Point compression for the trace zero subgroup over a small degree extension field
This work addresses the problem of reducing storage and bandwidth requirements for cryptographic applications using trace zero subgroups, though it is incremental as it builds on existing mathematical frameworks.
The authors tackled the problem of efficiently representing points on trace zero varieties of elliptic curves by deriving a new equation using Semaev's summation polynomials, resulting in an optimal-size representation compatible with scalar multiplication. They developed efficient compression and decompression algorithms specifically for small degree extension fields, with detailed implementations for cubic and quintic extensions.
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.