Applications of Arithmetical Geometry to Cryptographic Constructions

Gerhard Frey ()
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Gerhard Frey: Universität GH Essen, Institut für Experimentelle Mathematik

A chapter in Finite Fields and Applications, 2001, pp 128-161 from Springer

Abstract: Abstract Public key cryptosystems are very important tools for data transmission. Their performance and security depend on the underlying crypto primitives. In this paper we describe one such primitive: The Discrete Logarithm (DL) in cyclic groups of prime order (Section 1). To construct DL-systems we use methods from algebraic and arithmetic geometry and especially the theory of abelian varieties over finite fields. It is explained why Jacobian varieties of hyperelliptic curves of genus ≤ 4 are candidates for cryptographically “good” abelian varieties (Section 2). In the third section we describe the (constructive and destructive) role played by Galois theory: Local and global Galois representation theory is used to count points on abelian varieties over finite fields and we give some applications of Weil descent and Tate duality.

Keywords: Elliptic Curve; Finite Field; Elliptic Curf; Galois Group; Abelian Variety (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-56755-1_13

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DOI: 10.1007/978-3-642-56755-1_13

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