Advances in Mathematics of Communications (AMC)

Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves

Pages: 215 - 235, Volume 4, Issue 2, May 2010      doi:10.3934/amc.2010.4.215

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Anton Stolbunov - Department of Telematics, Norwegian University of Science and Technology, O.S. Bragstads plass 2a, N-7491 Trondheim, Norway (email)

Abstract: We propose a public-key encryption scheme and key agreement protocols based on a group action on a set. We construct an implementation of these schemes for the action of the class group $\mathcal{CL}(\mathcal{O}_K)$ of an imaginary quadratic field $K$ on the set $\mathcal{ELL}$p,n$(\mathcal{O}_K)$ of isomorphism classes of elliptic curves over $\mathbb{F}_p$ with $n$ points and the endomorphism ring $\mathcal{O}_K$. This introduces a novel way of using elliptic curves for constructing asymmetric cryptography.

Keywords:  Public-key cryptography, group action, elliptic curve, isogeny, ideal class group, finite field.
Mathematics Subject Classification:  94A60, 14G50.

Received: June 2009;      Revised: March 2010;      Available Online: May 2010.