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Abstrak :
Bintang boson statis dengan simetri bola merupakan solusi dari sistem persamaan Einstein yang terkopel dengan persamaan relativistik Klein-Gordon dan Proca dari medan skalar kompleks pada simetri lokal U(1). Perumusan sistem persamaan Einstein yang terkopel dengan persamaan relativistik Klein-Gordon dan Proca dilakukan di ruang lengkung yang kemudian metriknya dilimitkan ke ruang datar. Sistem persamaan tersebut diasumsikan tidak homogen sehingga menyebabkan tekanan pada arah tangensial berbeda dengan pada arah radial yang menunjukkan sifat anisotropis dari persamaan keadaan bintang boson. Penyelesaian sistem persamaan tersebut dilakukan secara numerik untuk melihat persamaan keadaan yang terbentuk pada bintang boson akibat adanya asumsi inhomogen tersebut. Perhitungan untuk kasus homogen juga dilakukan dengan menghilangkan suku-suku derivatif sistem persamaan tersebut di atas. Hasil dari perhitungan menunjukkan bahwa dengan asumsi inhomogen, syarat batas sistem persamaan tidak terpenuhi dan menyebabkan persamaan keadaan bintang yang terbentuk tidak stabil. Untuk kasus homogen, hasil yang didapatkan dapat menggambarkan keadaan bintang boson yang stabil. ......Spherically symmetric static boson stars are solutions of the system of equations of Einstein equation which is coupled to the Klein-Gordon and Proca equation with complex scalar field with U(1) gauge symmetry. We look for the system of equations of Einstein equation which is coupled to the Klein-Gordon and Proca equation in the curved space then we limit the metric to the flat space. The system of equations is assumed to be inhomogeneous so, the pressure in direction of tangential is different to the pressure in radial direction that shows the anisotropic equations of state (EOS).We find numerically solutions to see the equations of state which are formed in boson stars as the consequence of inhomogeneous assumption. We also find the solutions for the case of homogeneous assumption with omitting the derivative terms of that system of equations. The result shows that with the inhomogeneous assumption, the boundary conditions cannot be fulfilled and causes the EOS unstable. For the case of homogeneous assumption, the result can describe the stable EOS of the boson stars.

Abstrak :
ABSTRAK
Teknologi Internet of Things (IoT) menjadi salah satu kebutuhan yang terus meningkat, dan tentunya memunculkan risiko dan tantangan keamanan informasi. Mengingat serangan terhadap perangkat IoT juga semakin meningkat, aspek keamanan menjadi bagian utama dalam implementasi IoT salah satunya adalah teknik kriptografi. Dari segi teknik kriptografi, lightweight cryptography dibutuhkan untuk memenuhi aspek keamanan informasi sekaligus didesain untuk diterapkan pada perangkat IoT. PRESENT merupakan salah satu algoritma block cipher ultra lightweight yang banyak diimplementasikan karena telah teruji ringan dan cepat, serta termasuk dalam salah satu algoritma lightweight yang direkomendasikan pada standar ISO/IEC 29192-2. Namun beberapa peneliti telah melakukan analisis kelemahan algoritma PRESENT terhadap suatu cryptanalysis salah satunya adalah improbable differential cryptanalysis. Improbable differential cryptanalysis merupakan gabungan dari metode impossible differential characteristic dan differential characteristic. Metode improbable differential cryptanalysis ini memanfaatkan karakteristik unik berupa undisturbed bit dari s-box PRESENT untuk membentuk pola dalam melakukan analisis cryptanalysis. Oleh karena itu, pada penelitian ini dilakukan analisis dan pengembangan algoritma modifikasi PRESENT berdasarkan ketahanannya terhadap potensi improbable differential cryptanalysis. Modifikasi algoritma dilakukan dengan mengganti s-box PRESENT menggunakan 9 (sembilan) pilihan s-box yang meliputi 4 (empat) s-box SERPENT, s-box BORON, s-box KLEIN, s-box LED, s-box RECTANGLE, dan s-box NES. Analisa yang dilakukan menggunakan uji Strict Avalanche Criterion (SAC), uji Differential Approximation Probability (DAP), dan analisa terhadap probabilitas karakteristik improbable differential yang dapat dibentuk. Berdasarkan hasil penelitian, substitution box KLEIN menghasilkan nilai uji SAC dan DAP yang paling baik dibandingkan 9 s-box lainnya yaitu memiliki nilai SAC rata-rata sebesar 0.59375 dan nilai DAP tertinggi sebesar 0.25 sebanyak 15. Serta berdasarkan hasil analisa improbable differential, algoritma modifikasi PRESENT yang menggunakan s-box KLEIN memiliki probabilitas terendah yaitu sebesar . Hal ini menunjukkan bahwa algoritma modifikasi PRESENT menggunakan s-box KLEIN memiliki ketahanan yang lebih baik terhadap potensi dilakukannya improbable differential cryptanalysis.

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ABSTRACT
Internet of Things (IoT) technology increased for needed, and it raises risks and challenges of information security. Considering that attacks on IoT devices are increasing, security aspects become a part important in the implementation of IoT, one of which is cryptography. In terms of cryptography techniques, lightweight cryptography is needed to comply with the information security aspects and designed to be applied to IoT devices. PRESENT is one of the ultra-lightweight block cipher algorithms that has been implemented because it has been tested small and fast. PRESENT is included in one of the lightweight algorithms recommended in the ISO/IEC 29192-2 standard, but some researchers have analyzed algorithm weaknesses. Improbable differential cryptanalysis is a combination of impossible differential characteristics and differential characteristics. This improbable differential cryptanalysis method uses a unique characteristic consisting of the uninterrupted bits of the PRESENT s-box to create a pattern for conducting cryptanalysis. Therefore, in this research, an analysis of the PRESENT modification algorithm is based on its resistance to improbable differential cryptanalysis potential. Algorithm modification is done by replacing PRESENT S-box using 9 (nine) s-box options, which include 4 (four) SERPENT s-boxes, BORON s-boxes, KLEIN s-boxes, LED s-boxes, RECTANGLE s-boxes, and NES s-boxes. The analysis is performed using Strict Avalanche Criterion (SAC) test, Differential Approximation Probability (DAP) test, and analysis of the probability of improbable differential characteristics that can be formed. Based on the results of the research, KLEIN substitution box produces the best SAC and DAP test values compared to 9 other s-boxes, which have an average SAC value of 0.59375 and the highest DAP value of 0.25 of 15. And based on the results of improbable differential analysis, PRESENT modification algorithm which use the KLEIN s-box has the lowest probability of. This shows that the PRESENT modification algorithm using the KLEIN s-box has better resistance to the potential for improbable differential cryptanalysis.

Abstrak :
This book was inspired by the general observation that the great theories of modern physics are based on simple and transparent underlying mathematical structures – a fact not usually emphasized in standard physics textbooks – which makes it easy for mathematicians to understand their basic features. It is a textbook on quantum theory intended for advanced undergraduate or graduate students: mathematics students interested in modern physics, and physics students who are interested in the mathematical background of physics and are dissatisfied with the level of rigor in standard physics courses. More generally, it offers a valuable resource for all mathematicians interested in modern physics, and all physicists looking for a higher degree of mathematical precision with regard to the basic concepts in their field.