A Review on the Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Finite Fields Admitting a Weierstrass Point
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A Review on the Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Finite Fields Admitting a Weierstrass Point J. Espinosa García · L. Hernández Encinas · J. Muñoz Masqué
Published online: 8 August 2006 © Springer Science + Business Media B.V. 2006
Abstract The reduced equations for the isomorphism classes of hyperelliptic curves of genus 2 admitting a Weierstrass point over a finite field of arbitrary characteristic, are shown and the number of such classes is included. This work picks up in a unified way a series of previous results published by several authors by using different methodologies. These classifications are of interest in designing and implementing of hyperelliptic curve cryptosystems. Mathematics Subject Classifications (2000) 11G20 · 11T71 · 12E20 · 14H10 · 14H45 · 14H50 · 94A60. Key words finite fields · hyperelliptic curves · public-key cryptography.
1. Introduction In the last years, the main groups proposed in designing public key cryptosystems are the following: 1. 2. 3. 4.
The group of units of Zn , n being a integer [24, 41, 47, 62, 63, 72, 73, 75, 76]. A proper subgroup of the multiplicative group of a finite field [54, 55, 65]. The group of points on an elliptic curve defined over a finite field [38, 52]. The Jacobian of a hyperelliptic curve defined over a finite field [39].
J. Espinosa García (B) · L. Hernández Encinas · J. Muñoz Masqué Department of Information Processing and Coding, Applied Physics Institute, CSIC, C/ Serrano 144, E-28006 Madrid, Spain e-mail: [email protected] L. Hernández Encinas e-mail: [email protected] J. Muñoz Masqué e-mail: [email protected]
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Acta Appl Math (2006) 93: 299–318
5. The class group of an imaginary quadratic number field [11], and 6. The Jacobian of a superelliptic curve defined over a finite field [29]. The interest to consider such groups is based on the fact that their group law is easy to implement in software or in hardware and that the underlying computational problem in them is hard [28]. The two basic mathematical problems – nowadays assumed to be computationally unfeasible – on which the security of public key cryptosystems is based, are the integer factorization problem and the discrete logarithm problem (e.g., see [50, 3.2, 3.6]). The integer factorization problem is the following: Given a positive integer n, find its prime factorization; i.e., write n = p1e1 p2e2 · · · pkek , where p1 < . . . < pk are primes and each ei ≥ 1. As is well known, the security of the RSA cryptosystem [7, 8, 16, 32, 63, 67, 68] is based on this problem (cf. [50, 8.2]); namely, if a cryptanalyst can factor the RSA modulus n, then the private key of the cryptosystem can derived easily. The discrete logarithm problem (in short, DLP) is the following: Given a cyclic group G of order n, a generator α of G, and an element β ∈ G, find the integer x, 0 ≤ x ≤ n − 1, such that β = α x . The importance of the discrete logarithm problem to cryptography commenced with the invention of public-key cryptography by Diffie and Hellman [23] in 1976. Di
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