A Secure Family of Composite Finite Fields Suitable for Fast Implementation of Elliptic Curve Cryptography (original) (raw)
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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 ⋆
—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.
Attack Experiments on Elliptic Curves of Prime and Binary Fields
Emerging Technologies in Data Mining and Information Security, 2018
At the beginning the paper describes the basic properties of finite field arithmetic and elliptic curve arithmetic over prime and binary fields. Then it discusses the elliptic curve discrete logarithm problem and its properties. We study the Baby-Step, Giant-Step method, Pollard's rho method and Pohlig-Hellman method, known as general methods that can exploit the elliptic curve discrete logarithm problem, and describe in detail attack experiments using these methods over prime and binary fields. Finally, the paper discusses the expected running time of these attacks and suggests the strong elliptic curves that are not vulnerable to these attacks.
—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.
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The security of many cryptographic protocols depends on the di culty of solving the so-called \discrete logarithm" problem, in the multiplicative group of a nite eld. Although, in the general case, there are no polynomial time algorithms for this problem, constant improvements are being made { with the result that the use of these protocols require much larger key sizes, for a given level of security, than may be convenient.
THE DISCRETE LOG PROBLEM AND ELLIPTIC CURVE CRYPTOGRAPHY
In this paper, discrete log-based public-key cryptography is explored. Specifically, we first examine the Discrete Log Problem over a general cyclic group and algorithms that attempt to solve it. This leads us to an investigation of the security of cryptosystems based over certain specific cyclic groups: Fp, F × p , and the cyclic subgroup generated by a point on an elliptic curve; we ultimately see the highest security comes from using E(Fp) as our group. This necessitates an introduction of elliptic curves, which is provided. Finally, we conclude with cryptographic implementation considerations.
Cryptanalysis and improvement of an encryption scheme that uses elliptic curves over finite fields
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...
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Undergraduate Texts in Mathematics, 2014
Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves also figured prominently in the recent proof of Fermat's Last Theorem by Andrew Wiles. Originally pursued for purely aesthetic reasons, elliptic curves have recently been utilized in devising algorithms for factoring integers, primality proving, and in public-key cryptography. In this article, we aim to give the reader an introduction to elliptic curve cryptosystems, and to demonstrate why these systems provide relatively small block sizes, high-speed software and hardware implementations, and offer the highest strength-per-key-bit of any known public-key scheme.
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IJIREEICE, 2015
This paper provides an overview of elliptic curves and their use in cryptography. The purpose of this paper is an in-depth examination of the Elliptic Curve Discrete Logarithm (ECDLP) including techniques in attacking cryptosystems dependent on the ECDLP. The paper includes properties of elliptic curve and methods for various attacks.
On Assurance of Information Security using Elliptic Curves Cryptosystems
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.