Elliptic Curves with Low Embedding Degree

Motivated by the needs of the pairing based cryptography, Miyaji, Nakabayashi and Takano have suggested a construction of so-called MNT elliptic curves with low embedding degree. We give some heuristic arguments which suggest that there are only about z1/2+o(1) of MNT curves with complex multiplication discriminant up to z. We also show that there are very few finite fields over which elliptic curves with small embedding degree and small complex multiplication discriminant may exist (regardless of the way they are constructed). 1 

• Type: Journal
• Year: 2006
• Where: JOC
• Authors: Florian Luca, Igor Shparlinski
• Keywords: Complex Multiplication, Embedding Degree, JOC 2006, MNT Elliptic Curves