On the Fixed Points of an Elliptic-Curve Version of Self-Power Map

Abstract

Fixed points of the self-power map over a finite field have been studied in cryptology as a special case of modular exponentiation. In this note, we define an elliptic-curve version of the self-power map, enumerate the number of curves that contain at least one fixed point, and give its upper and lower bounds. Our result is a partial solution to the open question raised by Glebsky and Shparlinski in 2010.