Kummer versus Montgomery Face-off over Prime Order Fields

This paper makes a comprehensive comparison of the efficiencies of vectorized implementations of Kummer lines and Montgomery curves at various security levels. For the comparison, nine Kummer lines are considered, out of which eight are new, and new assembly implementations of all nine Kummer lines have been made. Seven previously proposed Montgomery curves are considered and new vectorized assembly implementations have been made for three of them. Our comparisons show that for all security levels, Kummer lines are consistently faster than Montgomery curves, though the speed-up gap is not much.