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We present hierarchical identity-based encryption schemes and signature schemes that have total collusion resistance on an arbitrary number of levels and that have chosen ciphertext security in the random oracle model assuming the difficulty of the Bilinear Diffie-Hellman problem .
— We apply results from algebraic coding theory to solve problems in cryptography, by using recent results on list decoding of error-correcting codes to efficiently find traitors who collude to create pirates. We produce schemes for which the TA (traceability) traitor tracing algorithm is very fast. We compare the TA and IPP (identifiable parent property)… (More)
We study the problem of finding efficiently computable non-degenerate multilinear maps from G n 1 to G 2 , where G 1 and G 2 are groups of the same prime order, and where computing discrete logarithms in G 1 is hard. We present several applications to cryptography, explore directions for building such maps, and give some reasons to believe that finding… (More)
We apply powerful, recently discovered techniques for the list decoding of error-correcting codes to the problem of efficiently tracing traitors. Much work has focused on methods for constructing such traceability schemes, but the complexity of the traitor tracing algorithms has received little attention. A widely used traitor tracing algorithm, the TA… (More)
We introduce the concept of torus-based cryptography, give a new public key system called CEILIDH, and compare it to other discrete log based systems including Lucas-based systems and XTR. Like those systems, we obtain small key sizes. While Lucas-based systems and XTR are essentially restricted to exponentiation, we are able to perform multiplication as… (More)
We give easy ways to distinguish between the twists of an ordinary elliptic curve E over Fp in order to identify one with p + 1 − 2U points, when p = U 2 + dV 2 with 2U, 2V ∈ Z and E is constructed using the CM method. This is useful for finding elliptic curves with a prescribed number of points, and is a new, faster, and easier way to implement the last… (More)
We give explicit examples of infinite families of elliptic curves E over Q with (nonconstant) quadratic twists over Q(t) of rank at least 2 and 3. We recover some results announced by Mestre, as well as some additional families. Suppose D is a squarefree integer and let r E (D) denote the rank of the quadratic twist of E by D. We apply results of Stewart… (More)
We apply the Cocks-Pinch method to obtain pairing-friendly composite order groups with prescribed embedding degree associated to ordinary elliptic curves, and we show that new security issues arise in the composite order setting.
If V is a commutative algebraic group over a field k, O is a com-mutative ring that acts on V , and I is a finitely generated free O-module with a right action of the absolute Galois group of k, then there is a commutative algebraic group I ⊗ O V over k, which is a twist of a power of V. These group varieties have applications to cryptography (in the cases… (More)
We give an overview of joint work with Karl Rubin on computing the number of points on reductions of elliptic curves with complex multiplication , including some of the history of the problem.