New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs (original) (raw)

Abstract

The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group \({\mathbb {G}}=\langle P\rangle \) of order p and some elements \(P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}\), an attacker can recover \(\alpha \in {\mathbb {Z}}_p^*\) in the case of \(d|(p\pm 1)\) with running time of \({\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)\) group operations by using \({\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})\) storage (\(i=\frac{1}{2}\) or 1 for \(d|(p-1)\) case or \(d|(p+1)\) case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points \(P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}\) and multiplicative cyclic group \(K=\langle k \rangle \) are given, where d is a prime, \(\varphi (d)\) is the order of K and \(\varphi \) is the Euler totient function, the secret key \(\alpha \in {\mathbb {Z}}_p^*\) can be solved in \({\mathcal {O}}(\sqrt{(p-1)/d}+d)\) group operations by using \({\mathcal {O}}(\sqrt{(p-1)/d})\) storage.

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Acknowledgments

We gratefully acknowledge the reviewers for the help comments an suggestions. This work is supported by the National Natural Science Foundation of China (Nos. 61309016, 61379150) and Open Project Program of the State Key Laboratory of Mathematical Engineering and Advanced Computing.

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Authors and Affiliations

  1. Air Force Engineering University, Xi’an, China
    Jiang Weng
  2. State Key Laboratory of Mathematical Engineering and Advanced Computing, Zhengzhou, China
    Jiang Weng & Yunqi Dou
  3. Basic Department, Army Aviation Institution, Beijing, China
    Chuangui Ma

Authors

  1. Jiang Weng
  2. Yunqi Dou
  3. Chuangui Ma

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Correspondence toJiang Weng.

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Weng, J., Dou, Y. & Ma, C. New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs.AAECC 28, 99–108 (2017). https://doi.org/10.1007/s00200-016-0301-z

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