is well known that the integer factorization problem is a computationally u201c difficult u201d or u201c hard u201d problem application_number US 95833710 A description the rsa analogous xz elliptic curve cryptography system and method provides for improved secure communication over an insecure channel using elliptic curve cryptography the method utilizes two stages of coordinate projections in the first of the two stages a projective coordinate is used to embed extra message data bits in the z coordinate as well as the x coordinate in the second stage a projective coordinate is used to remove a division operation at each iteration and also for randomizing the computation in order to provide a countermeasure against differential power analysis by u201c xz elliptic curve cryptography u201d the inventors refer to a cryptographic system invented by mohammad k ibrahim and described in u s pat no 7 379 546 issued may 27 2008 which is hereby incorporated by reference in its entirety conventional elliptic curve cryptography is based on embedding data bits in the z coordinate of an elliptic curve point only and hence the inventors refer to conventional elliptic curve cryptography as x coordinate embedding elliptic curve cryptography or x elliptic curve cryptography the system described in the.