Recently the study of models for elliptic curves alternative to the Weierstrass form such as the Edwards form of an elliptic curve, the Huff form and many others, attracted much attention and interest due to their cryptographic applications.

On the other hand, many cryptographic protocols such as some password-based authentication protocols, various signature schemes and others use hashing into elliptic curves. Mathematically, this last operation reduces to the problem of deterministic encoding to the points on a given elliptic curve over a finite field. This problem arises in a very natural manner in many cryptographic protocols, when one wants to hash messages into the group of points of an elliptic curves. Hashing into finite fields being easy, the main step is to encode a field elements into an elliptic curve.

In 2009 Th. Icart proposed a deterministic algorithm for hashing into the Weierstrass form of an elliptic curve over finite field and showed its utility for constructing hash functions in elliptic curve based cryptography.

In our work we propose a new algorithm for a direct hashing into different form of ellipic curves (Jacobi quartic, Huff, Edwards, etc), which do not use a transformation of the equation of the curve into the Weierstrass form. This algorithm is in constant-time which is essential for many cryptographic applications and can be applied to construct satisfactory hashing functions for elliptic curves.