Abstract
We investigate the complexity of breaking cryptosystems of which security is based on the discrete logarithm problem. We denote the algorithms of breaking the Diffie-Hellman’s key exchange scheme by DH, the Bellare-Micali’s non-interactive oblivious transfer scheme by BM, the ElGamal’s public-key cryptosystem by EG, the Okamoto’s conference- key sharing scheme by CONF, and the Shamir’s 3-pass key-transmission scheme by 3PASS, respectively. We show a relation among these cryp- tosystems that 3PASS ≤ FPm CONF ≤ FPm EG ≡ FPm DH, where ≤ FPm denotes the polynomial-time functionally many-to-one re- ducibility, i.e. a function version of the ≤ pm -reducibility. We further give some condition in which these algorithms have equivalent difficulty. Namely,
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1.
If the complete factorization of p − 1 is given, i.e. if the the dis- crete logarithm problem is a certified one, then these cryptosystems are equivalent w.r.t. expected polynomial-time functionally Turing reducibility.
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2.
If the underlying group is the Jacobian of an elliptic curve over Z p with a prime order, then these cryptosystems are equivalent w.r.t. polynomial-time functionally many-to-one reducibility.
We also discuss the complexity of several languages related to those computing problems.
A part of this work was done while the first author was working for Mitsubishi Electric Corp.
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Sakurai, K., Shizuya, H. (1995). Relationships among the Computational Powers of Breaking Discrete Log Cryptosystems. In: Guillou, L.C., Quisquater, JJ. (eds) Advances in Cryptology — EUROCRYPT ’95. EUROCRYPT 1995. Lecture Notes in Computer Science, vol 921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49264-X_28
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DOI: https://doi.org/10.1007/3-540-49264-X_28
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