A Note on Subgroup Security in Discrete Logarithm-Based Cryptography

Tadanori TERUYA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E104-A   No.1   pp.104-120
Publication Date: 2021/01/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.2020CIP0019
Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)
discrete logarithm-based cryptography,  ElGamal encryption,  membership check,  pairing-based cryptography,  subgroup security,  

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The membership check of a group is an important operation to implement discrete logarithm-based cryptography in practice securely. Since this check requires costly scalar multiplication or exponentiation operation, several efficient methods have been investigated. In the case of pairing-based cryptography, this is an extended research area of discrete logarithm-based cryptography, Barreto et al. (LATINCRYPT 2015) proposed a parameter choice called subgroup-secure elliptic curves. They also claimed that, in some schemes, if an elliptic curve is subgroup-secure, costly scalar multiplication or exponentiation operation can be omitted from the membership check of bilinear groups, which results in faster schemes than the original ones. They also noticed that some schemes would not maintain security with this omission. However, they did not show the explicit condition of what schemes become insecure with the omission. In this paper, we show a concrete example of insecurity in the sense of subgroup security to help developers understand what subgroup security is and what properties are preserved. In our conclusion, we recommend that the developers use the original membership check because it is a general and straightforward method to implement schemes securely. If the developers want to use the subgroup-secure elliptic curves and to omit the costly operation in a scheme for performance reasons, it is critical to carefully analyze again that correctness and security are preserved with the omission.