithm

Elliptic Curve Method (ECM) is a well-known method for factoring integers, which is usually used in the Number Field Sieve algorithm as a subroutine for factoring smaller integers than the targeted one. ECM is called many times and can be executed in parallel for different inputs. This method mainly consist of simple operations on elliptic curves. Thus, ECM is suitable for hardware implementations that can efficiently reduce computational time. This work describes a new, improved FPGA-based hardware accelerator for ECM, designed for large scale computations. Our accelerator can operate with an on board ARM processor or with an external host computer. This design can factor several numbers at once and can be easily ported to various FPGA boards. Different methods for improving results (e.g. the use of DSP blocks, cache-registers, reorganizing instruction order) are described and their performance is analyzed. As a result, one of the fastest hardware ECM units is achieved.

Several mathematical concepts from analytics number theory, especially the counting functions (CFs) are employed to solve some modern problems in mathematics, computer sciences, engineering and others. One of them is the factorization problem for integers into two primes. This work proposes an extended factorization problem using the CFs, as main part from analytics number theory, to factor a real number r into two primes p and q. The computations of the points doubling and addition on the elliptic curves E modulo a composite number n are also used in this extension to give new two versions of the factorization over real field R. On the first version, the primes are obtained from doubling and addition on elliptic curve points that are computed modulo composite number which is resulting as output of the CFs algorithm with the input r. Whereas, the primes, on the second version, are got directly from applying the CFs algorithm with r as input. The successful computation of the scalar multiplication kP on the elliptic curve E modulo primes are determined. The computational complexities of extended elliptic curve factorization versions are determined using the counting operations. These operations are elliptic curve and finite field operations. New numerical results on two versions are computed as study cases. The elliptic curve factorization algorithm over R (ECF-R) with two versions, is more efficient for factoring-based cryptography in comparing to the original ECM.

Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations modulo N and occasionally taking GCDs with N. These began with Pollard’s p − 1 p - 1 and Monte Carlo methods. More recently, Williams published a p + 1 p + 1 method, and Lenstra discovered an elliptic curve method (ECM). We present ways to speed all of these. One improvement uses two tables during the second phases of p ± 1 p \pm 1 and ECM, looking for a match. Polynomial preconditioning lets us search a fixed table of size n with n / 2 + o ( n ) n/2 + o(n) multiplications. A parametrization of elliptic curves lets Step 1 of ECM compute the x-coordinate of nP from that of P in about 9.3 log 2 {\log _2} n multiplications for arbitrary P.

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

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 cryptosystem of elliptic curve has gained more and more researchers' attention from home and abroad in recent years as it owns a short key and a fine security. This paper elaborates the definition of elliptic curve which is used to establish cryptosystem and also makes some explanation to the realization of the Add Point Operation on the elliptic curve together with the fast Add Point Operation of its corresponding coordinate system. In addition, the security of the elliptic curve cryptosystem is also analyzed starting from the theory of the elliptic curve in this paper. And the issue of production of the safe elliptic curve is discussed. Finally, the elliptic curve cryptosystem has aroused great attention of the researchers as the public key system of the next generation. The production arithmetic of the safe elliptic curve, the security analysis of the elliptic curve cryptosystem together with the fast arithmetic on the elliptic curve will be the hotspot and the issues that are remained to be solved in its future development.

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.

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.

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.

Group key agreement based on elliptic curve at ad hoc networks is studied. In order to meet the demand of efficiency and security, an improved key exchange scheme based on elliptic curve is proposed in this paper. In the scheme, elliptic curve cryptosystem and digital signature algorithm in elliptic curve field are combined to make a group key. The key is agreed on within all members who would participate in the conversation. The new improved scheme has not only the features of forward security, backward security and resisting attack, but also certain theoretical and practical values.

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.

The Enhanced and Secure RSA Key Generation Scheme (ESRKGS), introduced in 2014, aimed to improve RSA security by employing a modulus constructed from four prime factors. However, subsequent studies in 2016 revealed that this structure did not provide additional security over standard RSA. In response, a modified version of ESRKGS was proposed in 2021, incorporating dual encoding techniques using a masking parameter γ and double encryption. This study evaluates the security of the modified ESRKGS by simulating an attack scenario in which the adversary is assumed to know of ϕ(N ), enabling recovery of encrypted messages. Additionally, we implement Lenstra’s Elliptic Curve Method (ECM) to assess the factorization resistance of the four-prime modulus when ϕ(N ) is not known. Experimental results indicate that ECM can efficiently factor the modulus into its four constituent primes under practical time constraints. These findings demonstrate that, despite recent modifications, the ESRKGS variant remains vulnerable to factorization based attacks. This highlights the necessity for more rigorous cryptographic design principles in multiprime RSA systems and calls into question the long-term viability of ESRKGS-based schemes in high-security applications.

Various cryptosystems have been proposed whose security relies on the difficulty of factoring integers of the special form N = pq 2. To factor integers of that form, Peralta and Okamoto introduced a variation of Lenstra’s Elliptic Curve Method (ECM) of factorization, which is based on the fact that the Jacobi symbols (a/N) and (a/P) agree for all integers a coprime with q. We report on an implementation and extensive experiments with that variation, which have been conducted in order to determine the speed-up compared with ECM for numbers of general form.KeywordsPrime FactorElliptic CurveEquality CheckGreat Common DivisorDecimal DigitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We describe the complete factorization of the tenth Fermat number F 10 F_{10} by the elliptic curve method (ECM). F 10 F_{10} is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 , … , F 11 F_5, \dots , F_{11} .

Elliptic curve cryptosystems proved to be well suited for securing systems with constrained resources like embedded and portable devices. In a fault-based attack, errors are induced during the computation of a cryptographic primitive, and the results are collected to derive information about the secret key safely stored in the device. We introduce a novel attack methodology to recover the secret key employed in implementations of the Elliptic Curve Digital Signature Algorithm. Our attack exploits the information leakage induced when altering the execution of the modular arithmetic operations used in the signature primitive and does not rely on the underlying elliptic curve mathematical structure, thus being applicable to all standardized curves. We provide both a validation of the feasibility of the attack, even employing common off-the-shelf hardware to perform the required computations, and a low-cost countermeasure to counteract it.