1. A method implemented by a single instruction multiple data (SIMD) computing system, the method comprising: Identifying, by a SIMD computing system configured to perform elliptic-curve octupling using SIMD, the coordinates x, y, and z of a weighted projective point P on an elliptic curve; computing by the SIMD computing system the coordinates r, s, t for 8P from P, 8P comprising 12 sets of field multiplications, the set of field multiplications comprising one to four particular field multiplications, each particular set of the 12 sets of field multiplications executed by the SIMD computing devices in parallel according to an assigned time-step, wherein computing 8P is determined with a straight-line algorithm; initially receiving original data; determining 8P for respective points on the elliptic curve with the straight-line elliptic curve point octupling algorithm; encrypting or signing the original data based upon the results of the determining of 8P; and storing the coordinates r, s, t to a computer readable storage medium associated with the computing system for subsequent use by the computing system.