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Applications of Arithmetical Geometry to Cryptographic Constructions

In: Finite Fields and Applications

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  • Gerhard Frey

    (Universität GH Essen, Institut für Experimentelle Mathematik)

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.

Suggested Citation

  • Gerhard Frey, 2001. "Applications of Arithmetical Geometry to Cryptographic Constructions," Springer Books, in: Dieter Jungnickel & Harald Niederreiter (ed.), Finite Fields and Applications, pages 128-161, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-56755-1_13
    DOI: 10.1007/978-3-642-56755-1_13
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