Improving the Parallelized Pollard Rho Method for Computing Elliptic Curve Discrete Logarithms

Pollard rho method and its parallelized variant are at present known as the best generic algorithms for computing elliptic curve discrete logarithms. We design new iteration functions for the parallel rho method by exploiting the fact that for any two points P and Q, we can efficiently get P-Q when we compute P+Q. We present a careful analysis of the alternative rho method with new iteration functions. Compare to the previous parallel r-adding walk, generally the new method can reduce the size of the space that is being searched by a factor of 4 with the additional costs of 2 field multiplications and 1 squaring at each iteration step for computing elliptic curve discrete logarithms.