On Fast Calculation of Addition Chains for Isogeny-Based Cryptography

Addition chain calculations play a critical role in determining the e ciency of cryptosystems based on isogenies on elliptic curves. However, nding a minimal length addition chain is not easy; a generalized version of the problem, in which one must nd a chain that simultaneously forms each of a sequence of values, is NP-complete. For the special primes used in such cryptosystems, nding fast addition chains for nite eld arithmetic such as inversion and square root is also not easy. In this paper, we investigate the shape of smooth isogeny primes and propose new methods to calculate fast addition chains. Further, we also provide techniques to reduce the temporary register consumption of these large exponentials, applicable to both software and hardware implementations utilizing addition chains. Lastly, we utilize our procedures to compare multiple isogeny primes by the complexity of the addition chains.