Computing low-degree isogenies in genus 2 with the Dolgachev-Lehavi method

Let ell be a prime, and H a curve of genus 2 over a field k of characteristic not 2 or ell. If S is a maximal Weil-isotropic subgroup of Jac(H)[ell], then Jac(H)/S is isomorphic to the Jacobian of some (possibly reducible) curve X. We investigate the Dolgachev--Lehavi method for constructing the curve X, simplifying their approach and making it more explicit. The result, at least for ell=3, is an efficient and easily programmable algorithm suitable for number-theoretic calculations

Families of fast elliptic curves from Q-curves

We construct new families of elliptic curves over \FF_{p^2} with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) endomorphisms. Our construction is based on reducing \QQ-curves-curves over quadratic number fields without complex multiplication, but with isogenies to their Galois conjugates-modulo inert primes. As a first application of the general theory we construct, for every $p > 3$, two one-parameter families of elliptic curves over \FF_{p^2} equipped with endomorphisms that are faster than doubling. Like GLS (which appears as a degenerate case of our construction), we offer the advantage over GLV of selecting from a much wider range of curves, and thus finding secure group orders when $p$ is fixed. Unlike GLS, we also offer the possibility of constructing twist-secure curves. Among our examples are prime-order curves equipped with fast endomorphisms, with almost-prime-order twists, over \FF_{p^2} for $p = 2^{127}-1$ and $p = 2^{255}-19$

The Q-curve construction for endomorphism-accelerated elliptic curves

We give a detailed account of the use of $\mathbb{Q}$-curve reductions to construct elliptic curves over $\mathbb{F}\_{p^2}$ with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant--Lambert--Vanstone (GLV) and Galbraith--Lin--Scott (GLS) endomorphisms. Like GLS (which is a degenerate case of our construction), we offer the advantage over GLV of selecting from a much wider range of curves, and thus finding secure group orders when $p$ is fixed for efficient implementation. Unlike GLS, we also offer the possibility of constructing twist-secure curves. We construct several one-parameter families of elliptic curves over $\mathbb{F}\_{p^2}$ equipped with efficient endomorphisms for every p \textgreater{} 3, and exhibit examples of twist-secure curves over $\mathbb{F}\_{p^2}$ for the efficient Mersenne prime $p = 2^{127}-1$.Comment: To appear in the Journal of Cryptology. arXiv admin note: text overlap with arXiv:1305.540

Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians

The first step in elliptic curve scalar multiplication algorithms based on scalar decompositions using efficient endomorphisms-including Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) multiplication, as well as higher-dimensional and higher-genus constructions-is to produce a short basis of a certain integer lattice involving the eigenvalues of the endomorphisms. The shorter the basis vectors, the shorter the decomposed scalar coefficients, and the faster the resulting scalar multiplication. Typically, knowledge of the eigenvalues allows us to write down a long basis, which we then reduce using the Euclidean algorithm, Gauss reduction, LLL, or even a more specialized algorithm. In this work, we use elementary facts about quadratic rings to immediately write down a short basis of the lattice for the GLV, GLS, GLV+GLS, and Q-curve constructions on elliptic curves, and for genus 2 real multiplication constructions. We do not pretend that this represents a significant optimization in scalar multiplication, since the lattice reduction step is always an offline precomputation---but it does give a better insight into the structure of scalar decompositions. In any case, it is always more convenient to use a ready-made short basis than it is to compute a new one

Polymorphic Types in ACL2

This paper describes a tool suite for the ACL2 programming language which incorporates certain ideas from the Hindley-Milner paradigm of functional programming (as exemplified in popular languages like ML and Haskell), including a "typed" style of programming with the ability to define polymorphic types. These ideas are introduced via macros into the language of ACL2, taking advantage of ACL2's guard-checking mechanism to perform type checking on both function definitions and theorems. Finally, we discuss how these macros were used to implement features of Specware, a software specification and implementation system.Comment: In Proceedings ACL2 2014, arXiv:1406.123

Internet Radio: A New Engine for Content Diversity?

While traditional radio stations are subject to extensive government regulations, Internet radio stations remain largely unregulated. As Internet radio usage has increased certain stakeholders have begun to argue that these Internet radio broadcasters are providing significant and diverse programming to American audiences and that government regulation of spectrum-using radio station ownership may be further relaxed. One of the primary justifications for regulation of ownership has been to protect diversity in broadcasting. This study hypothesizes that Internet radio broadcasting does add diversity to the radio broadcasting industry and that it should be considered as relevant by regulators. This study evaluates the role of Internet radio broadcasters according to five criteria intended to gauge the level of diversity being delivered to listeners online. By measuring the levels of format, channel, ownership, location and language diversity among Internet radio stations, it is possible to draw benchmark lessons about the new medium's ability to provide Americans with diverse broadcasting options. The study finds that Internet radio broadcasters are in fact adding measurable diversity to the radio broadcasting industry. Internet broadcasters are providing audiences with access to an increasing number of stations, owners, formats, and language choices, and it is likely that technologies aiding in the mobility of access as well as broadband evolution will reinforce these findings.Comment: 29th TPRC Conference, 200

Stochastic continuity equations with conservative noise

The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the existence and uniqueness of entropy solutions under hypotheses on the velocity field which are weaker than those required in the deterministic setting. This extends related results of [Flandoli, Gubinelli, Priola; Invent. Math., 2010] applicable for linear multiplicative noise to a non-linear setting. The existence proof relies on a duality argument which makes use of the regularity theory for fully non-linear parabolic equations.Comment: 42 page