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Scalar

scalar

Table of Links Acknowledgements 1 Introduction to thesis 1.1 History and Evidence 1.2 Facts on dark matter 1.3 Candidates to dark matter 1.4 Dark matter detection 1.5 Outline of the thesis 2 Dark matter through ALP portal and 2.1 Introduction 2.2 Model 2.3 Existing constraints on ALP parameter space 2.4 Dark matter analysis 2.5 Summary 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3.2 Model 3.3 Theoretical and experimental constraints 3.4 Phenomenology of dark matter 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.6 Summary 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4.2 Model 4.3 Theoretical and experimental constraints 4.4 Dark Matter analysis 4.5 Summary 5 Summary Appendices A Standard model B Friedmann equations C Type I seasaw mechanism D Feynman diagrams in two-component DM model Bibliography 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM In this chapter, we study a pseudo-scalar DM in a generic U(1)𝑋 extension of SM. The results are based on the work: Shivam Gola, "Pseudo scalar dark matter in a generic U(1)𝑋 model, Phys.Lett.B 842 (2023) 137982” 4.1 Introduction WIMP-type dark matter is heavily constrained by direct experimental searches [32, 184]; therefore, to continue with WIMP, one has to find a way to evade the stringent direct detection bounds. A pseudo-scalar DM can naturally evade the strong direct detection constraints as it has derivative couplings that imply a momentum suppression in the tree-level DM-nucleon scattering matrix, which vanishes in the non-relativistic limit [185–192]. Hence, it is interesting to seek a pseudo-scalar particle as a WIMP dark matter. In this chapter, we consider a generic U(1)𝑋 model for pseudo-scalar DM and discuss its phenomenological implications in this study. The interesting aspect of the 𝑈(1)𝑋 models is that the three generations of right-handed neutrinos (RHNs) are required to eliminate the gauge and mixed gauge-gravity anomalies [144–147]. The RHNs mix with active neutrinos of SM via type I seesaw mechanism [71–74] to generate the required light neutrino masses and flavor mixing. A model that explains both the neutrino mass problem and the nature of dark matter would be a major step forward in high-energy physics [148–153]. The pseudo-scalar is not protected by any symmetry; therefore, for it to be a DM candidate, it must have a lifetime much greater than the age of the universe. We study the feasible parameter space allowed by the lifetime constraint on our pseudo-scalar particle, then we scan for the allowed parameter space by relic and direct detection bounds while respecting several other theoretical and experimental constraints. The chapter is organised as follows: In sec. 4.2, we introduce the model and discuss the details of the new fields and their interactions. In sec. 4.3, we discuss some of the relevant theoretical and experimental constraints. In sec. 4.4, we discuss relic density, direct detection, and other phenomenologically relevant studies. Finally, in sec. 4.5, we summarize the chapter. This paper is available on arxiv under CC BY 4.0 DEED license. Author:
(1) Shivam Gola, The Institute of Mathematical Sciences, Chennai. Table of Links Acknowledgements Acknowledgements 1 Introduction to thesis 1 Introduction to thesis 1.1 History and Evidence 1.1 History and Evidence 1.2 Facts on dark matter 1.2 Facts on dark matter 1.3 Candidates to dark matter 1.3 Candidates to dark matter 1.4 Dark matter detection 1.4 Dark matter detection 1.5 Outline of the thesis 1.5 Outline of the thesis 2 Dark matter through ALP portal and 2.1 Introduction 2 Dark matter through ALP portal and 2.1 Introduction 2.2 Model 2.2 Model 2.3 Existing constraints on ALP parameter space 2.3 Existing constraints on ALP parameter space 2.4 Dark matter analysis 2.4 Dark matter analysis 2.5 Summary 2.5 Summary 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3.2 Model 3.2 Model 3.3 Theoretical and experimental constraints 3.3 Theoretical and experimental constraints 3.4 Phenomenology of dark matter 3.4 Phenomenology of dark matter 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.6 Summary 3.6 Summary 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4.2 Model 4.2 Model 4.3 Theoretical and experimental constraints 4.3 Theoretical and experimental constraints 4.4 Dark Matter analysis 4.4 Dark Matter analysis 4.5 Summary 4.5 Summary 5 Summary 5 Summary Appendices Appendices A Standard model A Standard model B Friedmann equations B Friedmann equations C Type I seasaw mechanism C Type I seasaw mechanism D Feynman diagrams in two-component DM model D Feynman diagrams in two-component DM model Bibliography Bibliography 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM In this chapter, we study a pseudo-scalar DM in a generic U(1)𝑋 extension of SM. The results are based on the work: Shivam Gola, "Pseudo scalar dark matter in a generic U(1)𝑋 model, Phys.Lett.B 842 (2023) 137982” 4.1 Introduction WIMP-type dark matter is heavily constrained by direct experimental searches [32, 184]; therefore, to continue with WIMP, one has to find a way to evade the stringent direct detection bounds. A pseudo-scalar DM can naturally evade the strong direct detection constraints as it has derivative couplings that imply a momentum suppression in the tree-level DM-nucleon scattering matrix, which vanishes in the non-relativistic limit [185–192]. Hence, it is interesting to seek a pseudo-scalar particle as a WIMP dark matter. In this chapter, we consider a generic U(1)𝑋 model for pseudo-scalar DM and discuss its phenomenological implications in this study. The interesting aspect of the 𝑈(1)𝑋 models is that the three generations of right-handed neutrinos (RHNs) are required to eliminate the gauge and mixed gauge-gravity anomalies [144–147]. The RHNs mix with active neutrinos of SM via type I seesaw mechanism [71–74] to generate the required light neutrino masses and flavor mixing. A model that explains both the neutrino mass problem and the nature of dark matter would be a major step forward in high-energy physics [148–153]. The pseudo-scalar is not protected by any symmetry; therefore, for it to be a DM candidate, it must have a lifetime much greater than the age of the universe. We study the feasible parameter space allowed by the lifetime constraint on our pseudo-scalar particle, then we scan for the allowed parameter space by relic and direct detection bounds while respecting several other theoretical and experimental constraints. The chapter is organised as follows: In sec. 4.2, we introduce the model and discuss the details of the new fields and their interactions. In sec. 4.3, we discuss some of the relevant theoretical and experimental constraints. In sec. 4.4, we discuss relic density, direct detection, and other phenomenologically relevant studies. Finally, in sec. 4.5, we summarize the chapter. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Author: (1) Shivam Gola, The Institute of Mathematical Sciences, Chennai. Author: Author: (1) Shivam Gola, The Institute of Mathematical Sciences, Chennai.

A Pseudo-Scalar Dark Matter Case: An Introduction to Get You Up to Speed

Table of Links Acknowledgements 1 Introduction to thesis 1.1 History and Evidence 1.2 Facts on dark matter 1.3 Candidates to dark matter 1.4 Dark matter detection 1.5 Outline of the thesis 2 Dark matter through ALP portal and 2.1 Introduction 2.2 Model 2.3 Existing constraints on ALP parameter space 2.4 Dark matter analysis 2.5 Summary 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3.2 Model 3.3 Theoretical and experimental constraints 3.4 Phenomenology of dark matter 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.6 Summary 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4.2 Model 4.3 Theoretical and experimental constraints 4.4 Dark Matter analysis 4.5 Summary 5 Summary Appendices A Standard model B Friedmann equations C Type I seasaw mechanism D Feynman diagrams in two-component DM model Bibliography 3.2 Model In the next subsections, we discuss various parts of the lagrangian of the model, 3.2.1 Scalar Sector where 𝛼 is the mixing angle. The rotation matrix satisfies The real and imaginary components of 𝜒 have the following masses 3.2.2 Gauge sector To determine the gauge boson spectrum, we have to expand the scalar kinetic terms and replace 3.2.3 Yukawa sector The Yukawa sector of the model can be written in a gauge-invariant way as This paper is available on arxiv under CC BY 4.0 DEED license. Author:
(1) Shivam Gola, The Institute of Mathematical Sciences, Chennai. Table of Links Acknowledgements Acknowledgements 1 Introduction to thesis 1 Introduction to thesis 1.1 History and Evidence 1.1 History and Evidence 1.2 Facts on dark matter 1.2 Facts on dark matter 1.3 Candidates to dark matter 1.3 Candidates to dark matter 1.4 Dark matter detection 1.4 Dark matter detection 1.5 Outline of the thesis 1.5 Outline of the thesis 2 Dark matter through ALP portal and 2.1 Introduction 2 Dark matter through ALP portal and 2.1 Introduction 2.2 Model 2.2 Model 2.3 Existing constraints on ALP parameter space 2.3 Existing constraints on ALP parameter space 2.4 Dark matter analysis 2.4 Dark matter analysis 2.5 Summary 2.5 Summary 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3.2 Model 3.2 Model 3.3 Theoretical and experimental constraints 3.3 Theoretical and experimental constraints 3.4 Phenomenology of dark matter 3.4 Phenomenology of dark matter 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.6 Summary 3.6 Summary 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4.2 Model 4.2 Model 4.3 Theoretical and experimental constraints 4.3 Theoretical and experimental constraints 4.4 Dark Matter analysis 4.4 Dark Matter analysis 4.5 Summary 4.5 Summary 5 Summary 5 Summary Appendices Appendices A Standard model A Standard model B Friedmann equations B Friedmann equations C Type I seasaw mechanism C Type I seasaw mechanism D Feynman diagrams in two-component DM model D Feynman diagrams in two-component DM model Bibliography Bibliography 3.2 Model In the next subsections, we discuss various parts of the lagrangian of the model, 3.2.1 Scalar Sector where 𝛼 is the mixing angle. The rotation matrix satisfies The real and imaginary components of 𝜒 have the following masses 3.2.2 Gauge sector To determine the gauge boson spectrum, we have to expand the scalar kinetic terms and replace 3.2.3 Yukawa sector The Yukawa sector of the model can be written in a gauge-invariant way as This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Author: (1) Shivam Gola, The Institute of Mathematical Sciences, Chennai. Author: Author: (1) Shivam Gola, The Institute of Mathematical Sciences, Chennai.

A Phenomenological Study of WIMP Models: Scalar Sector, Gauge Sector, and More

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the random signal in frequency space (c.f. Eq. (2.9)) takes the following form where (and following the notation of [45]) Proceeding similarly as in Appendix B, the marginalized likelihood is which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We arrive at the following: where This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the random signal in frequency space (c.f. Eq. (2.9)) takes the following form where (and following the notation of [45]) Proceeding similarly as in Appendix B, the marginalized likelihood is which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We arrive at the following: where This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Exploring the Gradient of a Scalar in Dark Matter Velocity Analysis

Let be a cyclic group, is the element in the group, is the scalar, and the Multi-Scalar Multiplication (MSM) is the algorithm for calculating , the sum of multiple scalar multiplications. As the group operation is more complicated than the addition and multiplication of elements in the finite field, the MSM algorithm is employed to reduce the number of times of group operation as much as possible. Usually, is a cyclic group defined on the elliptic curve of on the finite field of , with the order being bits, and . If the group operation is conducted by the plain algorithm of fast exponentiation, the number of group operations needed for every is times on average, and for all is times in total. Next, we will introduce some optimization methods corresponding to larger . G P_i ∈ G a_i ∈ Z ∑_{i=1}^{n} a_{i} P_{i} G y^2=x^3+ax+b F_p | | G b ≈ 256 0 ≤ a_i< 2^b a_i.P_i 1.5bn a_i.P_i 1.5b n Optimization Based on the Windowing Technique We can split the bit scalar into c width windows: reduce the scalar to base system. b 2^c Here, } , so we reduce the original algorithm issue of -bit MSM into a smaller -bit issue. We can put the same scalars from into a bucket and write them in another form: Q_j=∑_{i=1}^{n a_{i j} P_{i} b c ∑_{i=1}^{n} a_{i j} P_{i} For example, when , it is calculated as: c = 3 When calculating , set the partial sum T_l=S_l+⋯+S_k, then ∑kS_k However, the operation results of can be obtained through the recurrence sequence . It means that only one group operation is needed for the operation for each . Under the MSM algorithm with cc-bit scalar, all requires times of group operation. All requires times and the sum of these requires times of group operation. Therefore, each window with the value of c requires times of group operations. The operation of } only requires the normal method of fast exponentiation by times of group operation. T_l T_l=T_l+1+S_l T_l S_k n−2^c+1 T_k 2^ 1 c− T_k ^c−2 2 n+2^c−2 ∑_{j=0}^{b / c-1} Q_{j} 2^{j c (b/c−1)(c+1)=b − c + b/c−1 In summary, we can conclude that the total number of group operations required by the method of the Windowing Technique of a width of is c Set , then the number of group operations is . When , the number of group operations reduces to of its original value. c = logn 2bn/logn n≈10^5 1/12 Optimization Based on Group Endomorphism For the cyclic group on the elliptic curve on the finite field F_p, if the following group endomorphism of φ can be found: there exists , satisfying holds for all the points on , then it is easy to prove that such an endomorphism is a multiplication map, namely there is a making hold for all the points of on , which means that whenever we know the coordinates of one point, we can change it to the coordinates of another point simply by multiplying a number in to both the values of abscissa and ordinate. The algorithm can be further optimized based on this important property. G y^2=x^3+ax+b α,β ∈ F_p φ(x,y)=(αx,βy) G λ φ(P)=λP P G F_p When . The reverse value of one point can be obtained only by taking the opposite number from the ordinate. Based on this feature, in the plain fast exponentiation algorithm, the scalar can be written into the non-adjacent form (NAF), namely and any two adjacent cannot be non-zero at the same time. Compared to the -bit scalars, the group operation number in the fast exponentiation algorithm can be reduced from the original average of times to times. This technique can also be used to optimize the Windowing Technique. After writing to λ = −1, α = 1, β = −1 e_{i}.2^{i}, e_{i} ∈ {−1,0,1} ∑ e_i b 3/2⋅b 4/3⋅b a_i a_i=∑_{j=0}^{b / c-1} a_{i, j} 2^{j c}, 0 <= a_{i, j}<2^{c} conduct the following operation: The result of and - can be obtained. Since the complexity of addition and subtraction in group operation of an elliptic curve is the same, we can put these terms into a bucket according to the absolute value of scalars. Therefore, the number of groups can be reduced from the original 2c to , and the overall number of group operations required is reduced from (1) to: a_i=∑_{j=0}^{b / c-1} a’_{i, j}^2^{j c} 2^{c-1} <= a’_{i, j}^ <= 2^{c-1} 2^c If the parameters of the elliptic curve are special, for example, the BLS curve can be written as , and , take third-order element α from F_p^x, then there is a corresponding that holds y^2=x^3+b p ≡ 1( mod3 ) λ λ( )=( ) x,y αx,y Algorithm 1: Signed Digits In addition, . Then the multiplication operation can be optimized as follows: λ3 ≡ 1(mod |G| ) From [1], we can find a small enough scalar | |, | |<√ to make the above equation true, which gives us a way to reduce the multiplication of bb bits into the multiplication of two bits. Use it into the Windowing Technique, and the number of group operations is reduced to m1 m2 G b/2 In summary, both the above two optimization methods can reduce the number of group operations by when . 5.5% n=10^5 References [1] Francesco Sica, Mathieu Ciet, and Jean-Jacques Quisquater. Analysis of the gallant-lambert-vanstone method based on eﬀicient endomorphisms: Elliptic and hyperelliptic curves. In International Workshop on Selected Areas in Cryptography, pages 21–36. Springer, 2002.

Optimization of Multi-Scalar Multiplication Algorithm: Sin7Y Tech Review (21)

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Authors:
(1) Sheng Liu, School of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, Hunan, China;
(2) Chengqing Li (Corresponding author), School of Computer Science, Xiangtan University, Xiangtan 411105, Hunan, China;
(3) Qiao Hu, School of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, Hunan, China. Authors: Authors: (1) Sheng Liu, School of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, Hunan, China; (2) Chengqing Li (Corresponding author), School of Computer Science, Xiangtan University, Xiangtan 411105, Hunan, China; (3) Qiao Hu, School of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, Hunan, China. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Description of two analyzed image encryption algorithms 2. Description of two analyzed image encryption algorithms 3. Cryptanalysis of IEATD 3. Cryptanalysis of IEATD 4. Cryptanalysis of IEACD 4. Cryptanalysis of IEACD 5. Conclusion, Acknowledgements, and References 5. Conclusion, Acknowledgements, and References Abstract Security is a key problem for the transmission, interchange and storage process of multimedia systems and applications. In 2018, M. Li et al. proposed in-depth security analysis on an image encryption algorithm based on a first-order time-delay system (IEATD) and gave a specific chosen-plaintext attack on it. Moreover, an enhanced version called as IEACD was designed to fix the reported security defects. This paper analyzes the essential structures of the two algorithms and evaluates their real security performances: 1) no efficient nonlinear operations are adopted to assure the sensibility of keystream; 2) the equivalent secret key of IEATD can be efficiently recovered from one known plain-image and the corresponding cipherimage; 3) IEACD can still be efficiently cracked with a chosen-plaintext attack. Both rigorous theoretical analyses and detailed experimental results are provided to demonstrate effectiveness of the advanced cryptanalytic methods. 1. Introduction Social media not only drive product discovery and purchase, but also incur serious concern on the security and privacy of the images shared by the Internet users. Due to the special properties of multimedia information, the modern text encryption standards, such as AES and Triple DES, cannot efficiently protect them in general. To cope with the challenge, a number of special image encryption algorithms, e.g. joint encryption and compression together, were proposed every year [1, 2, 3]. It is well known that cryptography (designing encryption algorithm) and cryptanalysis (security analysis of a given encryption algorithm) are two integral parts of cryptology. The cryptanalysis results facilitate the designers strengthen or replace flawed algorithms. Cryptanalysis of a given image encryption scheme also provides a special perspective for promoting some multimedia processing techniques, e.g. image recovery. Some image encryption algorithms like that proposed in [4, 2] are found to be insecure to different extents from the viewpoint of modern cryptology [5, 6, 7, 8]. The complex dynamics of a chaotic system demonstrated in an infinite-precision domain is very similar to the expected properties of a secure encryption scheme outlined ∗Corresponding author. Email address: DrChengqingLi@gmail.com (Chengqing Li) by Shannon in [9]. So, a large number of chaos-based encryption schemes were proposed in the past three decades [10, 11]. In [12], Ikeda adopted a one-variable differential difference equation to model light going around a ring cavity containing a nonlinear dielectric medium and found “chaotic” phenomena in the transmitted field. In [4], Mannai et al. introduced the equation’s variant as a chaos-based pseudorandom number generator (PRNG), where α and m are coefficients, and T is the positive delay time. The evolution of the dynamics is dependent on not only the present value x(t) but also earlier one x(t − T). To solve equation (1), it is discretized with the following way: 1) each interval T is divided into N subintervals h and each subinterval h is approximated with a scalar value, where T = N × h; 2) the N samples of each interval are considered as an N-dimension vector. In [4], an image encryption algorithm based on a time-delay Ikeda system (IEATD) was proposed. The designers of IEATD believed that utilizing the rich dynamics of a discretized Ikeda system and a new keystream generation mechanism associated with the average of all pixels of the plain-image can provide sufficient capacity to withstand known/chosen-plaintext attacks. In reality, the security strength of IEATD is very weak as its equivalent secret key can be obtained with only two chosen plain-images [13]. Meanwhile, M. Li et al. pointed out that two security defects exist in IEATD: 1) the regularity of the keystream and absence of position scrambling; 2) incapacity to resist differential attack [13]. To remedy the defects, they adopted much more complex encryption operations: permutation and crossover diffusion phases. In short, we call the enhanced image encryption algorithm using the crossover diffusion as IEACD. This paper focuses on security analysis of the two image encryption algorithms, IEATD [4] and IEACD [13]. We found that the authors of [13] did not notice a fatal drawback of the keystream generation mechanism: insensibility to minor change of a pixel. This leads to that IEACD still cannot withstand chosen-plaintext attack. Furthermore, there is improper keystream configuration in diffusion that almost discloses the whole keystream. The essential structures of the two algorithms cause that the equivalent secret key of IEATD and IEACD can be recovered with known-plaintext attack and chosen-plaintext attack, respectively. The rest of this paper is organized as follows. Section 2 concisely describes the encryption procedures of the encryption algorithm IEATD and its enhanced version IEACD. Then, Sec. 3 and Sec. 4 present the detailed cryptanalysis results on the two encryption algorithms, respectively. The last section concludes the paper. 2. Description of two analyzed image encryption algorithms 2.1. The framework of IEATD 2.1. The framework of IEATD 2.2. The enhanced elements of IEACD compared with IEATD 2.2. The enhanced elements of IEACD compared with IEATD To enhance the security level of IEATD, some extra operations were appended to withstand the chosen-plaintext attack reported in [13]. • The added secret sub-keys: a positive integer C, the initial condition q(0) ∈ (0, 1) and the control parameter β ∈ (3.5699456, 4] of Logistic map • The added secret sub-keys: • The modified encryption procedures: • The modified encryption procedures: 3. Cryptanalysis of IEATD 3.2. The chosen-plaintext attack proposed by M. Li et al. 3.2. The chosen-plaintext attack proposed by M. Li et al. To make the cryptanalysis of IEATD more complete, we briefly introduce the chosen-plaintext attack on IEATD proposed by M. Li et al. in [13] and comment its performance: 3.3. Known-plaintext attack on IEATD 3.3. Known-plaintext attack on IEATD This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv

Can One Plain Image Break an Entire Encryption Algorithm?

Caption: embeddings and its complexities Caption embeddings and its complexities READ PART-1 here (https://hackernoon.com/hallucination-by-design-how-embedding-models-misunderstand-language) READ PART-1 here (https://hackernoon.com/hallucination-by-design-how-embedding-models-misunderstand-language) https://hackernoon.com/hallucination-by-design-how-embedding-models-misunderstand-language Last month, I shared how embedding models hallucinate when handling simple language variations like negation and capitalization. The response was overwhelming – seems I'm not the only one who's been burnt by these issues. Today, I'm diving deeper into even more troubling blind spots I've discovered through testing. These are the kinds that keep me up at night and make me question everything about how we're building AI systems. This is the second part in the series on Hallucinations by Design. It is a continuation of our previous discussion on how embeddings hallucinate.  To get the most out of this article, I highly recommend reading the linked article first, as it lays the foundational concepts necessary to fully grasp the ideas explored here. By doing so, you'll have a seamless learning experience and a deeper understanding of the topic. how embeddings hallucinate Hypothetical vs. actual? Just details! Here's where things get truly disturbing. When I ran "If the treatment works, symptoms should improve" against "The treatment works and symptoms have improved", the similarity score hit 0.95. I sat staring at my screen in disbelief. One's speculating about potential outcomes; the other's reporting confirmed results! When I ran "If the treatment works, symptoms should improve" against "The treatment works and symptoms have improved", the similarity score hit 0.95. I hit this problem working on a clinical research document. The search couldn't distinguish between hypothesized treatment outcomes and verified results. Doctors searching for proven treatments were getting mixed results with unproven hypotheses. Do you think physicians making treatment decisions appreciate confusing speculation with evidence? I am sure I wouldn't want my medical care based on "might work" rather than "does work". Again, think about all the cases where distinguishing hypotheticals from facts is essential - scientific research, medical trials, legal precedents, and investment analyses. When your model conflates "if X then possibly Y" with "X happened and caused Y", you've completely misunderstood the epistemic status of the information. We're basically working with models that can't tell the difference between speculation and confirmation despite analyzing text where this distinction determines whether something is reliable information or mere conjecture. Temporal order? Whatever order! Embedding models see "She completed her degree before starting the job" and "She started her job before completing her degree" as NEARLY identical – ridiculous 0.97 similarity score. One's a traditional career path; the other's working while studying. Completely different situations! She completed her degree before starting the job" and "She started her job before completing her degree" as NEARLY identical – ridiculous 0.97 similarity score. I found this while building a resume screening system. The embeddings couldn't distinguish between candidates who finished their degrees before working and those who were still completing studies. Hiring managers wasted hours interviewing candidates who didn't meet their basic qualification requirements. Do you think busy recruiters appreciate having their time wasted with mismatched candidates? I am sure I wouldn't want my hiring pipeline filled with noise. Think about all the cases where sequence is crucial – medical treatment protocols, legal procedural requirements, cooking recipes, assembly instructions, and chemical formulations. When your model can't tell "A before B" from "B before A," you've lost fundamental causal relationships. We're basically working with models that treat time as an optional concept despite analyzing text that's full of critical sequential information. Quantitative thresholds vanish into thin air This one actually made me spill my coffee. Embedding models see "The company barely exceeded earnings expectations" and "The company significantly missed earnings expectations" as SHOCKINGLY similar – 0.93 similarity score. Exceeded versus missed! These mean opposite things in finance! "The company barely exceeded earnings expectations" and "The company significantly missed earnings expectations" as SHOCKINGLY similar – 0.93 similarity score. If you are building a financial news analysis system, the embeddings wouldn’t distinguish between positive and negative earnings surprises – literally the difference between stock prices going up or down. Investors making trading decisions based on our summaries were getting completely contradictory information. Do you think people risking actual money appreciate getting fundamentally wrong market signals? I am sure I wouldn't want my retirement account guided by such confusion. Now, think about all the cases where crossing a threshold changes everything – passing vs. failing grades, healthy vs. dangerous vital signs, profitable vs. unprofitable businesses, compliant vs. non-compliant regulatory statuses. Your model loses its ability to make meaningful distinctions when it cannot distinguish between barely meeting the target and completely missing it. We're basically working with models that don't understand the concept of thresholds despite analyzing text that's constantly discussing whether targets were met or missed. Scalar inversions get completely flipped The absurdity just keeps piling up. During testing, I found that "The meeting ran significantly shorter than planned" and "The meeting ran significantly longer than planned" scored a 0.96 similarity. I was in complete shock. These sentences describe completely opposite situations – time saved versus time wasted! "The meeting ran significantly shorter than planned" and "The meeting ran significantly longer than planned" scored a 0.96 similarity. I encountered this with project management documents. The search couldn't distinguish between schedule overruns and efficiencies. Managers searching for examples of time-saving techniques were getting shown projects with serious delays. Do you think executives tracking project timelines appreciate getting the exact opposite information they asked for? I am sure I would be furious if I were preparing for a board meeting with such backward data. Think about all the cases where direction on a scale is crucial – cost savings vs. overruns, performance improvements vs. degradations, health improvements vs. declines, and risk increases vs. decreases. When your model treats "much higher than" as interchangeable with "much lower than", you've lost the ability to track directional change. We're basically working with models that don't understand opposing directions despite analyzing text filled with comparative assessments. Domain-specific opposites look like synonyms Medical documents I couldn't believe what I was seeing in the healthcare tests. "The patient presents with tachycardia" versus "The patient presents with bradycardia" returned a 0.94 similarity score. For non-medical folks, that's like confusing a racing heart with one that's dangerously slow – conditions with opposite treatments! "The patient presents with tachycardia" versus "The patient presents with bradycardia" returned a 0.94 similarity score. I discovered this while working on a symptom-matching system for electronic health records. The embedding model couldn't distinguish between fundamentally different medical conditions that require opposite treatments. Physicians searching for cases similar to a patient with a racing heart were shown cases of patients with dangerously slow heartbeats. Do you think doctors making time-sensitive decisions appreciate getting contradictory clinical information? I am sure I wouldn't want my treatment based on the opposite of my actual condition. In the field of medicine, these distinctions can have significant consequences. Tachycardia might be treated with beta-blockers, while bradycardia might require a pacemaker – giving the wrong treatment could be fatal. We're basically working with models that can't distinguish between opposite medical conditions despite analyzing text where this distinction determines appropriate care. Legal documents The legal tests were just as bad. When comparing "Plaintiff bears the burden of proof" with "Defendant bears the burden of proof", the model returned a staggering 0.97 similarity. Let that sink in. These statements literally determine which side has to prove their case in court! Mixing these up could lose you your lawsuit. When comparing "Plaintiff bears the burden of proof" with "Defendant bears the burden of proof", the model returned a staggering 0.97 similarity. The search couldn't distinguish between fundamentally different legal standards and responsibilities. Lawyers researching precedents about plaintiff burdens were shown cases discussing defendant burdens. Do you think attorneys preparing for trial appreciate getting precisely backward legal standards? I am sure I wouldn't want my lawsuit built on completely inverted legal principles. In legal contexts, who bears the burden of proof often determines the outcome of a case. When your model can't distinguish which party has which responsibilities, you've undermined the entire basis of legal reasoning. We're basically working with models that confuse legal roles despite analyzing text where these distinctions define how justice functions. Units of measurement I had to run this test multiple times because I couldn't believe the results. "The procedure takes about 5 minutes" versus "The procedure takes about 5 hours" scored a whopping 0.97 similarity. Is this for real? That's a 60x time difference! Imagine waiting for your "5-minute" appointment that actually takes 5 hours. "The procedure takes about 5 minutes" versus "The procedure takes about 5 hours" scored a whopping 0.97 similarity I found this while building the same healthcare system. The embeddings couldn't distinguish between brief and lengthy procedures. Clinic managers trying to schedule short procedures were being shown lengthy operations that would block their surgery suites for entire days. Do you think medical facilities with tight scheduling constraints appreciate having their entire day's workflow disrupted? I am sure I wouldn't want my hospital running 60x behind schedule. Units of measurement fundamentally change meaning. When your model treats "5 minutes" and "5 hours" as essentially identical, you've lost the ability to understand magnitude. We're basically working with models that ignore units despite analyzing text where units determine whether something is trivial or significant. More measurement problems And it just gets worse from there. During the use of the same healthcare documents, I found "The tumor is 2 centimeters in diameter" and "The tumor is 2 inches in diameter" scored an alarming 0.98 similarity. For context, that's the difference between a potentially minor tumor and one that's 2.54x larger – often the threshold between "watch and wait" versus immediate surgery. "The tumor is 2 centimeters in diameter" and "The tumor is 2 inches in diameter" scored an alarming 0.98 similarity. The embeddings couldn't distinguish between metric and imperial measurements. Oncologists researching treatment options for small tumors were being shown cases of much larger growths. Do you think cancer specialists appreciate getting case studies that aren't remotely comparable to their patients? Even speed limits get confused. Models treat "Maintain speeds under 30 mph" and "Maintain speeds under 30 kph" as HIGHLY similar – a problematic 0.96 similarity score. That's the difference between 30 and 18.6 miles per hour – enough to determine whether an accident is fatal! Even speed limits get confused. Models treat "Maintain speeds under 30 mph" and "Maintain speeds under 30 kph" as HIGHLY similar – a problematic 0.96 similarity score. That's the difference between 30 and 18.6 miles per hour – enough to determine whether an accident is fatal! Converting between units isn't just a mathematical exercise – it fundamentally changes recommendations, safety parameters, and outcomes. We're basically working with models that think numbers without units are sufficient despite analyzing text where the units completely transform the meaning. The Truth and the Results Here is the comparison between msmarco-distilbert-base-tas-b, all-mpnet-base-v2, and open-ai-text-embedding-3-large, and you will notice that there is no significant difference between the output of these models. msmarco-distilbert-base-tas-b, all-mpnet-base-v2, and open-ai-text-embedding-3-large, and you will notice that there is no significant difference between the output of these models. ***msmarco-distilbert-base-tas-b embedding score across different test cases*** ***msmarco-distilbert-base-tas-b embedding score across different test cases*** ***all-mpnet-base-v2 embedding score across different test cases*** ***all-mpnet-base-v2 embedding score across different test cases*** ***openai-text-embedding-3-large embedding score across different test cases*** ***openai-text-embedding-3-large embedding score across different test cases*** Just to repeat.. Look, embeddings are amazingly useful despite these problems. I'm not advocating against using them, but rather, it's crucial to approach them cautiously. Here's my battle-tested advice after dozens of projects and countless failures: Test your model on real user language patterns before deployment. Not academic benchmarks, not sanitized test cases – actual examples of how your users communicate. We built a "linguistic stress test" toolkit that simulates common variations like negations, typos, and numerical differences. Every system we test fails in some areas – the question is whether those areas matter for your specific application.


Build guardrails around critical blind spots. Different applications have different can't-fail requirements. For healthcare, it's typically negation and entity precision. For finance, it's numbers and temporal relationships. For legal, it's conditions and obligations. Identify what absolutely can't go wrong in your domain, and implement specialized safeguards.


Layer different techniques instead of betting everything on embeddings. Our most successful systems combine embedding-based retrieval with keyword verification, explicit rule checks, and specialized classifiers for critical distinctions. This redundancy isn't inefficient; it's essential.


Be transparent with users about what the system can and can't do reliably. We added confidence scores that explicitly flag when a result might involve negation, numerical comparison, or other potential weak points. Users appreciate the honesty, and it builds trust in the system overall. Test your model on real user language patterns before deployment. Not academic benchmarks, not sanitized test cases – actual examples of how your users communicate. We built a "linguistic stress test" toolkit that simulates common variations like negations, typos, and numerical differences. Every system we test fails in some areas – the question is whether those areas matter for your specific application. Test your model on real user language patterns before deployment. Not academic benchmarks, not sanitized test cases – actual examples of how your users communicate. We built a "linguistic stress test" toolkit that simulates common variations like negations, typos, and numerical differences. Every system we test fails in some areas – the question is whether those areas matter for your specific application. Build guardrails around critical blind spots. Different applications have different can't-fail requirements. For healthcare, it's typically negation and entity precision. For finance, it's numbers and temporal relationships. For legal, it's conditions and obligations. Identify what absolutely can't go wrong in your domain, and implement specialized safeguards. Build guardrails around critical blind spots. Different applications have different can't-fail requirements. For healthcare, it's typically negation and entity precision. For finance, it's numbers and temporal relationships. For legal, it's conditions and obligations. Identify what absolutely can't go wrong in your domain, and implement specialized safeguards. Layer different techniques instead of betting everything on embeddings. Our most successful systems combine embedding-based retrieval with keyword verification, explicit rule checks, and specialized classifiers for critical distinctions. This redundancy isn't inefficient; it's essential. Layer different techniques instead of betting everything on embeddings. Our most successful systems combine embedding-based retrieval with keyword verification, explicit rule checks, and specialized classifiers for critical distinctions. This redundancy isn't inefficient; it's essential. Be transparent with users about what the system can and can't do reliably. We added confidence scores that explicitly flag when a result might involve negation, numerical comparison, or other potential weak points. Users appreciate the honesty, and it builds trust in the system overall. Be transparent with users about what the system can and can't do reliably. We added confidence scores that explicitly flag when a result might involve negation, numerical comparison, or other potential weak points. Users appreciate the honesty, and it builds trust in the system overall. **Here's the most important thing I've learnt:**these models don't understand language the way humans do – they understand statistical patterns. When I stopped expecting human-like understanding and started treating them as sophisticated pattern-matching tools with specific blind spots, my systems got better. Much better. The blind spots I've described aren't going away anytime soon – they're baked into how these models work. But if you know they're there, you can design around them. And sometimes, acknowledging a limitation is the first step toward overcoming it. Note: I have many more such cases found through experiments, and I will cover them in my subsequent post. Note Note The next continuation article will be coming out soon. Stay tuned!!

Hallucinations by Design (Part 2): The Silent Flaws of Embeddings & Why Your AI Is Getting It Wrong

ቅዠቶች በንድፍ (ክፍል 2)፡ የመክተት ፀጥታ ጉድለቶች እና የእርስዎ AI ለምን እየተሳሳተ ነው

Halucinacije po dizajnu (2. dio): tihe mane ugrađivanja i zašto vaš AI griješi

Hallucinaciones por diseño (Parte 2): Las carencias silenciosas de las incorporaciones y por qué tu IA se está equivocando

設計による幻覚（パート 2）：埋め込みの隠れた欠陥と AI が間違える理由

यो बर्ष महानगरको अभियान — सम्पदा संरक्षण तथा सम्बर्धन, पर्यटन प्रवर्धनका लागि पूर्वाधार Next Post

Halucynacje według projektu (część 2): ciche wady osadów i dlaczego Twoja sztuczna inteligencja robi to źle

Alucinações por Design (Parte 2): As falhas silenciosas dos embeddings e por que sua IA está errando

Галюцинации по дизайну (часть 2): молчаливые недостатки встраиваний и почему ваш ИИ ошибается

Hallucinations by Igishushanyo (Igice cya 2): Amakosa acecetse yo gushushanya & Impamvu AI yawe irimo Kubona nabi

ERROR: invalid target_lang (tk)

Задумані галюцинації (частина 2): тихі недоліки вбудовування та чому ваш штучний інтелект помиляється

Authors:
(1) Kinjal Basu, IBM Research;
(2) Keerthiram Murugesan, IBM Research;
(3) Subhajit Chaudhury, IBM Research;
(4) Murray Campbell, IBM Research;
(5) Kartik Talamadupula, Symbl.ai;
(6) Tim Klinger, IBM Research. Authors: Authors: (1) Kinjal Basu, IBM Research; (2) Keerthiram Murugesan, IBM Research; (3) Subhajit Chaudhury, IBM Research; (4) Murray Campbell, IBM Research; (5) Kartik Talamadupula, Symbl.ai; (6) Tim Klinger, IBM Research. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Background 2 Background 3 Symbolic Policy Learner 3 Symbolic Policy Learner 3.1 Learning Symbolic Policy using ILP 3.1 Learning Symbolic Policy using ILP 3.2 Exception Learning 3.2 Exception Learning 4 Rule Generalization 4 Rule Generalization 4.1 Dynamic Rule Generalization 4.1 Dynamic Rule Generalization 5 Experiments and Results 5.1 Dataset 5.1 Dataset 5.2 Experiments 5.2 Experiments 5.3 Results 5.3 Results 6 Related Work 6 Related Work 7 Future Work and Conclusion, Limitations, Ethics Statement, and References 7 Future Work and Conclusion, Limitations, Ethics Statement, and References 2 Background Text-based Reinforcement Learning: TBGs provide a challenging environment where an agent can observe the current state of the game and act in the world using only the modality of text. The agent perceives the state of the game only through natural language observations. Hence, TBGs can be modeled as a Partially Observable Markov Decision Process (POMDP) (S, A, O, T , E, r), where S is the set of states of the game, A is the natural language action space, O is the set of textual observations describing the current state, T are the conditional transition probabilities from one state to another, E are the conditional observation probabilities, r : S × A → R is a scalar reward function, which maps a state-action pair to the reward received by the agent. Text-based Reinforcement Learning: Inductive Logic Programming (ILP): ILP is a machine learning technique where the learned model is in the form of logic programming rules (Horn Clauses) that are comprehensible to humans. It allows the background knowledge to be incrementally extended without requiring the entire model to be re-learned. Additionally, the comprehensibility of symbolic rules makes it easier for users to understand and verify induced models and even edit them. Details can be found in the work of Muggleton and De Raedt (1994). Inductive Logic Programming (ILP): Answer Set Programming (ASP): An answer set program is a collection of rules of the form: Answer Set Programming (ASP): s(CASP) Engine: For this work, we have used s(CASP) ASP solver to predict an action. s(CASP) (Arias et al., 2018) is a query-driven, goal-directed implementation of ASP that includes constraint solving over reals. Goal-directed execution of s(CASP) is indispensable for automating commonsense reasoning, as traditional grounding and SATsolver based implementations of ASP may not be scalable. There are three major advantages of using the s(CASP) system: (i) s(CASP) does not ground the program, which makes our framework scalable, (ii) it only explores the parts of the knowledge base that are needed to answer a query, and (iii) it provides natural language justification (proof tree) for an answer (Arias et al., 2020). s(CASP) Engine: 3 Symbolic Policy Learner Deep reinforcement learning (DRL) has experienced great success by learning directly from high-dimensional sensory inputs, yet it suffers from a lack of interpretability. Interpretability of an agent’s action is of utmost importance in sequential decision-making problems, as it increases the transparency of black-box-style agents; it also helps RL researchers understand the high-level behavior of the system better. To make a system interpretable, one of the most widely used approaches is learning the agent’s policies symbolically. In our work, EXPLORER learns these symbolic policies in the form of logical rules represented in the ASP. An example of such a rule is - insert(X, fridge) :- apple(X) which can be translated as “X is insertable into a fridge if X is an apple” [2]. These learned ASP rules not only provide a better understanding of the system’s functionality but can also be used to predict the agent’s next action using an ASP solver. EXPLORER learns the rules iteratively (in an online manner) and applies the rules to predict an action in collaboration with the neural module. Our results show that this approach is very effective in terms of performance and interpretability. Partial Observability and Non-Monotonic Reasoning: EXPLORER works in a partially observable environment, where it needs to predict an action based on its prior knowledge. If EXPLORER fails, then it learns something new that will be applied in the next episode. The reasoning approach of EXPLORER is non-monotonic in nature: that is, what it believes currently may become false in the future with new evidence. We can model this using a non-monotonic logic programming paradigm that supports default rules and exception to defaults (Gelfond and Kahl, 2014). In this work, the belief of EXPLORER has been represented as an Answer Set Program in the form of default rules with exceptions. With the help of Inductive Logic Programming (ILP) (see Section 3.1) and Exception learner (see Section 3.2), these rules are learned by EXPLORER after each episode and then applied in the following episode. The agent uses an ASP solver to predict actions by utilizing the observation and the rules. Based on the outcome after applying the rules, the learned policies are updated with the exception (if needed), and new rules are learned as needed. Partial Observability and Non-Monotonic Reasoning: This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv [2] For ease of use, we retain action names as the predicate names; however, they are interpreted normally as adjectives.

Learning Logic in Games: How EXPLORER Combines NLP and RL for

Table of Links Abstract and 1. Introduction Clock in Orbit
2.1 Coordinate Time
2.2 Local Frame for the Moon


Clock Rate Differences Between Earth and Moon


Clocks at Earth-Moon Lagrance Points
4.1 Clock at Lagrange point L1
4.2. Clock at Lagrange point L2
4.3. Clock at Lagrange point L4 or L5


Conclusions Appendix 1: Fermi Coordinates with Origin at the Center of the Moon Appendix 2: Construction of Freely Falling Center of Mass Frame Appendix 3: Equations of Motion of Earth and Moon Appendix 4: Comparing Results in Rotating and Non-Rotating Coordinate Systems Acknowledgments and References APPENDIX 4: COMPARING RESULTS IN ROTATING AND NON-ROTATING COORDINATE SYSTEMS We calculate the fractional difference between a clock on the Moon’s surface and a clock on the Earth’s surface in three different coordinate systems. These are (1) the center-of-mass locally inertial system; (2) a rotating system in which the x−axis is along the Earth-Moon line; and (3) a translated, rotating system in which the Earth is at the origin of coordinates and the Earth-Moon line is in the x′ direction. We show that in all three coordinate systems, the fractional rate difference is the same. A Keplerian orbit is assumed for the Earth-Moon system. To simplify the calculations we assume that the clocks are on the surfaces of the respective bodies. This is an approximation that can be refined when the actual positions of the clocks are specified. 4.1 Center-Of-Mass inertial coordinate system The scalar invariant in the locally inertial frame whose origin is at the center of mass of the Earth-Moon system, neglecting tidal terms, is The difference in the last term represents a difference of squares of velocities. 4.2 Rotating center-of-mass coordinates Introduce a rotating system with an Earth-Moon line along the new x-axis: X = x cos(f) − y sin(f); Y = x sin(f) + y cos(f); T = t. Then The scalar invariant becomes For a clock on the Moon, For a clock on the Moon, there is no contribution from the Sagnac term. The proper time interval is Note that there is a significant contribution from the centrifugal potential. For a clock on Earth, The proper time interval is then The fractional proper time interval difference reduces exactly to the expression in Eq. (120). 4.3 Rotating coordinates with Earth at origin This implies the use of a coordinate system in which the Earth is not moving. This has to be a rotating coordinate system with its origin coinciding with the Earth’s center. Therefore translating the origin to the center of the Earth, with no change in the time variable, The scalar invariant becomes The potentials in the last term have been suppressed since they do not contribute to the order of this calculation. For a clock on the surface of the Moon, There is no contribution from the Sagnac term but there is a significant contribution from the centrifugal potential, representing the transverse velocity of the Moon. The radial velocity of the Moon comes from the spatial part of the metric. The proper time interval for such a clock is For a clock on the surface of the Earth, x′ = y′ = x′ = y′ = 0. The proper time interval is It is easily seen that the fractional proper time difference reduces to expressions that have been previously derived. Thus in all three coordinate systems, the fractional proper time difference is the same. Acknowledgments We would like to acknowledge the funding we received from the NASA grant NNH12AT81I. We are grateful to Elizabeth Donley, who carefully and critically reviewed the manuscript and provided valuable suggestions. We would also like to express our gratitude to Cheryl Gramling for initiating discussions on lunar time. We extend our sincere thanks to Roger Brown, Thomas Heavner, Judah Levine, Jeffrey Sherman, and Daniel Slichter for their review of the manuscript. This work is a contribution of NIST and is not subject to US copyright. [1] Artemis Plan: NASA’s Lunar Exploration Program Overview, https://www.nasa.gov/sites/default/files/atoms/files/ artemis plan-20200921.pdf, Sept. 2020 [2] Einstein, A. The Collected Papers of Albert Einstein, Volume 6: The Berlin Years: Writings, 1914-1917 Princeton University Press, 1996. [3] Park RS, Folkner WM, Williams JG, and Boggs DH, The JPL Planetary and lunar Ephemerides DE440 and DE441, AJ. 161 105, 2021 [4] Ashby N, Relativity in the Global Positioning System, Liv. Rev. Relativity 6, [Online Article]: cited Sept. 14, 2023, http://www.livingreviews.org/lrr-2003-1. [5] International Astronomical Union (IAU), Proceedings of the Twenty-Fourth General Assembly,” Transactions of the IAU, XXIVB,pp. 34-57,2000 [6] Ashby N and Bertotti B, Relativistic effects in local inertial frames, Phys. Rev. D 14(8), 1986 (Appendix), 2246-2259. [7] Ashby N and Allan D, Practical Implications of Relativity for a Global Time Scale, Radio Science 14, 1979, pp 649-669. [8] Bertone, S, Arnold, D, Girardin, V, Lasser, M, Meyer, U, J¨aggi, A, Assessing Reduced-Dynamic Parametrizations for GRAIL Orbit Determination and the Recovery of Independent lunar Gravity Field Solutions, Earth and Space Science, 2021, https://doi.org/10.1029/2020EA001454. [9] Ries JC, Eanes RJ, Shum CK, Watkins MM (20 March 1992). ”Progress in the determination of the gravitational coefficient of the Earth”. Geophysical Research Letters. 19 (6): 529–531. [10] CODATA 2006, physics.nist.gov/cuu/Constants [11] Mohr PJ, Taylor BN, and Newell DB, The CODATA recommended values of the fundamental physical constants: 2006, Rev. Mod. Phys., 80, pp. 633-730, 2008 [12] Konopliv AS, et al., The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL Primary Mission, J. Geophys. Res. Planets, 118, 1415–1434, 2013, doi:10.1002/jgre.20097 [13] Daher, H et al., Long-Term Earth-Moon Evolution With High-Level Orbit and Ocean Tide Models, Journal of Geophysical Research: Planets, 126(12), 2021, e2021JE006875. https://doi.org/10.1029/2021JE006875 [14] Petit G and Wolf P, Relativistic theory for time comparisons: a review Metrologia, 42 (3), 2005, S138 Authors:
(1) Neil Ashby, National Institute of Standards and Technology, Boulder, CO 80305 (Neil.Ashby@Colorado.Edu);
(2) Bijunath R. Patla, National Institute of Standards and Technology, Boulder, CO 80305 (bijunath.patla@nist.gov). This paper is available on arxiv under CC0 1.0 license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Clock in Orbit
2.1 Coordinate Time
2.2 Local Frame for the Moon Clock Rate Differences Between Earth and Moon Clocks at Earth-Moon Lagrance Points
4.1 Clock at Lagrange point L1
4.2. Clock at Lagrange point L2
4.3. Clock at Lagrange point L4 or L5 Conclusions Clock in Orbit 2.1 Coordinate Time 2.2 Local Frame for the Moon Clock in Orbit Clock in Orbit 2.1 Coordinate Time 2.1 Coordinate Time 2.2 Local Frame for the Moon 2.2 Local Frame for the Moon Clock Rate Differences Between Earth and Moon Clock Rate Differences Between Earth and Moon Clock Rate Differences Between Earth and Moon Clocks at Earth-Moon Lagrance Points 4.1 Clock at Lagrange point L1 4.2. Clock at Lagrange point L2 4.3. Clock at Lagrange point L4 or L5 Clocks at Earth-Moon Lagrance Points 4.1 Clock at Lagrange point L1 4.1 Clock at Lagrange point L1 4.2. Clock at Lagrange point L2 4.2. Clock at Lagrange point L2 4.3. Clock at Lagrange point L4 or L5 4.3. Clock at Lagrange point L4 or L5 Conclusions Conclusions Conclusions Appendix 1: Fermi Coordinates with Origin at the Center of the Moon Appendix 1: Fermi Coordinates with Origin at the Center of the Moon Appendix 2: Construction of Freely Falling Center of Mass Frame Appendix 2: Construction of Freely Falling Center of Mass Frame Appendix 3: Equations of Motion of Earth and Moon Appendix 3: Equations of Motion of Earth and Moon Appendix 4: Comparing Results in Rotating and Non-Rotating Coordinate Systems Appendix 4: Comparing Results in Rotating and Non-Rotating Coordinate Systems Acknowledgments and References Acknowledgments and References APPENDIX 4: COMPARING RESULTS IN ROTATING AND NON-ROTATING COORDINATE SYSTEMS We calculate the fractional difference between a clock on the Moon’s surface and a clock on the Earth’s surface in three different coordinate systems. These are (1) the center-of-mass locally inertial system; (2) a rotating system in which the x−axis is along the Earth-Moon line; and (3) a translated, rotating system in which the Earth is at the origin of coordinates and the Earth-Moon line is in the x′ direction. We show that in all three coordinate systems, the fractional rate difference is the same. A Keplerian orbit is assumed for the Earth-Moon system. To simplify the calculations we assume that the clocks are on the surfaces of the respective bodies. This is an approximation that can be refined when the actual positions of the clocks are specified. 4.1 Center-Of-Mass inertial coordinate system The scalar invariant in the locally inertial frame whose origin is at the center of mass of the Earth-Moon system, neglecting tidal terms, is The difference in the last term represents a difference of squares of velocities. 4.2 Rotating center-of-mass coordinates Introduce a rotating system with an Earth-Moon line along the new x-axis: X = x cos(f) − y sin(f); Y = x sin(f) + y cos(f); T = t. Then The scalar invariant becomes For a clock on the Moon, For a clock on the Moon, there is no contribution from the Sagnac term. The proper time interval is Note that there is a significant contribution from the centrifugal potential. For a clock on Earth, The proper time interval is then The fractional proper time interval difference reduces exactly to the expression in Eq. (120). 4.3 Rotating coordinates with Earth at origin This implies the use of a coordinate system in which the Earth is not moving. This has to be a rotating coordinate system with its origin coinciding with the Earth’s center. Therefore translating the origin to the center of the Earth, with no change in the time variable, The scalar invariant becomes The potentials in the last term have been suppressed since they do not contribute to the order of this calculation. For a clock on the surface of the Moon, There is no contribution from the Sagnac term but there is a significant contribution from the centrifugal potential, representing the transverse velocity of the Moon. The radial velocity of the Moon comes from the spatial part of the metric. The proper time interval for such a clock is For a clock on the surface of the Earth, x′ = y′ = x′ = y′ = 0. The proper time interval is It is easily seen that the fractional proper time difference reduces to expressions that have been previously derived. Thus in all three coordinate systems, the fractional proper time difference is the same. Acknowledgments We would like to acknowledge the funding we received from the NASA grant NNH12AT81I. We are grateful to Elizabeth Donley, who carefully and critically reviewed the manuscript and provided valuable suggestions. We would also like to express our gratitude to Cheryl Gramling for initiating discussions on lunar time. We extend our sincere thanks to Roger Brown, Thomas Heavner, Judah Levine, Jeffrey Sherman, and Daniel Slichter for their review of the manuscript. This work is a contribution of NIST and is not subject to US copyright. [1] Artemis Plan: NASA’s Lunar Exploration Program Overview, https://www.nasa.gov/sites/default/files/atoms/files/ artemis plan-20200921.pdf, Sept. 2020 [2] Einstein, A. The Collected Papers of Albert Einstein, Volume 6: The Berlin Years: Writings, 1914-1917 Princeton University Press, 1996. [3] Park RS, Folkner WM, Williams JG, and Boggs DH, The JPL Planetary and lunar Ephemerides DE440 and DE441, AJ. 161 105, 2021 [4] Ashby N, Relativity in the Global Positioning System, Liv. Rev. Relativity 6, [Online Article]: cited Sept. 14, 2023, http://www.livingreviews.org/lrr-2003-1. [5] International Astronomical Union (IAU), Proceedings of the Twenty-Fourth General Assembly,” Transactions of the IAU, XXIVB,pp. 34-57,2000 [6] Ashby N and Bertotti B, Relativistic effects in local inertial frames, Phys. Rev. D 14(8), 1986 (Appendix), 2246-2259. [7] Ashby N and Allan D, Practical Implications of Relativity for a Global Time Scale, Radio Science 14, 1979, pp 649-669. [8] Bertone, S, Arnold, D, Girardin, V, Lasser, M, Meyer, U, J¨aggi, A, Assessing Reduced-Dynamic Parametrizations for GRAIL Orbit Determination and the Recovery of Independent lunar Gravity Field Solutions, Earth and Space Science, 2021, https://doi.org/10.1029/2020EA001454. [9] Ries JC, Eanes RJ, Shum CK, Watkins MM (20 March 1992). ”Progress in the determination of the gravitational coefficient of the Earth”. Geophysical Research Letters. 19 (6): 529–531. [10] CODATA 2006, physics.nist.gov/cuu/Constants [11] Mohr PJ, Taylor BN, and Newell DB, The CODATA recommended values of the fundamental physical constants: 2006, Rev. Mod. Phys., 80, pp. 633-730, 2008 [12] Konopliv AS, et al., The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL Primary Mission, J. Geophys. Res. Planets, 118, 1415–1434, 2013, doi:10.1002/jgre.20097 [13] Daher, H et al., Long-Term Earth-Moon Evolution With High-Level Orbit and Ocean Tide Models, Journal of Geophysical Research: Planets, 126(12), 2021, e2021JE006875. https://doi.org/10.1029/2021JE006875 [14] Petit G and Wolf P, Relativistic theory for time comparisons: a review Metrologia, 42 (3), 2005, S138 Authors: (1) Neil Ashby, National Institute of Standards and Technology, Boulder, CO 80305 (Neil.Ashby@Colorado.Edu); (2) Bijunath R. Patla, National Institute of Standards and Technology, Boulder, CO 80305 (bijunath.patla@nist.gov). Authors: Authors: (1) Neil Ashby, National Institute of Standards and Technology, Boulder, CO 80305 ( Neil.Ashby@Colorado.Edu ); Neil.Ashby@Colorado.Edu (2) Bijunath R. Patla, National Institute of Standards and Technology, Boulder, CO 80305 ( bijunath.patla@nist.gov ). bijunath.patla@nist.gov This paper is available on arxiv under CC0 1.0 license. This paper is available on arxiv under CC0 1.0 license. available on arxiv available on arxiv

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