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  • Author: Steven D. Galbraith x
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Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated, since the full torsion subgroup has rank 2g. In this paper, we prove that distortion maps always exist for supersingular curves of genus g > 1. We also give several examples of curves of genus 2 with explicit distortion maps for embedding degrees 4, 5, 6, and 12.

Recent progress on pairing implementation has made certain pairings extremely simple and fast to compute. Hence, it is natural to examine if there are consequences for the security of pairing-based cryptography.

This paper gives a method to compute eta pairings on certain supersingular curves with a greatly simplified final exponentiation. The method does not lead to any improvement in the speed of pairing implementation. However, we show that it leads to a multivariate attack on the pairing inversion problem. We analyse this attack and show that it is infeasible for elliptic curves.


A self-pairing is a pairing computation where both inputs are the same group element. Self-pairings are used in some cryptographic schemes and protocols. In this paper, we show how to compute the Tate–Lichtenbaum pairing on a curve more efficiently than the general case. The speedup is obtained by using a simpler final exponentiation. We also discuss how to use this pairing in cryptographic applications.