Maike Massierer

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

Using Semaev’s summation polynomials, we derive a new equation for the

Trace zero varieties in cryptography : optimal representation and index calculus

{\mathbb {F}_q}

Index calculus in the trace zero variety

Fq-rational points of the trace zero variety of an elliptic curve defined over

Index calculus in the trace zero variety.

{\mathbb {F}_q}

Ramanujan graphs in cryptography.

Fq. 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.

Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication

We give an optimal-size representation for the elements of the trace zero subgroup of the Picard group of an elliptic or hyperelliptic curve of any genus, with respect to a field extension of any prime degree. The representation is via the coefficients of a rational function, and it is compatible with scalar multiplication of points. We provide efficient compression and decompression algorithms, and complement them with implementation results. We discuss in detail the practically relevant cases of small genus and extension degree, and compare with the other known compression methods.