The Montgomery ladder on binary elliptic curves

In this survey paper we present a careful analysis of the Montgomery ladder procedure applied to the computation of the constant-time point multiplication operation on elliptic curves defined over binary extension fields. We give a general view of the main improvements and formula derivations that several researchers have contributed across the years, since the publication of Peter Lawrence Montgomery seminal work in 1987. We also report a fast software implementation of the Montgomery ladder applied on a Galbraith-Lin-Scott (GLS) binary elliptic curve that offers a security level close to 128 bits. Using our software we can execute the ephemeral Diffie-Hellman protocol in just 95,702 clock cycles when implemented on an Intel Skylake machine running at 4 GHz

Two-photon laser spectroscopy of antiprotonic helium and the antiproton-to-electron mass ratio

Physical laws are believed to be invariant under the combined transformations of charge, parity and time reversal (CPT symmetry). This implies that an antimatter particle has exactly the same mass and absolute value of charge as its particle counterpart. Metastable antiprotonic helium is a three-body atom consisting of a normal helium nucleus, an electron in its ground state and an antiproton occupying a Rydberg state with high principal and angular momentum quantum numbers, respectively n and l, such that n~l+1 ~ 38. These atoms are amenable to precision laser spectroscopy, the results of which can in principle be used to determine the antiproton-to-electron mass ratio and to constrain the equality between the antiproton and proton charges and masses. Here we report two-photon spectroscopy of antiprotonic helium, in which two antiprotonic helium isotopes are irradiated by two counter-propagating laser beams. This excites nonlinear, two-photon transitions of the antiproton of the type (n,l) -> (n-2, l-2) at deep-ultraviolet wave-lengths (139.8, 193.0 and 197.0 nm), which partly cancel the Doppler broadening of the laser resonance caused by the thermal motion of the atoms. The resulting narrow spectral lines allowed us to measure three transition frequencies with fractional precisions of 2.3–5 parts in 10^9. By comparing the results with three-body quantum electrodynamics calculations, we derived an antiproton-to-electron mass ratio of 1,836.1526736(23), where the parenthetical error represents one standard deviation. This agrees with the proton-to-electron value known to a similar precision

High-precision comparison of the antiproton-to-proton charge-to-mass ratio

Invariance under the charge, parity, time-reversal (CPT) transformation$^{1}$ is one of the fundamental symmetries of the standard model of particle physics. This CPT invariance implies that the fundamental properties of antiparticles and their matter-conjugates are identical, apart from signs. There is a deep link between CPT invariance and Lorentz symmetry—that is, the laws of nature seem to be invariant under the symmetry transformation of spacetime—although it is model dependent$^{2}$. A number of high-precision CPT and Lorentz invariance tests—using a co-magnetometer, a torsion pendulum and a maser, among others—have been performed$^{3}$, but only a few direct high-precision CPT tests that compare the fundamental properties of matter and antimatter are available$^{4, 5, 6, 7, 8}$. Here we report high-precision cyclotron frequency comparisons of a single antiproton and a negatively charged hydrogen ion (H$^−$) carried out in a Penning trap system. From 13,000 frequency measurements we compare the charge-to-mass ratio for the antiproton $(q/m)_{\bar{p}}$ to that for the proton $(q/m)_{p}$ and obtain $(q/m)_{\bar{p}} / (q/m)_{p} - 1 = 1(69) \times 10^{-12}$. The measurements were performed at cyclotron frequencies of 29.6 megahertz, so our result shows that the CPT theorem holds at the atto-electronvolt scale. Our precision of 69 parts per trillion exceeds the energy resolution of previous antiproton-to-proton mass comparisons$^{7, 9}$ as well as the respective figure of merit of the standard model extension$^{10}$ by a factor of four. In addition, we give a limit on sidereal variations in the measured ratio of < 720 parts per trillion. By following the arguments of ref. 11, our result can be interpreted as a stringent test of the weak equivalence principle of general relativity using baryonic antimatter, and it sets a new limit on the gravitational anomaly parameter of $|\alpha_g - 1| < 8.7 × 10^{−7}$