Approaches for the Parallelization of Software Implementation of Integer Multiplication

In this paper there are considered several approaches for the increasing performance of software implementation of integer multiplication algorithm for the 32-bit & 64-bit platforms via parallelization. The main idea of algorithm parallelization consists in delayed carry mechanism using which authors have proposed earlier [11]. The delayed carry allows to get rid of connectivity in loop iterations for sums accumulation of products, which allows parallel execution of loops iterations in separate threads. Upon completion of sum accumulation threads, it is necessary to make corrections in final result via assimilation of carries. First approach consists in optimization of parallelization for the two execution threads and second approach is an evolution of the first approach and is oriented on three and more execution threads. Proposed approaches for parallelization allow increasing the total algorithm computational complexity, as for one execution thread, but decrease total execution time on multi-core CPU

Squaring Algorithms with Delayed Carry Method and Efficient Parallelization

Increasing amounts of information that needs to be protecting put in claims specific requirements for information security systems. The main goal of this paper is to find ways to increase performance of cryptographic transformation with public key by increasing performance of integers squaring. Authors use delayed carry mechanism and approaches of effective parallelization for Comba multiplication algorithm, which was previously proposing by authors. They use the idea of carries accumulation by addition products of multiplying the relevant machine words in columns. As a result, it became possible to perform addition of such products in the column independently of each other. However, independent accumulation of products and carries require correction of the intermediate results to account for the accumulated carries. Due to the independence of accumulation in the columns, it became possible to parallelize the process of products accumulation that allowed formulating several approaches. In this paper received theoretical estimates of the computational complexity for proposed squaring algorithms. Software implementations of algorithms in C++ allowed receiving practical results of the performance, which are not contrary to theoretical estimates. The authors first proposed applying the method of delayed carry and parallelization techniques for squaring algorithms, which was previously proposing for integer multiplication

Approaches for the performance increasing of software implementation of integer multiplication in prime fields

Authors have proposed the approach to increase performance of software implementation of finite field multiplication algorithm, for 32-bit and 64-bit platforms. The approach is based on delayed carry mechanism of significant bit in sum accumulating. This allows to avoid the requirement of taking into account the significant bit carry at the each iteration of the sum accumulation loop. The delayed carry mechanism reduces the total number of additions and gives the opportunity to apply the modern parallelization technologies

Techniques for Performance Improvement of Integer Multiplication in Cryptographic Applications

The problem of arithmetic operations performance in number fields is actively researched by many scientists, as evidenced by significant publications in this field. In this work, we offer some techniques to increase performance of software implementation of finite field multiplication algorithm, for both 32-bit and 64-bit platforms. The developed technique, called “delayed carry mechanism,” allows to preventing necessity to consider a significant bit carry at each iteration of the sum accumulation loop. This mechanism enables reducing the total number of additions and applies the modern parallelization technologies effectively

A study of CP violation in B-+/- -> DK +/- and B-+/- -> D pi(+/-) decays with D -> (KSK +/-)-K-0 pi(-/+) final states

A first study of CP violation in the decay modes $B^\pm\to [K^0_{\rm S} K^\pm \pi^\mp]_D h^\pm$ and $B^\pm\to [K^0_{\rm S} K^\mp \pi^\pm]_D h^\pm$, where $h$ labels a $K$ or $\pi$ meson and $D$ labels a $D^0$ or $\overline{D}^0$ meson, is performed. The analysis uses the LHCb data set collected in $pp$ collisions, corresponding to an integrated luminosity of 3 fb$^{-1}$. The analysis is sensitive to the CP-violating CKM phase $\gamma$ through seven observables: one charge asymmetry in each of the four modes and three ratios of the charge-integrated yields. The results are consistent with measurements of $\gamma$ using other decay modes

Study of the rare B-s(0) and B-0 decays into the pi(+) pi(-) mu(+) mu(-) final state

A search for the rare decays $B_s^0 \to \pi^+\pi^-\mu^+\mu^-$ and $B^0 \to \pi^+\pi^-\mu^+\mu^-$ is performed in a data set corresponding to an integrated luminosity of 3.0 fb$^{-1}$ collected by the LHCb detector in proton-proton collisions at centre-of-mass energies of 7 and 8 TeV. Decay candidates with pion pairs that have invariant mass in the range 0.5-1.3 GeV/$c^2$ and with muon pairs that do not originate from a resonance are considered. The first observation of the decay $B_s^0 \to \pi^+\pi^-\mu^+\mu^-$ and the first evidence of the decay $B^0 \to \pi^+\pi^-\mu^+\mu^-$ are obtained and the branching fractions are measured to be $\mathcal{B}(B_s^0 \to \pi^+\pi^-\mu^+\mu^-)=(8.6\pm 1.5\,({\rm stat}) \pm 0.7\,({\rm syst})\pm 0.7\,({\rm norm}))\times 10^{-8}$ and $\mathcal{B}(B^0 \to \pi^+\pi^-\mu^+\mu^-)=(2.11\pm 0.51\,({\rm stat}) \pm 0.15\,({\rm syst})\pm 0.16\,({\rm norm}) )\times 10^{-8}$, where the third uncertainty is due to the branching fraction of the decay $B^0\to J/\psi(\to \mu^+\mu^-)K^*(890)^0(\to K^+\pi^-)$, used as a normalisation.A search for the rare decays Bs0→π+π−μ+μ− and B0→π+π−μ+μ− is performed in a data set corresponding to an integrated luminosity of 3.0 fb−1 collected by the LHCb detector in proton–proton collisions at centre-of-mass energies of 7 and 8 TeV . Decay candidates with pion pairs that have invariant mass in the range 0.5–1.3 GeV/c2 and with muon pairs that do not originate from a resonance are considered. The first observation of the decay Bs0→π+π−μ+μ− and the first evidence of the decay B0→π+π−μ+μ− are obtained and the branching fractions, restricted to the dipion-mass range considered, are measured to be B(Bs0→π+π−μ+μ−)=(8.6±1.5 (stat)±0.7 (syst)±0.7(norm))×10−8 and B(B0→π+π−μ+μ−)=(2.11±0.51(stat)±0.15(syst)±0.16(norm))×10−8 , where the third uncertainty is due to the branching fraction of the decay B0→J/ψ(→μ+μ−)K⁎(892)0(→K+π−) , used as a normalisation.A search for the rare decays Bs0→π+π−μ+μ− and B0→π+π−μ+μ− is performed in a data set corresponding to an integrated luminosity of 3.0 fb−1 collected by the LHCb detector in proton–proton collisions at centre-of-mass energies of 7 and 8 TeV . Decay candidates with pion pairs that have invariant mass in the range 0.5–1.3 GeV/c2 and with muon pairs that do not originate from a resonance are considered. The first observation of the decay Bs0→π+π−μ+μ− and the first evidence of the decay B0→π+π−μ+μ− are obtained and the branching fractions, restricted to the dipion-mass range considered, are measured to be B(Bs0→π+π−μ+μ−)=(8.6±1.5 (stat)±0.7 (syst)±0.7(norm))×10−8 and B(B0→π+π−μ+μ−)=(2.11±0.51(stat)±0.15(syst)±0.16(norm))×10−8 , where the third uncertainty is due to the branching fraction of the decay B0→J/ψ(→μ+μ−)K⁎(892)0(→K+π−) , used as a normalisation.A search for the rare decays $B_s^0 \to \pi^+\pi^-\mu^+\mu^-$ and $B^0 \to \pi^+\pi^-\mu^+\mu^-$ is performed in a data set corresponding to an integrated luminosity of 3.0 fb$^{-1}$ collected by the LHCb detector in proton-proton collisions at centre-of-mass energies of 7 and 8 TeV. Decay candidates with pion pairs that have invariant mass in the range 0.5-1.3 GeV/$c^2$ and with muon pairs that do not originate from a resonance are considered. The first observation of the decay $B_s^0 \to \pi^+\pi^-\mu^+\mu^-$ and the first evidence of the decay $B^0 \to \pi^+\pi^-\mu^+\mu^-$ are obtained and the branching fractions, restricted to the dipion-mass range considered, are measured to be $\mathcal{B}(B_s^0 \to \pi^+\pi^-\mu^+\mu^-)=(8.6\pm 1.5\,({\rm stat}) \pm 0.7\,({\rm syst})\pm 0.7\,({\rm norm}))\times 10^{-8}$ and $\mathcal{B}(B^0 \to \pi^+\pi^-\mu^+\mu^-)=(2.11\pm 0.51\,({\rm stat}) \pm 0.15\,({\rm syst})\pm 0.16\,({\rm norm}) )\times 10^{-8}$, where the third uncertainty is due to the branching fraction of the decay $B^0\to J/\psi(\to \mu^+\mu^-)K^*(890)^0(\to K^+\pi^-)$, used as a normalisation

Observation of the B0 → ρ0ρ0 decay from an amplitude analysis of B0 → (π+π−)(π+π−) decays

Proton–proton collision data recorded in 2011 and 2012 by the LHCb experiment, corresponding to an integrated luminosity of 3.0 fb−1 , are analysed to search for the charmless B0→ρ0ρ0 decay. More than 600 B0→(π+π−)(π+π−) signal decays are selected and used to perform an amplitude analysis, under the assumption of no CP violation in the decay, from which the B0→ρ0ρ0 decay is observed for the first time with 7.1 standard deviations significance. The fraction of B0→ρ0ρ0 decays yielding a longitudinally polarised final state is measured to be fL=0.745−0.058+0.048(stat)±0.034(syst) . The B0→ρ0ρ0 branching fraction, using the B0→ϕK⁎(892)0 decay as reference, is also reported as B(B0→ρ0ρ0)=(0.94±0.17(stat)±0.09(syst)±0.06(BF))×10−6

Angular analysis of the B-0 -> K*(0) e(+) e(-) decay in the low-q(2) region

An angular analysis of the $B^0 \rightarrow K^{*0} e^+ e^-$ decay is performed using a data sample, corresponding to an integrated luminosity of 3.0 {\mbox{fb}^{-1}}, collected by the LHCb experiment in $pp$ collisions at centre-of-mass energies of 7 and 8 TeV during 2011 and 2012. For the first time several observables are measured in the dielectron mass squared ($q^2$) interval between 0.002 and 1.120${\mathrm{\,Ge\kern -0.1em V^2\!/}c^4}$. The angular observables $F_{\mathrm{L}}$ and $A_{\mathrm{T}}^{\mathrm{Re}}$ which are related to the $K^{*0}$ polarisation and to the lepton forward-backward asymmetry, are measured to be $F_{\mathrm{L}}= 0.16 \pm 0.06 \pm0.03$ and $A_{\mathrm{T}}^{\mathrm{Re}} = 0.10 \pm 0.18 \pm 0.05$, where the first uncertainty is statistical and the second systematic. The angular observables $A_{\mathrm{T}}^{(2)}$ and $A_{\mathrm{T}}^{\mathrm{Im}}$ which are sensitive to the photon polarisation in this $q^2$ range, are found to be $A_{\mathrm{T}}^{(2)} = -0.23 \pm 0.23 \pm 0.05$ and $A_{\mathrm{T}}^{\mathrm{Im}} =0.14 \pm 0.22 \pm 0.05$. The results are consistent with Standard Model predictions.An angular analysis of the B$^{0}$ → K^{*}^{0} e$^{+}$ e$^{−}$ decay is performed using a data sample, corresponding to an integrated luminosity of 3.0 fb$^{−1}$, collected by the LHCb experiment in pp collisions at centre-of-mass energies of 7 and 8 TeV during 2011 and 2012. For the first time several observables are measured in the dielectron mass squared (q$^{2}$) interval between 0.002 and 1.120 GeV$^{2}$ /c$^{4}$. The angular observables F$_{L}$ and A$_{T}^{Re}$ which are related to the K^{*}^{0} polarisation and to the lepton forward-backward asymmetry, are measured to be F$_{L}$ = 0.16 ± 0.06 ± 0.03 and A$_{T}^{Re}$  = 0.10 ± 0.18 ± 0.05, where the first uncertainty is statistical and the second systematic. The angular observables A$_{T}^{(2)}$ and A$_{T}^{Im}$ which are sensitive to the photon polarisation in this q$^{2}$ range, are found to be A$_{T}^{(2)}$  = − 0.23 ± 0.23 ± 0.05 and A$_{T}^{Im}$  = 0.14 ± 0.22 ± 0.05. The results are consistent with Standard Model predictions.An angular analysis of the $B^0 \rightarrow K^{*0} e^+ e^-$ decay is performed using a data sample, corresponding to an integrated luminosity of 3.0 {\mbox{fb}^{-1}}, collected by the LHCb experiment in $pp$ collisions at centre-of-mass energies of 7 and 8 TeV during 2011 and 2012. For the first time several observables are measured in the dielectron mass squared ($q^2$) interval between 0.002 and 1.120${\mathrm{\,Ge\kern -0.1em V^2\!/}c^4}$. The angular observables $F_{\mathrm{L}}$ and $A_{\mathrm{T}}^{\mathrm{Re}}$ which are related to the $K^{*0}$ polarisation and to the lepton forward-backward asymmetry, are measured to be $F_{\mathrm{L}}= 0.16 \pm 0.06 \pm0.03$ and $A_{\mathrm{T}}^{\mathrm{Re}} = 0.10 \pm 0.18 \pm 0.05$, where the first uncertainty is statistical and the second systematic. The angular observables $A_{\mathrm{T}}^{(2)}$ and $A_{\mathrm{T}}^{\mathrm{Im}}$ which are sensitive to the photon polarisation in this $q^2$ range, are found to be $A_{\mathrm{T}}^{(2)} = -0.23 \pm 0.23 \pm 0.05$ and $A_{\mathrm{T}}^{\mathrm{Im}} =0.14 \pm 0.22 \pm 0.05$. The results are consistent with Standard Model predictions

Precise measurements of the properties of the B-1(5721)(0,+) and B-2*(5747)(0,+) states and observation of B-+,B-0 pi(-,+) mass structures

Invariant mass distributions of $B^+\pi^-$ and $B^0\pi^+$ combinations are investigated in order to study excited B mesons. The analysis is based on a data sample corresponding to $3.0 fb^{-1}$ of $pp$ collision data, recorded by the LHCb detector at centre-of-mass energies of 7 and 8 TeV. Precise measurements of the masses and widths of the $B_1(5721)^{0,+}$ and $B_2^*(5747)^{0,+}$ states are reported. Clear enhancements, particularly prominent at high pion transverse momentum, are seen over background in the mass range $5850$-$6000$ MeV in both $B^+\pi^-$ and $B^0\pi^+$ combinations. The structures are consistent with the presence of four excited B mesons, labelled $B_J(5840)^{0,+}$ and $B_J(5960)^{0,+}$, whose masses and widths are obtained under different hypotheses for their quantum numbers.Invariant mass distributions of B$^{+}$ π$^{−}$ and B$^{0}$ π$^{+}$ combinations are investigated in order to study excited B mesons. The analysis is based on a data sample corresponding to 3.0 fb$^{−1}$ of pp collision data, recorded by the LHCb detector at centre-of-mass energies of 7 and 8 TeV. Precise measurements of the masses and widths of the B$_{1}$(5721)$^{0,+}$ and B$^{2}$(5747)$^{0,+}$ states are reported. Clear enhancements, particularly prominent at high pion transverse momentum, are seen over background in the mass range 5850-6000 MeV in both B$^{+}$ π$^{−}$ and B$^{0}$ π$^{+}$ combinations. The structures are consistent with the presence of four excited B mesons, labelled B$_{J}$ (5840)$^{0,+}$ and B$_{J}$ (5960)$^{0,+}$, whose masses and widths are obtained under different hypotheses for their quantum numbers.Invariant mass distributions of B+pi- and B0pi+ combinations are investigated in order to study excited B mesons. The analysis is based on a data sample corresponding to 3.0 fb-1 of pp collision data, recorded by the LHCb detector at centre-of-mass energies of 7 and 8 TeV. Precise measurements of the masses and widths of the B_1(5721)^(0,+) and B_2*(5747)^(0,+) states are reported. Clear enhancements, particularly prominent at high pion transverse momentum, are seen over background in the mass range 5850--6000 MeV in both B+pi- and B0pi+ combinations. The structures are consistent with the presence of four excited B mesons, labelled B_J(5840)^(0,+) and B_J(5960)^(0,+), whose masses and widths are obtained under different hypotheses for their quantum numbers