Journal of the Ramanujan Mathematical Society

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The Vector Decomposition Problem for Elliptic and Hyperelliptic Curves


Affiliations
1 Coordinated Science Laboratory, Department of Mathematics, University of Illinois at Urbana-Champaign, United States
2 Coordinated Science Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, United States
     

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The group of m-torsion points on an elliptic curve, for a prime number m, forms a two-dimensional vector space. It was suggested and proven by Yoshida that under certain conditions the vector decomposition problem (VDP) on a two-dimensional vector space is at least as hard as the computational Diffie-Hellman problem (CDHP) on a one-dimensional subspace. In this work we show that even though this assessment is true, it applies to the VDP for m-torsion points on an elliptic curve only if the curve is supersingular. But in that case the CDHP on the one-dimensional subspace has a known sub-exponential solution. Furthermore, we present a family of hyperelliptic curves of genus two that are suitable for the VDP.
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  • The Vector Decomposition Problem for Elliptic and Hyperelliptic Curves

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Authors

Iwan Duursma
Coordinated Science Laboratory, Department of Mathematics, University of Illinois at Urbana-Champaign, United States
Negar Kiyavash
Coordinated Science Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, United States

Abstract


The group of m-torsion points on an elliptic curve, for a prime number m, forms a two-dimensional vector space. It was suggested and proven by Yoshida that under certain conditions the vector decomposition problem (VDP) on a two-dimensional vector space is at least as hard as the computational Diffie-Hellman problem (CDHP) on a one-dimensional subspace. In this work we show that even though this assessment is true, it applies to the VDP for m-torsion points on an elliptic curve only if the curve is supersingular. But in that case the CDHP on the one-dimensional subspace has a known sub-exponential solution. Furthermore, we present a family of hyperelliptic curves of genus two that are suitable for the VDP.