On a Relation between the Ate Pairing and the Weil Pairing for Supersingular Elliptic Curves

The hyperelliptic curve Ate pairing provides an efficient way to compute a bilinear pairing on the Jacobian variety of a hyperelliptic curve. We prove that, for supersingular elliptic curves with embedding degree two, square of the Ate pairing is nothing but the Weil pairing. Using the formula, we develop an X-coordinate only pairing inversion method. However, the algorithm is still infeasible for cryptographic size problems

Still Wrong Use of Pairings in Cryptography

Several pairing-based cryptographic protocols are recently proposed with a wide variety of new novel applications including the ones in emerging technologies like cloud computing, internet of things (IoT), e-health systems and wearable technologies. There have been however a wide range of incorrect use of these primitives. The paper of Galbraith, Paterson, and Smart (2006) pointed out most of the issues related to the incorrect use of pairing-based cryptography. However, we noticed that some recently proposed applications still do not use these primitives correctly. This leads to unrealizable, insecure or too inefficient designs of pairing-based protocols. We observed that one reason is not being aware of the recent advancements on solving the discrete logarithm problems in some groups. The main purpose of this article is to give an understandable, informative, and the most up-to-date criteria for the correct use of pairing-based cryptography. We thereby deliberately avoid most of the technical details and rather give special emphasis on the importance of the correct use of bilinear maps by realizing secure cryptographic protocols. We list a collection of some recent papers having wrong security assumptions or realizability/efficiency issues. Finally, we give a compact and an up-to-date recipe of the correct use of pairings.Comment: 25 page

Effective Field Theory for Dilute Fermions with Pairing

Effective field theory (EFT) methods for a uniform system of fermions with short-range, natural interactions are extended to include pairing correlations, as part of a program to develop a systematic Kohn-Sham density functional theory (DFT) for medium and heavy nuclei. An effective action formalism for local composite operators leads to a free-energy functional that includes pairing by applying an inversion method order by order in the EFT expansion. A consistent renormalization scheme is demonstrated for the uniform system through next-to-leading order, which includes induced-interaction corrections to pairing.Comment: 31 pages, 10 figures, affiliation updated, paper unchange

Density Functional Theory: Methods and Problems

The application of density functional theory to nuclear structure is discussed, highlighting the current status of the effective action approach using effective field theory, and outlining future challenges.Comment: 10 pages, 14 figures, invited talk at INT workshop on Nuclear Forces and the Quantum Many-Body Problem, Seattle, October 200

What is liquid? Lyapunov instability reveals symmetry-breaking irreversibilities hidden within Hamilton's many-body equations of motion

Typical Hamiltonian liquids display exponential "Lyapunov instability", also called "sensitive dependence on initial conditions". Although Hamilton's equations are thoroughly time-reversible, the forward and backward Lyapunov instabilities can differ, qualitatively. In numerical work, the expected forward/backward pairing of Lyapunov exponents is also occasionally violated. To illustrate, we consider many-body inelastic collisions in two space dimensions. Two mirror-image colliding crystallites can either bounce, or not, giving rise to a single liquid drop, or to several smaller droplets, depending upon the initial kinetic energy and the interparticle forces. The difference between the forward and backward evolutionary instabilities of these problems can be correlated with dissipation and with the Second Law of Thermodynamics. Accordingly, these asymmetric stabilities of Hamilton's equations can provide an "Arrow of Time". We illustrate these facts for two small crystallites colliding so as to make a warm liquid. We use a specially-symmetrized form of Levesque and Verlet's bit-reversible Leapfrog integrator. We analyze trajectories over millions of collisions with several equally-spaced time reversals.Comment: 13 pages and 11 figures, prepared for Douglas Henderson's 80th Birthday Symposium at Brigham Young University in August 2014 revised to incorporate referee's suggestions as an acknowledgmen

Nonlinear Magneto-Optical Response of ss- and dd-Wave Superconductors

The nonlinear magneto-optical response of $s$- and $d$-wave superconductors is discussed. We carry out the symmetry analysis of the nonlinear magneto-optical susceptibility in the superconducting state. Due to the surface sensitivity of the nonlinear optical response for systems with bulk inversion symmetry, we perform a group theoretical classification of the superconducting order parameter close to a surface. For the first time, the mixing of singlet and triplet pairing states induced by spin-orbit coupling is systematically taken into account. We show that the interference of singlet and triplet pairing states leads to an observable contribution of the nonlinear magneto-optical Kerr effect. This effect is not only sensitive to the anisotropy of the gap function but also to the symmetry itself. In view of the current discussion of the order parameter symmetry of High-T$_c$ superconductors, results for a tetragonal system with bulk singlet pairing for various pairing symmetries are discussed.Comment: 21 pages (REVTeX) with 8 figures (Postscript

Superconductivity in ferromagnetic metals and in compounds without inversion centre

The symmetry properties and the general overview of the superconductivity theory in the itinerant ferromagnets and in materials without space parity are presented. The basic notions of unconventional superconductivity are introduced in broad context of multiband superconductivity which is inherent property of ferromagnetic metals or metals without centre of inversion.Comment: 38 pages, no figure

Topological superconducting phases from inversion symmetry breaking order in spin-orbit-coupled systems

We analyze the superconducting instabilities in the vicinity of the quantum-critical point of an inversion symmetry breaking order. We first show that the fluctuations of the inversion symmetry breaking order lead to two degenerate superconducting (SC) instabilities, one in the $s$-wave channel, and the other in a time-reversal invariant odd-parity pairing channel (the simplest case being the same as the of $^3$He-B phase). Remarkably, we find that unlike many well-known examples, the selection of the pairing symmetry of the condensate is independent of the momentum-space structure of the collective mode that mediates the pairing interaction. We found that this degeneracy is a result of the existence of a conserved fermionic helicity, $\chi$, and the two degenerate channels correspond to even and odd combinations of SC order parameters with $\chi=\pm1$. As a result, the system has an enlarged symmetry $U(1)\times U(1)$, with each $U(1)\times U(1)$ corresponding to one value of the helicity $\chi$. Because of the enlarged symmetry, this system admits exotic topological defects such as a fractional quantum vortex, which we show has a Majorana zero mode bound at its core. We discuss how the enlarged symmetry can be lifted by small perturbations, such as the Coulomb interaction or Fermi surface splitting in the presence of broken inversion symmetry, and we show that the resulting superconducting state can be topological or trivial depending on parameters. The $U(1)\times U(1)$ symmetry is restored at the phase boundary between the topological and trivial SC states, and allows for a transition between topologically distinct SC phases without the vanishing of the order parameter. We present a global phase diagram of the superconducting states and discuss possible experimental implications.Comment: 14 pages, 5 figures, to match the published versio

Turning a Band Insulator Into an Exotic Superconductor

Understanding exotic, non s--wave--like states of Cooper pairs is important and may lead to new superconductors with higher critical temperatures and novel properties. Their existence is known to be possible but has always been thought to be associated with non--traditional mechanisms of superconductivity where electronic correlations play an important role. Here we use a first principles linear response calculation to show that in doped Bi$_{2}$Se$_{3}$ an unconventional p--wave--like state can be favored via a conventional phonon--mediated mechanism, as driven by an unusual, almost singular behavior of the electron--phonon interaction at long wavelengths. This may provide a new platform for our understanding superconductivity phenomena in doped band insulators.Comment: Published versio

Glide reflection symmetry, Brillouin zone folding and superconducting pairing for the P4/nmmP4/nmm space group

Motivated by the studies of the superconducting pairing states in the iron-based superconductors, we analyze the effects of Brillouin zone folding procedure from a space group symmetry perspective for a general class of materials with the $P4/nmm$ space group. The Brillouin zone folding amounts to working with an effective one-Fe unit cell, instead of the crystallographic two-Fe unit cell. We show that the folding procedure can be justified by the validity of a glide reflection symmetry throughout the crystallographic Brillouin zone and by the existence of a minimal double degeneracy along the edges of the latter. We also demonstrate how the folding procedure fails when a local spin-orbit coupling is included although the latter does not break any of the space group symmetries of the bare Hamiltonian. In light of these general symmetry considerations, we further discuss the implications of the glide reflection symmetry for the superconducting pairing in an effective multi-orbital $t-J_{1}-J_{2}$ model. We find that the $P4/nmm$ space group symmetry allows only pairing states with even parity under the glide reflection and zero total momentum