Public Key Cryptography – PKC 2009

Volume 5443 of the series Lecture Notes in Computer Science pp 443-462

Fast Multibase Methods and Other Several Optimizations for Elliptic Curve Scalar Multiplication

  • Patrick LongaAffiliated withDepartment of Electrical and Computer Engineering, University of Waterloo
  • , Catherine GebotysAffiliated withDepartment of Electrical and Computer Engineering, University of Waterloo


Recently, the new Multibase Non-Adjacent Form (mbNAF) method was introduced and shown to speed up the execution of the scalar multiplication with an efficient use of multiple bases to represent the scalar. In this work, we first optimize the previous method using fractional windows, and then introduce further improvements to achieve additional cost reductions. Moreover, we present new improvements in the point operation formulae. Specifically, we reduce further the cost of composite operations such as quintupling and septupling of a point, which are relevant for the speed up of multibase methods in general. Remarkably, our tests show that, in the case of standard elliptic curves, the refined mbNAF method can be as efficient as Window-w NAF using an optimal fractional window size. Thus, this is the first published method that does not require precomputations to achieve comparable efficiency to the standard window-based NAF method using precomputations. On other highly efficient curves as Jacobi quartics and Edwards curves, our tests show that the refined mbNAF currently attains the highest performance for both scenarios using precomputations and those without precomputations.


Elliptic curve cryptosystem scalar multiplication multibase non-adjacent form double base number system fractional window