Journal of Cryptographic Engineering

, Volume 8, Issue 3, pp 201–210 | Cite as

Montgomery inversion

  • Erkay Savaş
  • Çetin Kaya Koç
Special Issue on Montgomery Arithmetic


Multiplicative inversion in finite fields is an essential operation in many cryptographic applications such as elliptic curve and pairing-based cryptography. While the classical extended Euclidean algorithm involves expensive division operations, the binary extended Euclidean and Kaliski’s algorithms use simple shift, addition and subtraction operations. The Montgomery inverse operation is applied when the Montgomery multiplication operation is used for fast arithmetic. As the inversion operation is applied to sensitive data, a constant-time inversion algorithm is useful against a class of side-channel attacks. In this paper, we show different ways of computing the Montgomery inverse of a given integer and compare their complexity in terms of the number of additions/subtractions and shift operations. We also propose a simple parallel algorithm to compute Montgomery inverse, which can be useful in multi-core processors where data sharing among cores is relatively inexpensive. Finally, we propose two efficient constant-time Montgomery inversion algorithms, which are useful as countermeasures against side-channel attacks.


Multiplicative inverse Extended Euclidean algorithm Montgomery inverse Constant-time implementation Parallelization 



We thank the anonymous reviewers for their comments and recommendations.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Sabancı UniversityIstanbulTurkey
  2. 2.University of California Santa BarbaraSanta BarbaraUSA

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