New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs

Autor: Jiang Weng, Yunqi Dou, Chuangui Ma
Rok vydání: 2016
Předmět:
Zdroj: Applicable Algebra in Engineering, Communication and Computing. 28:99-108
ISSN: 1432-0622
0938-1279
DOI: 10.1007/s00200-016-0301-z
Popis: 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 $${\mathbb {G}}=\langle P\rangle $$G=?P? of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$P,?P,?2P,?,?dP?G, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$??Zp? in the case of $$d|(p\pm 1)$$d|(p±1) with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$O((p±1)/d+di) group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$O(max{(p±1)/d,d}) storage ($$i=\frac{1}{2}$$i=12 or 1 for $$d|(p-1)$$d|(p-1) case or $$d|(p+1)$$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 $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$P,?P,?kP,?k2P,?k3P,?,?k?(d)-1P?G and multiplicative cyclic group $$K=\langle k \rangle $$K=?k? are given, where d is a prime, $$\varphi (d)$$?(d) is the order of K and $$\varphi $$? is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$??Zp? can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$O((p-1)/d+d) group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$O((p-1)/d) storage.
Databáze: OpenAIRE
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