Injective Encodings to Binary Ordinary Elliptic Curves

Abstract

Representing points of elliptic curves in a way that no pattern can be detected by sensors in the transmitted data is a crucial problem in elliptic curve cryptography. One of the methods that we can represent points of the elliptic curves in a way to be indistinguishable from random bit strings is using injective encoding function. So far, several injective encodings to elliptic curves have been presented, but the previous encoding functions have not supported the binary elliptic curves. More precisely, the only injective encoding to binary elliptic curves was given for Hessian curves, the family of elliptic curves with a point of order 3. In this paper, we propose approaches for constructing injective encoding algorithms to the ordinary binary elliptic curves \(y^2+xy=x^3+ax^2+b\) with \(\mathrm {Tr}(a)=1\) as well as those with \(\mathrm {Tr}(a+1)=0\).