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Extra resources for Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings
Moreover, since there is no non-generic algorithm for computing discrete logarithms on elliptic curves, it is possible to reach a high security level while using relatively short keys. However, in  Menezes, Okamoto and Vanstone showed that some special elliptic curves, called supersingular curves, are weaker than general elliptic curves. On these special curves, some additional properties allow an attacker to transport the discrete logarithm problem to a ﬁnite ﬁeld where more eﬃcient algorithms are available for discrete logarithm computation.
This idea of Hierarchical IBE is examined in  and a partial solution is proposed that works when collusion between the sub-authorities to break the scheme is limited. Remark: In identity based encryption, the key generation authority implicitly get the capability of an escrow agent. This shows that escrowed encryption protocols arise quite naturally when using pairings. This capability of pairing based cryptography was ﬁrst described in . Pairings and signatures. While identity based signatures and identiﬁcation protocols do not require the use of pairings, they can also be implemented by using pairings (see [8,18,31]).
27. A. Menezes, T. Okamoto, and S. Vanstone. Reducing elliptic curve logarithms to logarithms in a ﬁnite ﬁeld. IEEE Transaction on Information Theory, 39:1639– 1646, 1993. 28. V. Miller. Short programs for functions on curves. Unpublished manuscript, 1986. 29. V. Miller. Use of elliptic curves in cryptography. In H. Williams, editor, Advances in Cryptology — CRYPTO’85, volume 218 of Lecture Notes in Comput. , pages 417–428. Springer, 1986. 30. T. Okamoto and D. Pointcheval. The gap problems: a new class of problems for the security of cryptographic primitives.