Constructing Pairing-Friendly Elliptic Curves under Embedding Degree 1 for Securing Critical Infrastructures

Abstract

Information confidentiality is an essential requirement for cyber security in critical infrastructure. Identity-based cryptography, an increasingly popular branch of cryptography, is widely used to protect the information confidentiality in the critical infrastructure sector due to the ability to directly compute the user’s public key based on the user’s identity. However, computational requirements complicate the practical application of Identity-based cryptography. In order to improve the efficiency of identity-based cryptography, this paper presents an effective method to construct pairing-friendly elliptic curves with low hamming weight 4 under embedding degree 1. Based on the analysis of the Complex Multiplication(CM) method, the soundness of our method to calculate the characteristic of the finite field is proved. And then, three relative algorithms to construct pairing-friendly elliptic curve are put forward. 10 elliptic curves with low hamming weight 4 under 160 bits are presented to demonstrate the utility of our approach. Finally, the evaluation also indicates that it is more efficient to compute Tate pairing with our curves, than that of Bertoni et al.

Security analysis of elliptic curves with embedding degree 1 proposed in PLOS ONE 2016

“…In this paper, we point out that there is a serious issue with the security and efficiency claims given by Wang et al [35]. In short, their elliptic curves are insecure and should not be used for cryptographic purposes.…”

“…The hardness of finite field discrete logarithm problems is one of the most important bases of security; therefore, their elliptic curves should not be used for cryptographic purposes. In addition, we present security and efficiency analyses of the method of Wang et al [35]. Based on these analyses, we conclude that the claims of security and efficiency given by Wang et al [35] are flawed.…”

“…Recently, Wang et al [35] proposed a method for obtaining elliptic curves with embedding degree 1 in order to instantiate pairing-based cryptographic schemes for securing critical infrastructures. They presented some experimental results and claimed that their pairing implementation of an elliptic curve generated with their method is faster than another secure and efficient implementation presented by Bertoni et al [36, 37].…”

“…In short, their elliptic curves are insecure and should not be used for cryptographic purposes. Wang et al [35] presented several elliptic curves generated by their method. The bit lengths of orders of these curves are 160 bits and 189 bits, and these elliptic curves are described in the files [38, 39] of their supporting information.…”

Parallel point-multiplication architecture using combined group operations for high-speed cryptographic applications

“…Several ASIC-Based ECC processors have been proposed over the binary fields in the literature [ 10 , 18 , 26 – 33 ]. ECC can be used for modern practical applications like mobile services [ 34 ], authentication for identity protection for smart grid, wireless sensor and mesh networks [ 35 – 37 ], biometric-based authentication [ 38 ], identity-based cryptography [ 39 ], and session initiation protocol [ 40 ].…”