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Security analysis of the strong Diffie-Hellman problem. (English) Zbl 1129.94017
Vaudenay, Serge (ed.), Advances in cryptology – EUROCRYPT 2006. 25th annual international conference on the theory and applications of cryptographic techniques, St. Petersburg, Russia, May 28 – June 1, 2006. Proceedings. Berlin: Springer (ISBN 3-540-34546-9/pbk). Lecture Notes in Computer Science 4004, 1-11 (2006).
Summary: Let \(g\) be an element of prime order \(p\) in an abelian group and \(\alpha\in {{\mathbb Z}}_p\). We show that if \(g, g^{\alpha }\), and \(g^{\alpha^d}\) are given for a positive divisor \(d\) of \(p-1\), we can compute the secret \(\alpha \) in \(O(\log p \cdot (\sqrt{p/d}+\sqrt d))\) group operations using \(O(\max\{\sqrt{p/d},\sqrt d\})\) memory. If \(g^{\alpha^i} (i=0,1,2,\dots, d)\) are provided for a positive divisor \(d\) of \(p+1, \alpha \) can be computed in \(O(\log p \cdot (\sqrt{p/d}+d))\) group operations using \(O(\max\{\sqrt{p/d},\sqrt d\})\) memory. This implies that the strong Diffie-Hellman problem and its related problems have computational complexity reduced by \(O(\sqrt d)\) from that of the discrete logarithm problem for such primes.
Further we apply this algorithm to the schemes based on the Diffie-Hellman problem on an abelian group of prime order \(p\). As a result, we reduce the complexity of recovering the secret key from \(O(\sqrt p)\) to \(O(\sqrt{p/d})\) for Boldyreva’s blind signature and the original ElGamal scheme when \(p-1\) (resp. \(p+1)\) has a divisor \(d \leq p^{1/2}\) (resp. \(d \leq p^{1/3})\) and \(d\) signature or decryption queries are allowed.
For the entire collection see [Zbl 1108.94002].

94A60 Cryptography
94A62 Authentication, digital signatures and secret sharing
Full Text: DOI
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