Designs, Codes and Cryptography

, Volume 75, Issue 2, pp 335–357

Point compression for the trace zero subgroup over a small degree extension field


DOI: 10.1007/s10623-014-9921-0

Cite this article as:
Gorla, E. & Massierer, M. Des. Codes Cryptogr. (2015) 75: 335. doi:10.1007/s10623-014-9921-0


Using Semaev’s summation polynomials, we derive a new equation for the \({\mathbb {F}_q}\)-rational points of the trace zero variety of an elliptic curve defined over \({\mathbb {F}_q}\). Using this equation, we produce an optimal-size representation for such points. Our representation is compatible with scalar multiplication. We give a point compression algorithm to compute the representation and a decompression algorithm to recover the original point (up to some small ambiguity). The algorithms are efficient for trace zero varieties coming from small degree extension fields. We give explicit equations and discuss in detail the practically relevant cases of cubic and quintic field extensions.


Elliptic curve cryptographyPairing-based cryptographyDiscrete logarithm problemTrace zero varietyEfficient representationPoint compressionSummation polynomials

Mathematics Subject Classification


Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institut de mathématiquesUniversité de NeuchâtelNeuchâtelSwitzerland
  2. 2.Mathematisches InstitutUniversität BaselBaselSwitzerland