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"Elliptic Curve Digital Signature Algorithm (ECDSA) adalah algoritma penandatanganan digital yang menggunakan elliptic curve. ECDSA terdiri dari tiga tahapan, pembentukan kunci, pembentukan tanda tangan digital, dan algoritma verifikasi tanda tangan digital. ECDSA digunakan pada transaksi dengan Bitcoin. Ada penelitian terkait ECDSA weak randomness yang dapat dieksploitasi untuk mengungkap kunci privat pengguna. ECDSA weak randomness adalah melakukan generating pada bilangan random yang tidak aman secara kriptografi. Ada beberapa modifikasi yang dilakukan untuk mencegah eksploitasi ECDSA weak randomness. Namun, modifikasi tersebut rentan terhadap Rho method attack yang bisa memecahkan Elliptic Curve Discrete Logarithm Problem (ECDLP). Jika ECDLP diselesaikan, kunci privat pengguna dapat terungkap. Oleh karena itu, pada penelitian ini, akan dikonstruksi ECDSA yang tahan terhadap eksploitasi ECDSA weak randomness dan Rho method attack dengan menggunakan tiga kunci privat.

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Elliptic Curve Digital Signature Algorithm (ECDSA) is a digital signature algorithm that utilizes an elliptic curve. ECDSA consists of three steps, which are key generation, signature generation, and verification algorithm. ECDSA is used on Bitcoin transactions to generate the user’s public key, private key, and signature, and also to verify a Bitcoin user’s signature. There are some researches on ECDSA weak randomness which can be exploited by attackers to reveal the user’s private key, and causes thefts of the user’s money. ECDSA weak randomness is generating a random number that is not cryptographically secure. Some modifications of ECDSA to overcome this problem have been done, such as generating the digital signature by using two private keys. Although those modified algorithms overcome ECDSA weak randomness exploitations, it is not resistant to the Rho method attack which can solve elliptic curve discrete logarithm problem (ECDLP). In case ECDLP can be solved, the user’s private key can be revealed. Therefore, in this research, we modify an ECDSA algorithm that overcomes the exploitation of ECDSA weak randomness and is also resistant to the Rho method attack by using three private keys."

"ABSTRAKTujuan utama dari keamanan komunikasi adalah mencegah data pesan dari akses pihak ketiga. Solusi dari masalah tersebut adalah menerapkan enkripsi End-to-end. Algoritma enkripsi Elliptic Curve Cryptography (ECC) cocok digunakan sebagai metode enkripsi End-to-end pada sistem komunikasi digital termasuk aplikasi chat. ECC adalah jenis kriptografi kunci publik yang mendasarkan keamanannya pada permasalahan matematis dari kurva eliptik, ECC mempunyai tingkat keamanan yang setara dengan RSA menggunakan kunci berukuran lebih kecil. Penelitian ini membahas perancangan dan implementasi dari algoritma enkripsi dan dekripsi yang menerapkan kriptografi ECC sebagai metode enkripsi End-to-end pada aplikasi chat. Algoritma ini menggunakan operasi matematika dan nilai parameter dari kurva eliptik untuk menghasilkan kunci serta ciphertextnya dan juga melakukan dekripsi. Dari hasil ujicoba dan analisa implementasi enkripsi dan dekripsi menggunakan algoritma ECC, algoritma tersebut cocok digunakan sebagai metode enkripsi End-to-end, karena ukuran kunci yang relatif ringan dan operasi matematika yang menggunakan parameter khusus yang berbeda-beda. Hasil pengujian enkripsi dan dekripsi menggunakan parameter kurva eliptik secp192r1 dan secp192k1 yang mempunyai ukuran kunci sekitar 2¹⁹² bit, menghasilkan kunci dan ciphertext paling ringan dan paling cepat diproses, dengan angka nilai keduanya berukuran sepanjang 58 digit.

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ABSTRACTThe main purpose of communication security is to prevent message data from third party access. The solution to this problem is to implement End-to-end encryption. Elliptic Curve Cryptography (ECC) encryption algorithm is suitable for use as an End-to-end encryption method on digital communication systems including chat applications. ECC is a type of public key cryptography that bases its security on mathematical problems from elliptic curves, ECC has a level of security equals to RSA while using smaller keys. This study discusses the design and implementation of encryption and decryption algorithms that apply ECC cryptography as an End-to-end encryption method in chat applications. This algorithm uses mathematical operations and parameter values from the elliptic curve to generate keys and their ciphertext and also decrypt them. From the results of testing and analyzing the implementation of encryption and decryption using the ECC algorithm, the algorithm is suitable for use as an End-to-end encryption method, because of the relatively light key size and mathematical operations that use different parameters. The results of encryption and decryption testing using elliptic curve parameters secp192r1 and secp192k1 which have a key size of about 2¹⁹² bits, produce the lightest and fastest to process ciphertext and keys, with the value of both measuring 58 digits.

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"A concise survey of the current state of knowledge in 1972 about solving elliptic boundary-value eigenvalue problems with the help of a computer. This volume provides a case study in scientific computing -- the art of utilizing physical intuition, mathematical theorems and algorithms, and modern computer technology to construct and explore realistic models of problems arising in the natural sciences and engineering."

"The Finite Element Method for Elliptic Problems is the only book available that analyzes in depth the mathematical foundations of the finite element method. It is a valuable reference and introduction to current research on the numerical analysis of the finite element method, as well as a working textbook for graduate courses in numerical analysis. It includes many useful figures, and there are many exercises of varying difficulty.Although nearly 25 years have passed since this book was first published, the majority of its content remains up-to-date. Chapters 1 through 6, which cover the basic error estimates for elliptic problems, are still the best available sources for material on this topic. The material covered in Chapters 7 and 8, however, has undergone considerable progress in terms of new applications of the finite element method; therefore, the author provides, in the Preface to the Classics Edition, a bibliography of recent texts that complement the classic material in these chapters."

"Mengukur tingkat polusi udara berarti mengukur tingkat konsentrasipolutan dalam udara. Pengukuran tersebut dilakukan dengan pendekatan modelmatematika sederhana berupa Persamaan Diferensial Parsial Order Dua.Berdasarkan model sederhana tersebut yang diperoleh dari Model DanishEulerian, akan dicari solusi eksaknya (jika ada) dan solusi numeriknya. Adapunmetode eksak yang digunakan adalah Metode Transformasi Laplace pada ruang.Sedangkan untuk memperoleh solusi numeriknya, digunakan Metode CrankNicolson."