Abstract. We present an implementation of elliptic curves and of hyperelliptic curves of genus 2 and 3 over prime fields. To achieve a fair comparison between the different types of groups, we developed an ad-hoc arithmetic library, designed to remove most of the overheads that penalize implementations of curve-based cryptography over prime fields. These overheads get worse for smaller fields, and thus for larger genera for a fixed group size. We also use techniques for delaying modular reductions to reduce the amount of modular reductions in the formulae for the group operations. The result is that the performance of hyperelliptic curves of genus 2 over prime fields is much closer to the performance of elliptic curves than previously thought. For groups of 192 and 256 bits the difference is about 14 % and 15 % respectively.

Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of hyperelliptic curves.

We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.

We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. For a genus g hyperelliptic curve defined over F2 n , the average-case time complexity is O(g ) and the average-case space complexity is O(g ), whereas the worst-case time and space complexities are O(g ) and ) respectively.

Abstract. In 1986, D. V. Chudnovsky and G. V. Chudnovsky proposed to use formulae coming from Theta functions for the arithmetic in Jacobians of genus 2 curves. We follow this idea and derive fast formulae for the scalar multiplication in the Kummer surface associated to a genus 2 curve, using a Montgomery ladder. Our formulae can be used to design very efficient genus 2 cryptosystems that should be faster than elliptic curve cryptosystems in some hardware configurations.

Abstract. The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2-adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field Q(i p 75 + 12 √ 17), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1, j2, j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestre’s algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography. 1

Abstract. We present probabilistic algorithms which, given a genus 2 curve C defined over a finite field and a quartic CM field K, determine whether the endomorphism ring of the Jacobian J of C is the full ring of integers in K. In particular, we present algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups J[ℓ d] for prime powers ℓ d. We use these algorithms to create the first implementation of Eisenträger and Lauter’s algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem [EL], and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute p 3 curves for many small primes p. 1.

We propose a novel scheme for selective distribution of content, encoded as documents, that preserves the privacy of the users to whom the documents are delivered and is based on an efficient and novel group key management scheme. Our document broadcasting approach is based on access control policies specifying which users can access which documents, or subdocuments. Based on such policies, a broadcast document is segmented into multiple subdocuments, each encrypted with a different key. In line with modern attribute-based access control, policies are specified against identity attributes of users. However our broadcasting approach is privacy-preserving in that users are granted access to a specific document, or subdocument, according to the policies without the need of providing in clear information about their identity attributes to the document publisher. Under our approach, not only does the document publisher not learn the values of the identity attributes of users, but it also does not learn which policy conditions are verified by which users, thus inferences about the values of identity attributes are prevented. Moreover, our key management scheme on which the proposed broadcasting approach is based is efficient in that it does not require to send the decryption keys to the users along with the encrypted document. Users are able to reconstruct the keys to decrypt the authorized portions of a document based on subscription information they have received from the document publisher. The scheme also efficiently handles new subscription of users and revocation of subscriptions.

Abstract. We apply deformation theory to compute zeta functions in a family of Ca,b curves over a finite field of small characteristic. The method combines Denef and Vercauteren’s extension of Kedlaya’s algorithm to Ca,b curves with Hubrechts ’ recent work on point counting on hyperelliptic curves using deformation. As a result, it is now possible to generate Ca,b curves suitable for use in cryptography in a matter of minutes. 1

Abstract. A pairing-friendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large prime-order subgroup. In this paper we construct pairing-friendly genus 2 curves over finite fields Fq whose Jacobians are ordinary and simple, but not absolutely simple. We show that constructing such curves is equivalent to constructing elliptic curves over Fq that become pairing-friendly over a finite extension of Fq. Our main proof technique is Weil restriction of elliptic curves. We describe adaptations of the Cocks-Pinch and Brezing-Weng methods that produce genus 2 curves with the desired properties. Our examples include a parametric family of genus 2 curves whose Jacobians have the smallest recorded ρ-value for simple, nonsupersingular abelian surfaces. 1.