Elliptic curves with the montgomery-form and their cryptographic applications

Katsuyuki Okeya, Hiroyuki Kurumatani, Kouichi Sakurai

Research output: Chapter in Book/Report/Conference proceedingConference contribution

56 Citations (Scopus)

Abstract

We show that the elliptic curve cryptosystems based on the Montgomery-form EM: BY2 = X3+ AX2 +X are immune to the timing-attacks by using our technique of randomized projective coordinates, while Montgomery originally introduced this type of curves for speeding up the Pollard and Elliptic Curve Methods of integer factorization [Math. Comp. Vol.48, No.177, (1987) pp.243-264]. However, it should be noted that not all the elliptic curves have the Montgomery-form, because the order of any elliptic curve with the Montgomery-form is divisible by “4”. Whereas recent ECC-standards [NIST,SEC-1] recommend that the cofactor of elliptic curve should be no greater than 4 for cryptographic applications. Therefore, we present an efficient algorithm for generating Montgomery-form elliptic curve whose cofactor is exactly “4”. Finally, we give the exact consition on the elliptic curves whether they can be represented as a Montgomery-form or not. We consider divisibility by “8” for Montgomery-form elliptic curves. We implement the proposed algorithm and give some numerical examples obtained by this.

Original languageEnglish
Title of host publicationPublic Key Cryptography - 3rd International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000, Proceedings
EditorsHideki Imai, Yuliang Zheng
PublisherSpringer Verlag
Pages238-257
Number of pages20
ISBN (Print)3540669671, 9783540669678
DOIs
Publication statusPublished - Jan 1 2000
Event3rd International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000 - Melbourne, Australia
Duration: Jan 18 2000Jan 20 2000

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume1751
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other3rd International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000
CountryAustralia
CityMelbourne
Period1/18/001/20/00

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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    Okeya, K., Kurumatani, H., & Sakurai, K. (2000). Elliptic curves with the montgomery-form and their cryptographic applications. In H. Imai, & Y. Zheng (Eds.), Public Key Cryptography - 3rd International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000, Proceedings (pp. 238-257). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1751). Springer Verlag. https://doi.org/10.1007/978-3-540-46588-1_17