Efficient Simultaneous Inversion in Parallel and Application to Point Multiplication in ECC

  • Pradeep Kumar Mishra
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3822)


Inversion is the costliest of all finite field operations. Some algorithms require computation of several finite field elements simultaneously (elliptic curve factorization for example). Montgomery’s trick is a well known technique for performing the same in a sequential set up with little scope for parallelization. In the current work we propose an algorithm which needs almost same computational resources as Montgomery’s trick, but can be easily parallelized. Our algorithm uses binary tree structures for computation and using 2 r − 1 multipliers, it can simultaneously invert 2 r elements in 2r multiplication rounds and one inversion round. We also describe how the algorithm can be used when 2, 4, ... number of multipliers are available. To exhibit the utility of the method, we apply it to obtain a parallel algorithm for elliptic curve point multiplication. The proposed method is immune to side-channel attacks and compares favourably to many parallel algorithms existing in literature.


Elliptic Curve Cryptosystems Scalar Multiplication parallel algorithm Montgomery ladder simultaneous inversion 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Pradeep Kumar Mishra
    • 1
  1. 1.Centre for Information Security and Cryptography (CISaC)University of CalgaryCalgaryCanada

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