Journal ArticleDOI
Reducing elliptic curve logarithms to logarithms in a finite field
TLDR
The main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarathm problem in the multiplicative group of an extension of the underlying finite field, thus providing a probabilistic subexponential time algorithm for the former problem.Abstract:
Elliptic curve cryptosystems have the potential to provide relatively small block size, high-security public key schemes that can be efficiently implemented. As with other known public key schemes, such as RSA and discrete exponentiation in a finite field, some care must be exercised when selecting the parameters involved, in this case the elliptic curve and the underlying field. Specific classes of curves that give little or no advantage over previously known schemes are discussed. The main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying finite field. For the class of supersingular elliptic curves, the reduction takes probabilistic polynomial time, thus providing a probabilistic subexponential time algorithm for the former problem. >read more
Citations
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Book ChapterDOI
Identity-Based Encryption from the Weil Pairing
Dan Boneh,Matthew K. Franklin +1 more
TL;DR: This work proposes a fully functional identity-based encryption scheme (IBE) based on the Weil pairing that has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational Diffie-Hellman problem.
Journal ArticleDOI
Identity-Based Encryption from the Weil Pairing
Dan Boneh,Matthew K. Franklin +1 more
TL;DR: This work proposes a fully functional identity-based encryption (IBE) scheme based on bilinear maps between groups and gives precise definitions for secure IBE schemes and gives several applications for such systems.
Book ChapterDOI
Short Signatures from the Weil Pairing
Dan Boneh,Ben Lynn,Hovav Shacham +2 more
TL;DR: A short signature scheme based on the Computational Diffie-Hellman assumption on certain elliptic and hyperelliptic curves is introduced, designed for systems where signatures are typed in by a human or signatures are sent over a low-bandwidth channel.
Book
Guide to Elliptic Curve Cryptography
TL;DR: This guide explains the basic mathematics, describes state-of-the-art implementation methods, and presents standardized protocols for public-key encryption, digital signatures, and key establishment, as well as side-channel attacks and countermeasures.
Journal ArticleDOI
The Elliptic Curve Digital Signature Algorithm (ECDSA)
TL;DR: The ANSI X9.62 ECDSA is described and related security, implementation, and interoperability issues are discussed, and the strength-per-key-bit is substantially greater in an algorithm that uses elliptic curves.
References
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Journal ArticleDOI
Elliptic curve cryptosystems
TL;DR: The question of primitive points on an elliptic curve modulo p is discussed, and a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point is given.
Book ChapterDOI
Use of Elliptic Curves in Cryptography
TL;DR: In this paper, an analogue of the Diffie-Hellmann key exchange protocol was proposed, which appears to be immune from attacks of the style of Western, Miller, and Adleman.
Journal ArticleDOI
Approximate formulas for some functions of prime numbers
Journal ArticleDOI
Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p
TL;DR: A deterministic algorithm to compute the number of F^-points of an elliptic curve that is defined over a finite field Fv and which is given by a Weierstrass equation is presented.
Book ChapterDOI
Discrete logarithms in finite fields and their cryptographic significance
TL;DR: This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2n), finding that in order to be safe from attacks using these algorithms, the value of n for which GF( 2n) is used in a cryptosystem has to be very large and carefully chosen.