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

*Elisa Gorla and Maike Massierer*

**Abstract: **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.

**Category / Keywords: **public-key cryptography / elliptic curve cryptography, pairing-based cryptography, discrete logarithm problem, trace zero variety, efficient representation, point compression, summation polynomials

**Original Publication**** (in the same form): **Designs, Codes and Cryptography
**DOI: **10.1007/s10623-014-9921-0

**Date: **received 1 Mar 2014

**Contact author: **maike massierer at inria fr

**Available format(s): **PDF | BibTeX Citation

**Version: **20140303:112603 (All versions of this report)

**Short URL: **ia.cr/2014/158

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