AN APPROACH TO ELLIPTIC CURVES AND DISCRETE LOGARITHMIC PROBLEM.

Kuwait Journal of Science, 2021

In this paper, we cryptanalyzed a recently proposed encryption scheme that uses elliptic curves over a finite field. The security of the proposed scheme depends upon the elliptic curve discrete logarithm problem. Two secret keys are used to increase the security strength of the scheme as compared to traditionally used schemes that are based on one secret key. In this scheme, if an adversary gets one secret key then he is unable to get the contents of the original message without the second secret key. Our analysis shows that the proposed scheme is not secure and unable to provide the basic security requirements of the encryption scheme. Due to our successful cryptanalysis, an adversary can get the contents of the original message without the knowledge of the secret keys of the receiver. To mount the attack, Mallory first gets the transmitted ciphertext and then uses public keys of the receiver and global parameters of the scheme to recover the associated plaintext message. To overco...

Advances in Science, Technology and Engineering Systems Journal, 2017

A P)) (Card(E a,b (F p))−3) n +Card(E a,b (F p)) and the execution time to solve the problem of discrete logarithm in E a,b (A P) is Ω(N), sch that the execution time to solve the problem of discrete logarithm in E a,b (F p) is O(√ N). The motivation for this work came from search for new groups with intractable (DLP) discrete logarithm problem is therefore of great importance.

1999

Recently, in 1995, elliptic curves played an important role in proving, by Andrew Wiles, Fermat's Last Theorem (formulated in 1635) , what could be considered as one of the most important mathematical achievements of the last 50 years. Elliptic curves have also close relation to BSD Conjecture (Birch and Swinnerton-Dyer Conjecture), one of the Millennium problems of the Clay Mathematics institute. Elliptic curves are currently behind practically most preferred methods of cryptographic security. Elliptic curves are also a basis of very important factorization method. prof. Jozef Gruska IV054 8. Elliptic curves cryptography and factorization 7/86

2021

Elliptic curve is a set of two variable points on polynomials of degree 3 over a field acted by an addition operation that forms a group structure. The motivation of this study is that the mathematics behind that elliptic curve to the applicability within a cryptosystem. Nowadays, pair- ings bilinear maps on elliptic curve are popular to construct cryptographic protocol pairings help to transform a discrete logarithm problem on an elliptic curve to the discrete logarithm problem in nite elds. The purpose of this paper is to introduce elliptic curve, bilinear pairings on elliptic curves as based on pairing cryptography. Also this investigation serves as a basis in guiding anyone interested to understand one of the applications of group theory in cryptosystem.

International Journal of Multidisciplinary Studies

From the earliest days of history, the requirement for methods of secret communication and protection of information had been present. Cryptography is such an important field of science developed to facilitate secret communication and safeguard information. Cryptography is based on mathematics. It is an application of different disciplines such as Algebra, Number Theory, Graph Theory etc. Private key cryptography and Public key cryptography are the two main types of cryptography. Public key cryptosystems offer more security and convenience for the users. The main objective of this study is to explore the possibilities of further improvement of Elliptic Curve Cryptography (ECC) by studying the mathematical aspects behind the "Elliptic curve cryptosystem" which is one of the latest of this kind and develop a computer program to generate the cyclic subgroup of a given elliptic curve defined over a finite field ℤ , where p is a prime, which is the major requirement to perform ECC and then use the same to illustrate how data security is achieved from this. For an elliptic curve defined over a field, the points on an elliptic curve naturally form an abelian group. Elliptic curve arithmetic can be employed to develop a variety of Elliptic curve cryptographic schemes such as key exchange, encryption, digital signatures and specific construction of a keyed-Hash Message Authentication Code (HMAC) which are illustrated through this study. Moreover this study proposes an improvement for the encryption of a message through utilization of a concept in "Coding Theory" of Abstract algebra which offers an additional shield for the transmitted message.

Proceedings of the Second International Conference on Cryptology in India Progress in Cryptology, 2001

introduced a family of binary finite fields which are composite extensions of F2 and on which arithmetic operations can be performed more quickly than on prime extensions of F2 of the same size. We present here a fast approach to elliptic curve cryptography using a distinguished subset of the set of Silverman fields F 2 N = F h n . This approach leads to a theoretical computation speedup over fields of the same size, using a standard point of view (cf. ). We also analyse their security against prime extension fields F2p , where p is prime, following the method of Menezes and Qu . We conclude that our fields do not present any significant weakness towards the solution of the elliptic curve discrete logarithm problem and that often the Weil descent of Galbraith-Gaudry-Hess-Smart (GGHS) does not offer a better attack on elliptic curves defined over F 2 N than on those defined over F2p , with a prime p of the same size as N . A noteworthy example is provided by F 2 226 : a generic elliptic curve Y 2 + XY = X 3 + αX 2 + β defined over F 2 226 is as prone to the GGHS Weil descent attack as a generic curve defined on the NIST field F 2 233 . Elliptic curve cryptography was introduced in 1986 independently by Koblitz [10] and Miller as a rich context where one can apply cryptographic protocols based on the discrete logarithm problem in a multiplicative group G: given a, b ∈ G such that b = a d , find d. However, the rich structure of elliptic curves made possible a wide variety of attacks that must be avoided in the design of elliptic curve ⋆

International Journal of Computer Applications, 2012

Public key cryptography systems are based on sound mathematical foundations that are designed to make the problem hard for an intruder to break into the system. Number theory and algebraic geometry, namely the theory of elliptic curves defined over finite fields, has found applications in cryptology. The basic reason for this is that elliptic curves over finite fields provide an inexhaustible supply of finite abelian groups which, even when large, are amenable to computation because of their rich structure. The first level is the mathematical background concerning the needed tools from algebraic geometry and arithmetic. This paper introduces the elementary algebraic structures and the basic facts on number theory in finite fields. It includes the minimal amount of mathematical background necessary to understand the applications to cryptology. Elliptic curves are intimately connected with the theory of modular forms, in more than one ways. The paper gives a brief introduction to modular arithmetic, which is the core arithmetic of almost all public key algorithms.. The ultimate goal of the paper is to completely understand the structure of the points on the elliptic curve over any field F and being able to find them.

IACR Cryptol. ePrint Arch., 2019

In this paper, we intend to study the geometric meaning of the discrete logarithm problem defined over an Elliptic Curve. The key idea is to reduce the Elliptic Curve Discrete Logarithm Problem (EC-DLP) into a system of equations. These equations arise from the interesection of quadric hypersurfaces in an affine space of lower dimension. In cryptography, this interpretation can be used to design attacks on EC-DLP. Presently, the best known attack algorithm having a sub-exponential time complexity is through the implementation of Summation Polynomials and Weil Descent. It is expected that the proposed geometric interpretation can result in faster reduction of the problem into a system of equations. These overdetermined system of equations are hard to solve. We have used F4 (Faugere) algorithms and got results for primes less than 500,000. Quantum Algorithms can expedite the process of solving these over-determined system of equations. In the absence of fast algorithms for computing s...

1999

Enge, Andreas. Elliptie eurves and their applieations to eryptography an introduetion / by Andreas Enge. p. em. Includes bibliographieal referenees and index.

Lecture Notes in Computer Science, 2001

introduced a family of binary finite fields which are composite extensions of F2 and on which arithmetic operations can be performed more quickly than on prime extensions of F2 of the same size. We present here a fast approach to elliptic curve cryptography using a distinguished subset of the set of Silverman fields F 2 N = F h n . This approach leads to a theoretical computation speedup over fields of the same size, using a standard point of view (cf. ). We also analyse their security against prime extension fields F2p , where p is prime, following the method of Menezes and Qu . We conclude that our fields do not present any significant weakness towards the solution of the elliptic curve discrete logarithm problem and that often the Weil descent of Galbraith-Gaudry-Hess-Smart (GGHS) does not offer a better attack on elliptic curves defined over F 2 N than on those defined over F2p , with a prime p of the same size as N . A noteworthy example is provided by F 2 226 : a generic elliptic curve Y 2 + XY = X 3 + αX 2 + β defined over F 2 226 is as prone to the GGHS Weil descent attack as a generic curve defined on the NIST field F 2 233 . Elliptic curve cryptography was introduced in 1986 independently by Koblitz [10] and Miller as a rich context where one can apply cryptographic protocols based on the discrete logarithm problem in a multiplicative group G: given a, b ∈ G such that b = a d , find d. However, the rich structure of elliptic curves made possible a wide variety of attacks that must be avoided in the design of elliptic curve ⋆

2008

En els darrers anys, la criptografia amb corbes el.líptiques ha adquirit una importància creixent, fins a arribar a formar part en la actualitat de diferents estàndards industrials. Tot i que s'han dissenyat variants amb corbes el.líptiques de criptosistemes clàssics, com el RSA, el seu màxim interès rau en la seva aplicació en criptosistemes basats en el Problema del Logaritme Discret,

2013

Elliptic curve cryptography is an asymmetric key cryptography. The points on two dimensional elliptic curve are used for declaration of data encryption & decryption. It include public key generation on the elliptic curve and private key generation to decrypt the data. The present paper deals with an overview of Elliptic curve cryptography (ECC) and its implementation through coordinate geometry for data encryption. We introduce a new approach in the form of cardan's method to find points on X axis at elliptic curve over finite field and form public key cryptographic system and finally we define two dimensional alphabetic table and description in the form of algorithm to use it for plain text encryption.

Journal of Internet Technology and Secured Transaction, 2012

We present in this paper an important area of information security emerged in the last decades, namely Elliptic Curves Cryptosystems (ECC). Compared to traditional public-key cryptosystems like RSA or Diffie-Hellman, ECC offers equivalent security with smaller key sizes; these result in faster computations, lower power consumption, as well as memory and bandwidth savings. ECC are more and more considered as an attractive public-key cryptosystem for mobile/wireless environments. ECC are especially useful for mobile devices, which are typically limited in terms of their CPU, power and network connectivity. ECC are the next frontier in the use of security mechanisms by providing good security margins with lower computational cost. ECC's domain is an important field emerged in information security. The elliptic curves (EC) are used for conceiving efficient factorization algorithms and for proving the primality. They are used in public key cryptosystems and in pseudorandom bit generators, too. The elliptic curves were also applied in Codes Theory, where they were used to create very good error protected codes. In this paper, our aim is to examine the security, implementation and performance of ECC applications on various mobile devices. Also, our goal is to compare ECC and conventional PKC performances. Doing these, we want to prove that ECC could become the next-generation of PKC.

—This paper begins by describing basic properties of finite field and elliptic curve cryptography over prime field and binary field. Then we discuss the discrete logarithm problem for elliptic curves and its properties. We study the general common attacks on elliptic curve discrete logarithm problem such as the Baby Step, Giant Step method, Pollard's rho method and Pohlig-Hellman method, and describe in detail experiments of these attacks over prime field and binary field. The paper finishes by describing expected running time of the attacks and suggesting strong elliptic curves that are not susceptible to these attacks.