This document provides an introduction to ECDSA and its use in Bitcoin. It begins by explaining how Bitcoin uses a blockchain to record transactions and ensure their authenticity through digital signatures. ECDSA is the algorithm used to generate these signatures. The document then discusses public key cryptography and how ECDSA works. It introduces concepts like elliptic curves, finite fields, point addition/doubling, and scalar multiplication which are important to understanding how ECDSA generates public/private key pairs to digitally sign transactions on the Bitcoin blockchain.
The document describes Identity-Based Encryption (IBE), including the key algorithms involved: Setup, Extract, Encrypt, Decrypt. It explains that IBE allows encrypting messages for arbitrary string identities like email addresses, without needing a public key. The PKG runs Setup to generate parameters and a master key, and Extract to generate private keys from identities. Encrypt uses an identity and message to create a ciphertext, while Decrypt uses the private key to recover the message. Applications include key revocation and delegation. Security relies on bilinear pairings on elliptic curves.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
Implementation of Elliptic Curve Digital Signature Algorithm Using Variable T...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Data Security Using Elliptic Curve CryptographyIJCERT
Cryptography technique is used to provide data security. In existing cryptography technique the key generation takes place randomly. Key generation require shared key. If shared key is access by unauthorized user then security becomes disoriented. Hence existing problems are alleviated to give more security to data. In proposed system a algorithm called as Elliptic Curve Cryptography is used. The ECC generates the key by using the point on the curve. The ECC is used for generating the key by using point on the curve and encryption and decryption operation takes place through curve. In the proposed system the encryption and key generation process takes place rapidly.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
This document discusses the implementation of Elliptic Curve Digital Signature Algorithm (ECDSA) using variable text message encryption methods. It begins with an abstract that outlines ECDSA, its advantages over other digital signature algorithms like smaller key size, and implementation of ECDSA over elliptic curves P-192 and P-256 with variable size text message, fixed size text message, and text based message encryption. It then provides details on elliptic curve cryptography, the elliptic curve discrete logarithm problem, finite fields, and domain parameters for ECDSA.
Improved security system using steganography and elliptic curve crypto...atanuanwesha
The main objectives of the project is to make the data safe and secure and transmit the data in such a way that it is not possible for anyone to detect the data . Steganography is concealing the secret message in non secret image. Whereas Encryption is converting data into code to prevent unauthorized access .Steganography as well as cryptography has its own disadvantage. Our objective is to implement both the procedures to enforce tight security and to prevent evesdropping etc.
Elliptic curve cryptography (ECC) uses elliptic curves over finite fields to provide public-key encryption and digital signatures. ECC requires significantly smaller key sizes than other cryptosystems like RSA to provide equivalent security. This allows for faster computations and less storage requirements, making ECC ideal for constrained environments like smartphones. ECC relies on the difficulty of solving the elliptic curve discrete logarithm problem to provide security.
This document provides an overview of number theory and its applications to asymmetric key cryptography. It begins with definitions of prime numbers, relatively prime numbers, and modular arithmetic. It then covers the Euclidean algorithm for finding the greatest common divisor of two numbers, Fermat's and Euler's theorems, and the Chinese Remainder Theorem. The document concludes with an introduction to public key cryptography, including the basic principles, requirements, and the RSA algorithm as a widely used example of an asymmetric encryption scheme.
The document describes Identity-Based Encryption (IBE), including the key algorithms involved: Setup, Extract, Encrypt, Decrypt. It explains that IBE allows encrypting messages for arbitrary string identities like email addresses, without needing a public key. The PKG runs Setup to generate parameters and a master key, and Extract to generate private keys from identities. Encrypt uses an identity and message to create a ciphertext, while Decrypt uses the private key to recover the message. Applications include key revocation and delegation. Security relies on bilinear pairings on elliptic curves.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
Implementation of Elliptic Curve Digital Signature Algorithm Using Variable T...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Data Security Using Elliptic Curve CryptographyIJCERT
Cryptography technique is used to provide data security. In existing cryptography technique the key generation takes place randomly. Key generation require shared key. If shared key is access by unauthorized user then security becomes disoriented. Hence existing problems are alleviated to give more security to data. In proposed system a algorithm called as Elliptic Curve Cryptography is used. The ECC generates the key by using the point on the curve. The ECC is used for generating the key by using point on the curve and encryption and decryption operation takes place through curve. In the proposed system the encryption and key generation process takes place rapidly.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
This document discusses the implementation of Elliptic Curve Digital Signature Algorithm (ECDSA) using variable text message encryption methods. It begins with an abstract that outlines ECDSA, its advantages over other digital signature algorithms like smaller key size, and implementation of ECDSA over elliptic curves P-192 and P-256 with variable size text message, fixed size text message, and text based message encryption. It then provides details on elliptic curve cryptography, the elliptic curve discrete logarithm problem, finite fields, and domain parameters for ECDSA.
Improved security system using steganography and elliptic curve crypto...atanuanwesha
The main objectives of the project is to make the data safe and secure and transmit the data in such a way that it is not possible for anyone to detect the data . Steganography is concealing the secret message in non secret image. Whereas Encryption is converting data into code to prevent unauthorized access .Steganography as well as cryptography has its own disadvantage. Our objective is to implement both the procedures to enforce tight security and to prevent evesdropping etc.
Elliptic curve cryptography (ECC) uses elliptic curves over finite fields to provide public-key encryption and digital signatures. ECC requires significantly smaller key sizes than other cryptosystems like RSA to provide equivalent security. This allows for faster computations and less storage requirements, making ECC ideal for constrained environments like smartphones. ECC relies on the difficulty of solving the elliptic curve discrete logarithm problem to provide security.
This document provides an overview of number theory and its applications to asymmetric key cryptography. It begins with definitions of prime numbers, relatively prime numbers, and modular arithmetic. It then covers the Euclidean algorithm for finding the greatest common divisor of two numbers, Fermat's and Euler's theorems, and the Chinese Remainder Theorem. The document concludes with an introduction to public key cryptography, including the basic principles, requirements, and the RSA algorithm as a widely used example of an asymmetric encryption scheme.
We allow Eve to modify DH parameters as well as public keys of Alice and Bob. This allows Eve to derive the secret key and break the DH crypto system. We demonstrate that the DH key exchange algorithm should not be used without digital signatures.
Linear cryptanalysis is a method used to break encryption standards like DES. It involves finding linear approximations between plaintext, ciphertext, and key bits that hold with probability greater than 50%. These approximations are used to determine partial key bits using maximum likelihood algorithms on known or ciphertext-only data. For S-DES, the method finds a linear expression involving S-box inputs/outputs that predicts a key bit with 78% accuracy, allowing recovery of multiple key bits.
Aplikasi Bilangan Kompleks - Analisis Sinyal [PAPER]Ghins GO
Check out!
Website : https://ghinsblog.blogspot.com
Youtube : Ghins GO Math
APLIKASI BILANGAN KOMPLEKS
BAB 1 Pendahuluan
1. Latar belakang
2. Rumusan Masalah
3. Tujuan
BAB 2 Pembahasan
1. Bilangan Kompleks
2. Sinyal
BAB 3 Aplikasi
1. Analisis Sinyal
2. Analisis Fourier
3. Analisis Frekuensi Sinyal Periodik Waktu Diskrit
4. Penerapan Analisis Sinyal
BAB 4 Kesimpulan
DAFTAR PUSTAKA
[1] Gumilang, Muhammad. 2015. "Aplikasi Bilangan Kompleks dalam Analisis Sinyal". https://adoc.pub/aplikasi-bilangan-kompleks-dalam-analisis-sinyal.html, diakses 14 Mei 2021 pukul 18.11.
[2] Haryono, Stefanus Agus. 2015."Penggunaan Bilangan Kompleks dalam Pemrosesan Signal".https://pdfslide.tips/documents/penggunaan-bilangan-kompleks-dalam-pemrosesan-signalinformatikasteiitbacidrinaldimuniraljabargeometri2015aa.html, diakses 13 Mei 2021 pukul 20.37.
[5] Ratnadewi. dkk. 2019. "MATEMATIKA TEKNIK". Bandung: Rekayasa Sains.
Selengkapnya:
https://ghinsblog.blogspot.com/2021/08/variabel-kompleks-aplikasi-bilangan.html
Elliptic curve cryptography (ECC) uses elliptic curves over finite fields for encryption, digital signatures, and key exchange. It provides the same security as RSA or discrete logarithm schemes but with smaller key sizes (e.g. 256-bit ECC vs. 3072-bit RSA). ECC algorithms are also faster and use less energy than other schemes. While ECC offers advantages, security relies on using cryptographically strong elliptic curves and there is no deterministic method to encode messages as curve points.
This document provides an overview of binary repetition codes and the Hamming distance as a simple error-correcting code. It discusses how coding theory studies error correction, cryptography and data compression applications of codes. Binary repetition codes represent messages as repeated bits and use majority voting to decode received bits and correct errors introduced over noisy channels. The document demonstrates generating codewords from messages, encoding messages, calculating the Hamming distance between codewords, and simulating encoding and transmission over a channel with Sage.
This document provides an overview of elliptic curve cryptography (ECC). It begins with background on ECC, describing how it was independently proposed in 1985 as an approach to asymmetric cryptography. It then covers the basics of asymmetric cryptosystems and how ECC compares to RSA and Diffie-Hellman. The document goes on to explain elliptic curves over real and finite numbers, how points are added and doubled on elliptic curves, and how this relates to discrete logarithm problems. It discusses implementations of ECC for cryptography and comparisons to RSA in terms of key size and performance. Finally, it covers efficient implementations of ECC for smart cards.
The International Journal of Engineering and Sciencetheijes
This document summarizes a research paper that proposes a block cipher involving a key matrix and key bunch matrix supplemented with permutation. The cipher encrypts a plaintext matrix using modular multiplication with the key matrices over 256. It adds a permutation function that circularly rotates and swaps bits in the plaintext matrix in each round. Cryptanalysis showed the cipher cannot be broken by general attacks. The decryption uses the inverse key matrix and multiplicative inverse of the encryption key bunch matrix.
Disk-based data structures are needed to store large datasets that exceed main memory capacity. Hard disks allow large storage but have slow random access. B-trees address this by storing data in pages that can be read sequentially, keeping the tree shallow. Keys in B-tree nodes separate the ranges of keys in subtrees, and nodes have between t-1 and 2t-1 keys. Operations on B-trees like search, insert, and delete involve reading and writing nodes to disk and splitting or merging nodes as needed to maintain the B-tree properties.
Can we reveal the RSA private exponent d from its public key <e, n>? We study this question for two specific cases: e = 3 and e = 65537. Using demos, we verify that RSA reveals the most significant half of the private exponent d when the public exponent e is small. For example, for 2048-bit RSA, the most significant 1024 bits are revealed!
Error control codes are necessary for transmission and storage of large volumes of date sensitive to errors. BCH codes and Reed Solomon codes are the most important class of multiple error correcting codes for binary and non-binary channels respectively. Peterson and later Berlekamp and Massey discovered powerful algorithms which became viable with the help of new digital technology. Use of Galois fields gave a structured approach to designing of these codes. This presentation deals with above in a very structured and systematic manner.
This document presents a block cipher that incorporates concepts from the Hill cipher and previous block ciphers developed by the authors. The cipher uses a key matrix K and encryption key bunch matrix E to encrypt plaintext P into ciphertext C. Decryption uses the inverse of K and a decryption key bunch matrix D to recover P from C. The cipher is strengthened by including Mix() and Imix() functions that diffuse bits during encryption and decryption rounds. Cryptanalysis shows the cipher is unbreakable against known attacks due to the diffusion achieved by superimposing Hill cipher and previous block cipher concepts. In 3 sentences or less, this document proposes and analyzes a block cipher combining aspects of Hill cipher and previous work, using key matrices for
The document discusses attribute-based encryption (ABE) schemes, including Key-Policy ABE (KP-ABE) and Ciphertext-Policy ABE (CP-ABE). It defines the components of KP-ABE and CP-ABE, including setup, encryption, key generation, and decryption algorithms. It also describes the security models and proves the selective security of the GPSW KP-ABE scheme and correctness of the Waters CP-ABE scheme under the decisional bilinear Diffie-Hellman assumption. The document outlines the KP-ABE and CP-ABE constructions and security proofs in detail.
This document contains a 30 question mid-semester exam for a data structures and algorithms course. The exam covers topics like asymptotic analysis, sorting algorithms, hashing, binary search trees, and recursion. It provides multiple choice questions to test understanding of algorithm time complexities, worst-case inputs, and recursive functions. Students are instructed to attempt all questions in the 2 hour time limit and notify the proctor if any electronic devices other than calculators are used.
This document summarizes an article about implementing the RSA encryption/decryption algorithm on an FPGA. It begins with an overview of cryptography and the RSA algorithm. It then describes the key steps in RSA - key generation, encryption, and decryption. The main mathematical operations required for RSA are also summarized - modular addition, multiplication, and exponentiation. The document then presents the design of a 32-bit RSA decryption engine in VHDL, along with synthesis results showing its resource usage and maximum clock frequency on an FPGA. It concludes that an RSA decryption engine can be efficiently implemented on an FPGA using limited resources.
Composite Field Multiplier based on Look-Up Table for Elliptic Curve Cryptogr...Marisa Paryasto
This document discusses implementing elliptic curve cryptography using composite fields. It proposes using a 299-bit key represented in the composite field GF((213)23) instead of the conventional GF(2299). This breaks the finite field multiplication into smaller chunks by dividing the field into a ground field and extension field. A lookup table is used for multiplication in the ground field GF(213) while a classic multiplier is used for the extension field GF(23). This composite field approach aims to provide better time and area efficiency for implementation on FPGAs compared to a single large multiplier. The document provides background on elliptic curves, finite fields, and previous work on composite field representations.
The document summarizes key concepts in social network analysis including metrics like degree distribution, path lengths, transitivity, and clustering coefficients. It also discusses models of network growth and structure like random graphs, small-world networks, and preferential attachment. Computational aspects of analyzing large networks like calculating shortest paths and the diameter are also covered.
Ijcatr03051008Implementation of Matrix based Mapping Method Using Elliptic Cu...Editor IJCATR
Elliptic Curve Cryptography (ECC) gained a lot of attention in industry. The key attraction of ECC over RSA is that it
offers equal security even for smaller bit size, thus reducing the processing complexity. ECC Encryption and Decryption methods can
only perform encrypt and decrypt operations on the curve but not on the message. This paper presents a fast mapping method based on
matrix approach for ECC, which offers high security for the encrypted message. First, the alphabetic message is mapped on to the
points on an elliptic curve. Later encode those points using Elgamal encryption method with the use of a non-singular matrix. And the
encoded message can be decrypted by Elgamal decryption technique and to get back the original message, the matrix obtained from
decoding is multiplied with the inverse of non-singular matrix. The coding is done using Verilog. The design is simulated and
synthesized using FPGA.
The document discusses binary search trees and their properties. It explains that a binary search tree is a binary tree where every node's left subtree contains values less than the node's value and the right subtree contains greater values. Operations like search, insert, delete can be done in O(h) time where h is the height of the tree. The height is O(log n) for balanced trees but can be O(n) for unbalanced trees. The document also provides examples of using a binary search tree to sort a set of numbers in O(n log n) time by building the BST and doing an inorder traversal.
This document discusses elliptic curve cryptography and its underlying mathematics. It begins by providing historical context on the development of cryptography and highlights elliptic curves being suggested in 1985 as an encryption system. It then covers key mathematical concepts such as Weierstrass elliptic curve equations, properties of elliptic curves like point addition and doubling, and how elliptic curves form algebraic groups. Finally, it introduces the concepts of prime moduli, rings, and fields which are important for implementing elliptic curve cryptography in practice.
Novel encryption algorithm and software development ecc and rsaSoham Mondal
Awarded 2nd prize in the event Papier (scientific paper presentation) conducted by Jadavpur University Electrical Engineering Department, named Convolution, under the aegis of IET and IEEE Signal Processing Society in 2018
The document discusses the objectives and concepts of cryptography. The four main objectives are confidentiality, data integrity, authentication, and non-repudiation. It describes symmetric-key cryptography which uses a single secret key for encryption and decryption, and asymmetric key cryptography which uses different keys for encryption and decryption. It also provides an overview of elliptic curve cryptography, including how it works and some benefits over RSA in providing equivalent security with smaller key sizes.
Implementation of Elliptic Curve Digital Signature Algorithm Using Variable T...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
We allow Eve to modify DH parameters as well as public keys of Alice and Bob. This allows Eve to derive the secret key and break the DH crypto system. We demonstrate that the DH key exchange algorithm should not be used without digital signatures.
Linear cryptanalysis is a method used to break encryption standards like DES. It involves finding linear approximations between plaintext, ciphertext, and key bits that hold with probability greater than 50%. These approximations are used to determine partial key bits using maximum likelihood algorithms on known or ciphertext-only data. For S-DES, the method finds a linear expression involving S-box inputs/outputs that predicts a key bit with 78% accuracy, allowing recovery of multiple key bits.
Aplikasi Bilangan Kompleks - Analisis Sinyal [PAPER]Ghins GO
Check out!
Website : https://ghinsblog.blogspot.com
Youtube : Ghins GO Math
APLIKASI BILANGAN KOMPLEKS
BAB 1 Pendahuluan
1. Latar belakang
2. Rumusan Masalah
3. Tujuan
BAB 2 Pembahasan
1. Bilangan Kompleks
2. Sinyal
BAB 3 Aplikasi
1. Analisis Sinyal
2. Analisis Fourier
3. Analisis Frekuensi Sinyal Periodik Waktu Diskrit
4. Penerapan Analisis Sinyal
BAB 4 Kesimpulan
DAFTAR PUSTAKA
[1] Gumilang, Muhammad. 2015. "Aplikasi Bilangan Kompleks dalam Analisis Sinyal". https://adoc.pub/aplikasi-bilangan-kompleks-dalam-analisis-sinyal.html, diakses 14 Mei 2021 pukul 18.11.
[2] Haryono, Stefanus Agus. 2015."Penggunaan Bilangan Kompleks dalam Pemrosesan Signal".https://pdfslide.tips/documents/penggunaan-bilangan-kompleks-dalam-pemrosesan-signalinformatikasteiitbacidrinaldimuniraljabargeometri2015aa.html, diakses 13 Mei 2021 pukul 20.37.
[5] Ratnadewi. dkk. 2019. "MATEMATIKA TEKNIK". Bandung: Rekayasa Sains.
Selengkapnya:
https://ghinsblog.blogspot.com/2021/08/variabel-kompleks-aplikasi-bilangan.html
Elliptic curve cryptography (ECC) uses elliptic curves over finite fields for encryption, digital signatures, and key exchange. It provides the same security as RSA or discrete logarithm schemes but with smaller key sizes (e.g. 256-bit ECC vs. 3072-bit RSA). ECC algorithms are also faster and use less energy than other schemes. While ECC offers advantages, security relies on using cryptographically strong elliptic curves and there is no deterministic method to encode messages as curve points.
This document provides an overview of binary repetition codes and the Hamming distance as a simple error-correcting code. It discusses how coding theory studies error correction, cryptography and data compression applications of codes. Binary repetition codes represent messages as repeated bits and use majority voting to decode received bits and correct errors introduced over noisy channels. The document demonstrates generating codewords from messages, encoding messages, calculating the Hamming distance between codewords, and simulating encoding and transmission over a channel with Sage.
This document provides an overview of elliptic curve cryptography (ECC). It begins with background on ECC, describing how it was independently proposed in 1985 as an approach to asymmetric cryptography. It then covers the basics of asymmetric cryptosystems and how ECC compares to RSA and Diffie-Hellman. The document goes on to explain elliptic curves over real and finite numbers, how points are added and doubled on elliptic curves, and how this relates to discrete logarithm problems. It discusses implementations of ECC for cryptography and comparisons to RSA in terms of key size and performance. Finally, it covers efficient implementations of ECC for smart cards.
The International Journal of Engineering and Sciencetheijes
This document summarizes a research paper that proposes a block cipher involving a key matrix and key bunch matrix supplemented with permutation. The cipher encrypts a plaintext matrix using modular multiplication with the key matrices over 256. It adds a permutation function that circularly rotates and swaps bits in the plaintext matrix in each round. Cryptanalysis showed the cipher cannot be broken by general attacks. The decryption uses the inverse key matrix and multiplicative inverse of the encryption key bunch matrix.
Disk-based data structures are needed to store large datasets that exceed main memory capacity. Hard disks allow large storage but have slow random access. B-trees address this by storing data in pages that can be read sequentially, keeping the tree shallow. Keys in B-tree nodes separate the ranges of keys in subtrees, and nodes have between t-1 and 2t-1 keys. Operations on B-trees like search, insert, and delete involve reading and writing nodes to disk and splitting or merging nodes as needed to maintain the B-tree properties.
Can we reveal the RSA private exponent d from its public key <e, n>? We study this question for two specific cases: e = 3 and e = 65537. Using demos, we verify that RSA reveals the most significant half of the private exponent d when the public exponent e is small. For example, for 2048-bit RSA, the most significant 1024 bits are revealed!
Error control codes are necessary for transmission and storage of large volumes of date sensitive to errors. BCH codes and Reed Solomon codes are the most important class of multiple error correcting codes for binary and non-binary channels respectively. Peterson and later Berlekamp and Massey discovered powerful algorithms which became viable with the help of new digital technology. Use of Galois fields gave a structured approach to designing of these codes. This presentation deals with above in a very structured and systematic manner.
This document presents a block cipher that incorporates concepts from the Hill cipher and previous block ciphers developed by the authors. The cipher uses a key matrix K and encryption key bunch matrix E to encrypt plaintext P into ciphertext C. Decryption uses the inverse of K and a decryption key bunch matrix D to recover P from C. The cipher is strengthened by including Mix() and Imix() functions that diffuse bits during encryption and decryption rounds. Cryptanalysis shows the cipher is unbreakable against known attacks due to the diffusion achieved by superimposing Hill cipher and previous block cipher concepts. In 3 sentences or less, this document proposes and analyzes a block cipher combining aspects of Hill cipher and previous work, using key matrices for
The document discusses attribute-based encryption (ABE) schemes, including Key-Policy ABE (KP-ABE) and Ciphertext-Policy ABE (CP-ABE). It defines the components of KP-ABE and CP-ABE, including setup, encryption, key generation, and decryption algorithms. It also describes the security models and proves the selective security of the GPSW KP-ABE scheme and correctness of the Waters CP-ABE scheme under the decisional bilinear Diffie-Hellman assumption. The document outlines the KP-ABE and CP-ABE constructions and security proofs in detail.
This document contains a 30 question mid-semester exam for a data structures and algorithms course. The exam covers topics like asymptotic analysis, sorting algorithms, hashing, binary search trees, and recursion. It provides multiple choice questions to test understanding of algorithm time complexities, worst-case inputs, and recursive functions. Students are instructed to attempt all questions in the 2 hour time limit and notify the proctor if any electronic devices other than calculators are used.
This document summarizes an article about implementing the RSA encryption/decryption algorithm on an FPGA. It begins with an overview of cryptography and the RSA algorithm. It then describes the key steps in RSA - key generation, encryption, and decryption. The main mathematical operations required for RSA are also summarized - modular addition, multiplication, and exponentiation. The document then presents the design of a 32-bit RSA decryption engine in VHDL, along with synthesis results showing its resource usage and maximum clock frequency on an FPGA. It concludes that an RSA decryption engine can be efficiently implemented on an FPGA using limited resources.
Composite Field Multiplier based on Look-Up Table for Elliptic Curve Cryptogr...Marisa Paryasto
This document discusses implementing elliptic curve cryptography using composite fields. It proposes using a 299-bit key represented in the composite field GF((213)23) instead of the conventional GF(2299). This breaks the finite field multiplication into smaller chunks by dividing the field into a ground field and extension field. A lookup table is used for multiplication in the ground field GF(213) while a classic multiplier is used for the extension field GF(23). This composite field approach aims to provide better time and area efficiency for implementation on FPGAs compared to a single large multiplier. The document provides background on elliptic curves, finite fields, and previous work on composite field representations.
The document summarizes key concepts in social network analysis including metrics like degree distribution, path lengths, transitivity, and clustering coefficients. It also discusses models of network growth and structure like random graphs, small-world networks, and preferential attachment. Computational aspects of analyzing large networks like calculating shortest paths and the diameter are also covered.
Ijcatr03051008Implementation of Matrix based Mapping Method Using Elliptic Cu...Editor IJCATR
Elliptic Curve Cryptography (ECC) gained a lot of attention in industry. The key attraction of ECC over RSA is that it
offers equal security even for smaller bit size, thus reducing the processing complexity. ECC Encryption and Decryption methods can
only perform encrypt and decrypt operations on the curve but not on the message. This paper presents a fast mapping method based on
matrix approach for ECC, which offers high security for the encrypted message. First, the alphabetic message is mapped on to the
points on an elliptic curve. Later encode those points using Elgamal encryption method with the use of a non-singular matrix. And the
encoded message can be decrypted by Elgamal decryption technique and to get back the original message, the matrix obtained from
decoding is multiplied with the inverse of non-singular matrix. The coding is done using Verilog. The design is simulated and
synthesized using FPGA.
The document discusses binary search trees and their properties. It explains that a binary search tree is a binary tree where every node's left subtree contains values less than the node's value and the right subtree contains greater values. Operations like search, insert, delete can be done in O(h) time where h is the height of the tree. The height is O(log n) for balanced trees but can be O(n) for unbalanced trees. The document also provides examples of using a binary search tree to sort a set of numbers in O(n log n) time by building the BST and doing an inorder traversal.
This document discusses elliptic curve cryptography and its underlying mathematics. It begins by providing historical context on the development of cryptography and highlights elliptic curves being suggested in 1985 as an encryption system. It then covers key mathematical concepts such as Weierstrass elliptic curve equations, properties of elliptic curves like point addition and doubling, and how elliptic curves form algebraic groups. Finally, it introduces the concepts of prime moduli, rings, and fields which are important for implementing elliptic curve cryptography in practice.
Novel encryption algorithm and software development ecc and rsaSoham Mondal
Awarded 2nd prize in the event Papier (scientific paper presentation) conducted by Jadavpur University Electrical Engineering Department, named Convolution, under the aegis of IET and IEEE Signal Processing Society in 2018
The document discusses the objectives and concepts of cryptography. The four main objectives are confidentiality, data integrity, authentication, and non-repudiation. It describes symmetric-key cryptography which uses a single secret key for encryption and decryption, and asymmetric key cryptography which uses different keys for encryption and decryption. It also provides an overview of elliptic curve cryptography, including how it works and some benefits over RSA in providing equivalent security with smaller key sizes.
Implementation of Elliptic Curve Digital Signature Algorithm Using Variable T...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
A SURVEY ON ELLIPTIC CURVE DIGITAL SIGNATURE ALGORITHM AND ITS VARIANTScsandit
The Elliptic Curve Digital Signature Algorithm (ECDSA) is an elliptic curve variant of the
Digital Signature Algorithm (DSA). It gives cryptographically strong digital signatures making
use of Elliptic curve discrete logarithmic problem. It uses arithmetic with much smaller
numbers 160/256 bits instead of 1024/2048 bits in RSA and DSA and provides the same level of
security. The ECDSA was accepted in 1999 as an ANSI standard, and was accepted in 2000 as
IEEE and NIST standards. It was also accepted in 1998 as an ISO standard. Many cryptologist
have studied security aspects of ECDSA and proposed different variants. In this paper, we
discuss a detailed analysis of the original ECDSA and all its available variants in terms of the
security level and execution time of all the phases. To the best of our knowledge, this is a unique
attempt to juxtapose and compare the ECDSA with all of its variants.
International Journal of Computational Engineering Research(IJCER) ijceronline
This document presents an implementation of an Elliptic Curve Diffie-Hellman (ECDH) key exchange protocol using VB.NET. ECDH is based on the elliptic curve discrete logarithm problem and allows two parties to generate a shared secret key over an insecure channel. The implementation uses an elliptic curve group over the field F29 with parameters a=1 and b=1. It demonstrates the steps to generate and exchange public keys between two users to compute the same shared secret key. This allows encryption of messages using a symmetric key algorithm. ECDH is suitable for applications requiring security where resources are limited, as smaller key sizes provide the same level of security as larger keys in other cryptosystems.
Elliptic Curves as Tool for Public Key Cryptographyinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document summarizes a research paper that proposes a new public key cryptosystem based on the difficulty of inverting the function F(x) = (a × x)Mod(2p)Div(2q). The cryptosystem includes a key exchange algorithm, public key encryption algorithm, and digital signature algorithm. The document analyzes the efficiency and security of the cryptosystem, showing it has O(n) faster time complexity than RSA and Diffie-Hellman. It also reduces breaking the cryptosystem to solving difficult SAT instances or sets of multivariate polynomial equations over F(2). Python implementations of the key exchange and signature algorithms are provided in appendices.
Low Power FPGA Based Elliptical Curve CryptographyIOSR Journals
Abstract: Cryptography is the study of techniques for ensuring the secrecy and authentication of the information. The development of public-key cryptography is the greatest and perhaps the only true revolution in the entire history of cryptography. Elliptic Curve Cryptography is one of the public-key cryptosystem showing up in standardization efforts, including the IEEE P1363 Standard. The principal attraction of elliptic curve cryptography compared to RSA is that it offers equal security for a smaller key-size, thereby reducing the processing overhead. As a Public-Key Cryptosystem, ECC has many advantages such as fast speed, high security and short key. It is suitable for the hardware of implementation, so ECC has been more and more focused in recent years. The hardware implementation of ECC on FPGA uses the arithmetic unit that has small area, small storage unit and fast speed, and it is an extremely suitable system which has limited computation ability and storage space.[1][2] The modular arithmetic division operations are carried out using conditional successive subtractions, thereby reducing the area. The system is implemented on Vertex-Pro XCV1000 FPGA. Index Terms – VHDL, FSM, FPGA, Elliptic Curve Cryptography.
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Cryptography is the study of techniques for ensuring the secrecy and authentication of the
information. The development of public-key cryptography is the greatest and perhaps the only true revolution in
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up in standardization efforts, including the IEEE P1363 Standard. The principal attraction of elliptic curve
cryptography compared to RSA is that it offers equal security for a smaller key-size, thereby reducing the
processing overhead. As a Public-Key Cryptosystem, ECC has many advantages such as fast speed, high
security and short key. It is suitable for the hardware of implementation, so ECC has been more and more
focused in recent years. The hardware implementation of ECC on FPGA uses the arithmetic unit that has small
area, small storage unit and fast speed, and it is an extremely suitable system which has limited computation
ability and storage space.[1][2] The modular arithmetic division operations are carried out using conditional
successive subtractions, thereby reducing the area. The system is implemented on Vertex-Pro XCV1000 FPGA
Cryptography is the combination of Mathematics and Computer science. Cryptography is used for encryption and decryption of data using mathematics. Cryptography transit the information in an illegible manner such that only intended recipient will be able to decrypt the information
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AUTHENTICATED PUBLIC KEY ENCRYPTION SCHEME USING ELLIPTIC CURVE CRYPTOGRAPHYijujournal
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An Introduction to ECDSA and it's use in Bitcoin (1)
1. An introduction to ECDSA and it’s use in Bitcoin
Hakeem Hunte
1358508
1 Introduction
"The secure system of verifying transactions that could transform the way we exchange goods and
services around the world" [11]. Bitcoin is a "peer-to-peer" decentralised cyrptocurrency initially
launched by Satoshi Nakamoto (alias) (2009) [8]. To “posses” a bitcoin means the holder of the bitcoin
has the capacity to transfer it to someone else whilst recording that particular transaction in the
blockchain. A blockchain is a record of all the transactions made using bitcoin. Effectively, anyone
who holds bitcoin also has a copy of the blockchain, making it near to impossible to forge transactions,
as each transaction has been signed and verified. This is implemented through the Elliptic Curve
Digital Signature Algorithm (ECDSA) [8]. To understand this monitored transaction of a bitcoin is
the focus of the project.
1.1 Introduction to Cryptography
Cryptography is related with the design of algorithms, primarily introduced to provide secrecy or
authenticity of messages [5]. Cryptography’s foundations lie in number theory, in which the first
tangible form was introduced by Ron Rivest, Adi Shamir and Len Adleman (RSA) in 1977, based on
the difficulty of factorizing an integer problem [3]. Bitcoin utilises the ECDSA, proposed by Scott
Vanston (1992) [2] as the foundations of signing and verifying transactions. ECDSA is based off of
Elliptic Curve Cryptography (ECC) which was presented by Victor S. Miller and Neal Koblitz in 1985
[6]. Bitcoin uses this mechanism, rather than RSA, as ECDSA have a more smaller key size, leading
to faster computations which overall is very beneficial for computing systems with limited processing
powers that want to optimize security [10].
The structure of the project is as follows: Firstly, we will gain an understanding of Public Key
Cryptography which underpins the ECDSA. Secondly, we will explore how the bitcoin system works,
in terms of the bitcoin wallet. Following this we will analyse elliptic curves, and how their features
are used in ECC, finally finishing the ECDSA.
2 Preliminary Defintions
We start with preliminary definitions in which would be useful throughout this project.
Definition 2.1. A hash function is a computable map f: x → h, from a large input x to a more
compact output h. It is also impractical to find two different inputs being x and z s.t. f(x) = f(z)
[10].
Definition 2.2. (Characteristic of field) Let K be a field. Let n ∈ N s.t n > 0. The smallest n > 0
s.t.
n · 1 = 1 + 1 + . . . + 1
n times
= 0,
1
2. 3 PUBLIC KEY CRYPTOGRAPHIC SCHEMES 2
is called the characteristic of K.
Definition 2.3. Given a natural number n ≥ 2 and there exists a, b ∈ Z that have the same remainder
when divided by n then we say that a is congruent to b modulo n. This is denoted as a ≡ b (mod n) [10].
Definition 2.4. Let p be a prime number. The integers modulo p, consisting of the integers 0, 1, 2, .., p − 1
with addition multiplication performed modulo p, is a finite field of order p. This field is denoted as
Fp, where p is called the modulus of Fp [10].
Definition 2.5. A group G is a finite set of elements with a binary operation called addition denoted
+. For a set G to be called a group, it must satisfy the four properties if a, b ∈ G then [10]:
1. Closure. If a, b ∈ G then a + b ∈ G
2. Associativity. If a, b ∈ G then (a + b) + c = a + (b + c)
3. Identity. a ∈ G then a + 0 = a
4. inverse. If a, b ∈ G then for ever a there exists b s.t a + b = 0
Definition 2.6. A subgroup is a subset H of elements of a group G in which satisfies the four group
requirements [9].
3 Public Key Cryptographic Schemes
A public key scheme is when a key pair is selected and the task of acquiring the private key from the
public key is an arduous computational problem. The scheme consists of two keys: the public key,
which is seen by everybody, and the private key which can decrypt [5]. The main idea of this system
is that the holder of public key is unable to obtain the private key. So essentially, the cryptographic
function
f: P → C, (1)
is simple to calculate whilst having the encryption key KE. But, compartively it is extremely difficult
to calculate the reverse [10].
f−1
: C → P. (2)
4 How Bitcoins Work
4.1 Bitcoin wallet and address
For a user to use bitcoin, the user downloads an application in which serves as the wallet. This wallet
adopts public key cryptography based off elliptic curves explored later. In this case, the private key
is equivalent to a PIN number, enabling the user to access and authorise transactions. The public
key is equivalent to a bank account number derived through elliptic curve multiplication. When the
public key is combined with a cryptographic hash function1, the bitcoin address is produced. Each
address has it’s own balance of bitcoin, where transactions are fundamentally the exchange of bitcoin
(messages) between these adresses [8].
1
The cryptographic function used is the RIPEMD-160 hash
3. 5 THE ELLIPTIC CURVE 3
Figure 1: The private key, public key and address [8]
4.2 The transactions
A transaction also contains digital a signature in which is provided as the proof of a transfer of bitcoin,
adding validity to the transaction. The owner of this bitcoin provides this digital signature utilising
their private key. This digital signature proves that they own these bitcoins and can then go on to
spend them in a new transaction. Following the verficiation of the digital signature, the transaction
can occur [8].
5 The Elliptic Curve
As stated earlier, the bitcoin wallet creates a public key and a private key to then verify transactions
[8]. The public key is mathematically acquired from the private key using the properties of elliptic
curves over a finite field. In this section we are going to explore elliptic curves over R and the finite
field along with their properties respectively. This will give us insight into how the features are used
in ECDSA.
Definition 5.1. Let K be a field of characteristic = 2, 3 and let x3 + ax + b where a, b ∈ K and does
not contain multiple roots. An elliptic curve E over K with the points (x, y) in which satisfy:
y2
= x3
+ Ax + B. (3)
5.1 Elliptic Curve over R
Definition 5.2. An elliptic curve E is the graph of an equation of the form
y2
= x3
+ Ax + B, (4)
where A, B ∈ Z, where 4A3 + 27B2 = 0. This condition is to ensure that the all points on the curve
are well defined. The point at infinity ∞ is also defined on the curve. At the top and bottom of the y
axis [6].
5.1.1 The group Law
This section describes the fundamental properties in which make the elliptic curve useful for cryptog-
raphy, we will first introduce them, then later in the project describe how the features are implemented.
Definition 5.3. The group Law. We define a group over Elliptic Curve E as
1. Elements of the group are points on an elliptic curve
2. Identity is the point infinity
4. 5 THE ELLIPTIC CURVE 4
3. The inverse of a point P ∈ E is symmetric about the x axis
4. Addition of 3 aligned distinct points P, Q, R ∈ E is given as P + Q + R = 0.
We can write P + Q = −R , and from this we can derive the method for adding up two points on a
curve, which is crucial for point addition and point multiplication, explained later in the paper.
Case 1 Let P = (xp, yp) and −P = (xq, yq) where xp = xq but yp = yq, the line through P and −P
is vertical, therefore the line intersects E in ∞. Yielding P + (−P) = ∞.
Case 2 In the case Q = ∞. The line through the point P and ∞ is a vertical line, intersecting the
elliptic curve E at the point R, which is just a reflection of P across the x axis. Therefore P +∞ = R,
But we are back at P. Therefore P + ∞ = P
5.2 Point Addition demonstrated algebraically [6]
Case 3
Let P = (xp, yp) and Q = (xq, yq) and R = (xr, yr) s.t. P + Q = R. We assume that P, Q are points
on E, and that P and Q are distinct points We want to express R in terms of P and Q. We denote
the line (L) intersecting the two points P, Q has a slope m
m =
yp − yq
xp − xq
,
the equation of L then becomes
y = m(x − xp) + yp,
intersecting with the elliptic curve
(m(x − xp) + yp)2
= x3
+ Ax + B.
We rearrange this to then get
0 = x3
− m2
x2
+ . . . .
We already know two roots, being xp and xq, as P and Q are on the both E and L. We use the fact
that if we have a polynomial with roots r, s, t [6], then
x3
+ ax2
+ bx + c = (x − r)(x − s)(x − t) = x3
− (r + s + t)x2
+ . . .
Therefore we get,
r + s + t = −a.
Knowing the two roots r and s, we then get the third root as t = −a − r − s. So in our case we
get
x = m2
− xp − xq and yr = m(xp − xr) + yp.
Reflecting across the x axis we get the point R = (xr, yr)
xr = m2
− xp − xq and yr = m(xp − xr) − yp.
5.3 Point Doubling demonstrated algebraically [6]
Case 4 Consider the case P = Q = (xp, yp). As the two points on the curve are extremely close to
each other, the intercepting line of the points estimate a tangent line (L). The slope m of L is found
through implicit differentiation on equation (4) whereby:
2y
dy
dx
= 3x2
+ A, therefore m =
dy
dx
=
3x2
p + A
2yp
.
5. 5 THE ELLIPTIC CURVE 5
Figure 2: Point addition over R Figure 3: Point doubling over R
We assume that yp = 0 The equation of L is:
y = m(x − xp) + yp,
we obtain the cubic equation
0 = x3
− m2
x2
+ . . . .
On this occasion, we know only one root, being xp, but this is a double root as L is a tangent to the
Elliptic Curve E at P, utilising the same technique earlier, obtaining:
xr = m2
− 2xp,
yr = m(xp − xr) − yp.
Definition 5.4. Let n ∈ Z, and P a point on a given Elliptic Curve E then
nP = P + P + . . . + P,
n times
(5)
is the operation on an Elliptic Curve called the scalar multiplication
Computing kP for a very large integer k can be an arduous task. It becomes inefficient to repeatedly
add P to itself [5]. Therefore the properties of point multiplication and point addition can be used.
This is called successive doubling can be used to speed this process, extremely critical when the
public key is computed. Taken an example from [16]: k = 23
23P = 2(2(2(2P) + P) + P) + P. (6)
5.4 Elliptic Curve over a Finite Field Fp
Elliptic Curve Cryptography makes use of Elliptic curves over finite fields, where all the variables and
coefficients are strictly within the finite field of size prime number p. Bitcoin uses elliptic curve curves
frequently in it’s systems, whereby calculating these arithmetic’s over the real numbers is infeasible
due to the inclusion of rational numbers. Standard computers do not have enough space to store all the
digits and compute these commands efficiently. An elliptic curve is graphically represented differently
over a finite field demonstrated in Figure 4. The operations are more difficult to see visually, but the
underlying arithmetic in which we derived for a Elliptic Curve over R still holds.
6. 5 THE ELLIPTIC CURVE 6
Figure 4: Elliptic Curve in finite field [7] Figure 5: Point Multiplication [7]
Definition 5.5. Let p be a prime number, and let Fp denote the field of integers modulo p. An elliptic
curve E over Fp (E(Fp))is defined by an equation of the form
y2
= x3
+ ax + b. (7)
Where a, b ∈ Fp satisfy 4a3 + 27b2 ≡ 0 (mod p). This is to ensure the curve does not contain singu-
larities which in essence ensures that the elliptic curve is smooth and each point is defined. The point
at infinity, denoted by ∞, is also said to be on the curve.
Definition 5.6. E be an elliptic curve and kP, where k ∈ Z and P is a point on the elliptic curve
E defined over the field Fp with p being a prime number then we define (8) as the operation point
multiplication
kP = P + P + . . . + P.
k times
(8)
Point multiplication has interesting properties over the finite field exemplified in the following example.
This point multiplication operation dominates the execution time of many elliptic curve cryptographic
algorithms used to generate the public key [4].
Example 2 [14] Let E be the Elliptic Curve over Fp denoted as y2 ≡ x3 + 2x + 3 mod 97 and
the point G = (3, 6), then calculating the multiples using the features of point addition and
point doubling defined earlier we find that
0G = ∞, G = (3, 6), 2G = (80, 10), 3G = (80, 87), 4G = (3, 91)
5G = ∞, 6G = (3, 6), 7G = (80, 10), 8G = (80, 87), 9G = (3, 91) . . .
5.5 Order of Groups
In this section, we are going to understand the mathematics involved in forming the parameters used
in ECDSA. We are going to determine the size of the the subgroup of an Elliptic Curve over a finite
field, and understand the mathematics involved. Bitcoin uses a fixed set parameters for the ECDSA
algorithm, but this section is used to increase our understanding of how these parameters are formed.
The topic of counting points on an Elliptic Curve is extremely complex and beyond the scope of this
report, but to develop your understanding my recommended literature is Elliptic Curves - Number
Theory and Cryptography by Lawerence C. Washington.
7. 5 THE ELLIPTIC CURVE 7
Definition 5.7. Let E be an elliptic curve defined over Fp. The number of points in E(Fp), denoted
#E(Fp), is called the order of E [9].
Definition 5.8. Let E be an elliptic curve over the finite field Fp. Then the order of E(Fp) satisfies
the following inequality (Hasse)
| p + 1 − #E(Fp) |≤ 2
√
p. (9)
Proof. Proof omitted see [5] p.46
Theorem 5.1. The points on an elliptic curve together will have cyclic subgroups. Under certain
conditions all points on an elliptic curve form a cyclic group [9].
Proof. Proof omitted
Definition 5.9. A group G which contains an element α with maximum order ord(α) = | G | is said
to be cyclic. Elements with the maximum order are called the Generator point [9].
Proof. Proof omitted
Definition 5.10. Let k ∈ Z and P ∈ E(Fp). We let P be the generator point. We denote the order of
P as the smallest positive integer k s.t. kP = ∞.
Proof. Proof omitted
With this being said determined from Example 2, we can observe that the order of the generator
point G is 5.
5.5.1 Subgroup order
To achieve maximum efficiency of the algorithm, and to determining the coefficients of the ECDSA,
it is important to determine the order of the subgroup generated by the generator point G. We are
going to need Lagrange’s Theorem, one of group theory’s first great discoveries. Lagrange’s Theorem
first appeared in 1770-1771, and was proven by the German mathematician Carl Friedrich Gauss in
1801 over the field Zp with p being a prime number. Proof from [15]
Lemma 5.2. We suppose that G is a group and H is a subgroup G. Let x, y ∈ G. Then the following
hold:
1. Suppose that h ∈ H. Then Hh = He = H, whereby e is the identity element of G.
2. y ∈ Hx i.f.f Hx = Hy
3. H(xy) = (Hx)y whereby (Hx)y = {wy | w ∈ Hx}
4. For ∀ h in H, Hhx = Hx
Definition 5.11. For a group G, a subset group H of G and each g ∈ G we define a left coset of H
in G as the set
gH = {gh | h ∈ H} (10)
Lemma 5.3. Suppose that G is a group and H ≤ G. We define a relation on G by x ∼ y ⇔ xy−1 ∈ H.
Then ∼ is an equivalence relation and the right cosets are the equivalence classes.
Lemma 5.4. Suppose H ≤ G. Then
1. for all x, y ∈ G, then either Hx ∩ Hy = ∅ or Hx = Hy and
8. 6 ELLIPTIC CURVE CRYPTOGRAPHY 8
2. G = z∈H Hz
Lemma 5.5. Let G be a group G, and H be a subgroup of G and x ∈ G, the map
H −→ Hx, (11)
defined
h −→ hx, (12)
is a bijection. Resulting in | H |=| Hx |
Theorem 5.6. (Lagrange’s Theorem) Let G be a finite group.
1. Let H be a subgroup G. Then the order of H divides the order of G
2. Let g ∈ G. Then the order of g divides the order of G
Proof. Since the equivalence classes partition G, we find elements x1, x2, . . . , xr s.t.
G =
r
i=1
Hxi, (13)
whereby Hxi ∩ Hxj for i = j. The subsets of Hxi are disjoint. As the subsets are disjoint Lemma
4.2.1 gives
| G |=
r
i=1
Hxi =
r
i=1
| Hxi |=
r
i=1
| H |= r | H |, (14)
therefore, | G | is a multiple of | H | whenever H ≤ G, therefore the theorem is proved.
This is an important result, as it allows us to calculate the parameters used in ECDSA. With this said,
the larger the order of the sub-group of the finite field, the more difficult it is for the cryptosystem
to be infiltrated. Essentially, from knowing the order of the point, we can then calculate the group
order using (9). Strictly speaking, by finding a multiple of the group point within the Hasse’s Interval,
this value will equal #E(Fp). Relating to generating a hardy cryptosystem, a high order of a curve is
chosen, following this we choose a high divisor of this order, and then finally find a suitable generator
point which produces a cyclic subgroup order equal to the divisor. Taking an example from [4]:
Example Let E be the elliptic curve y2 = x3 − 10x + 21 over the finite field F557 The point
(2, 3) is shown to have order 189. Hasse’s Theorem suggests that 511 ≤ #E(Fp) ≤ 605 .
Calculating the multiples of 189 in this range is 3 · 189. Therefore #E(Fp) = 567.
6 Elliptic Curve Cryptography
After gaining understanding the use of cyclical subgroups and generator points, we are going to explore
the specific set of constraints in which bitcoin uses. Bitcoin uses a set of parameters defined in the
secp256k1 2 [1]. The elliptic curve y2 = x3 + Ax + B s.t.
a = 0000000000000000000000000000000000000000000000000000000000000000,
b = 0000000000000000000000000000000000000000000000000000000000000007,
defined over the prime finite field Fp, where
p = 2256 − 232 − 29 − 28 − 27 − 26 − 24 − 1.
2
secp256k1 refers to the parameters of the ECDSA curve used in Bitcoin, and is defined in Standards for Efficient
Cryptography (SEC)
9. 6 ELLIPTIC CURVE CRYPTOGRAPHY 9
The generator point in compressed 3 form
G = 02 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B
16F81798.
The order n of G
n = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE ABAAEDCE6 AF48A03B
BFD25E8C D0364141
and the cofactor h defined as #E(Fp)/n
h = 1.
As the generator point is specified within the parameters, and is constant for all bitcoin, the only point
in which changes is the unique point Q due to the change in the random integer d, as demonstrated
in the following subsection.
6.1 Key Generation
Before we introduce the ECDSA, we are going to understand Elliptic Curve Cryptography. Elliptic
Curve Cryptography was first proposed by Koblitz and Miller in 1985 [6]. We are going to investigate
the Elliptic Curve Discrete Logarithm Problem in which ECC it’s based from and then further our
understanding of how to create a public and private key used in ECDSA.
Definition 6.1. The Elliptic Curve Discrete logarithm Problem (ECDLP) Given an Elliptic Curve,
E represented as (4) defined over a finite field Fp of order n. We define the points, P and a point Q0
on E. The problem is defined to find k ∈ Z s.t. k ∈ [0, n − 1], s.t. Q0 = kP.
The resistance of the elliptic curve discrete logarithm problem is the foundations for all ECC schemes
[6]. The best algorithms to solve the problem having a running time of πn
2 steps.4 [6].
Firstly, a random private key is generated dA less than the order of the cyclic group and is of empirical
importance that the integer dA is random. The public key Q is generated by multiplying the generator
point G, dA number of times.
Q = dAG. (15)
Figure 6: Graphic representation of Point Multiplication of a generator point (G) [13]
Using the successive doubling method, it is comparatively simple to find Q from dA, however the
reverse is extremely difficult to compute, hence why the public key can be publicly shared without the
3
Further information on compressed points can be fouund at "http://bitcoin.stackexchange.com/questions/3059/what-
is-a-compressed-bitcoin-key"
4
The best algorithm to solve the ECDLP is the Pollard’s rho attack, with a total running time of πn
2
10. 7 ECDSA 10
risk of the private key being exposed. A graphical example is demonstrated in Figure 6, where dA =5
and Q = 32G. Note that this is a simple example. Given dA is usually a very large number, we can
visualise it being extremely difficult to determine dA given only G and Q.
7 ECDSA
The Elliptic Curve Digital Signature Algorithm (ECDSA) is a representation of DSA, introduced in
1992 by Scott Vanston [2]. The ECDSA is defined by the parameters D = (p, a, b, G, n, h), in which
the bitcoin utilises secp256k1 standard parameters defined earlier. We also have a private and public
key pair, in which is generated the same way as (15).
Suppose Alice and Bob would send a message (i.e. transaction to each other), the following algorithm
demonstrates how a digital signature and verification would occur [8]. A hash function H is combined
with the message 5. The algorithm is referenced from [6].
Algorithm 1 ECDSA Signature Generation
Input :Domain parameter D = (p, a, b, G, n, h), secret key d , message m
Output Signature (r, s)
1: Select k ∈ [1, n − 1], k ∈ N
2: Compute kG = (x1, y1) s.t. x1 ∈ Z
3: Compute r = x1 mod n . If r=0, then return to step 1.
4: Compute e = H(m)
5: Compute s = k−1(e + dr) (mod n). If s=0 then go to step 1.
6: return (r, s).
Algorithm 2 ECDSA Signature Verification
Input: Domain parameter D = (p, a, b, G, n, h), public key Q, message m, signature (r, s)
Output: Acceptance of rejection of the signature
1: Verify 0 < r < n and 0 < s < n , if not you reject the signature
2: Compute e = H(m)
3: Compute w = s−1 mod n
4: Compute the two integers : u1 = we (mod n) and u2 = rw (mod n)
5: Compute the point X using the result of the computations : X = u1G + u2Q
6: If X = ∞ then you reject the signature
7: Convert the x co-ordinate of X, x1 to an integer v = x1 (mod n)
8: v = r, accept the signature
7.1 Proof of Correctness
Proof. If the signature (r, s) from the message m was created by a honest signee of the message. We
must then rearrange X to get
X = u1G + u2Q,
= u1G + u2dAG,
= G(u1 + u2dA),
5
The cryptographic function used is the ’Secure Hash Algorithm’
11. 8 CONCLUSION 11
utilising the definitions of u1 and u2
X = G(u1 + u2dA),
= (s−1
e + s−1
rdA)G,
= s−1
(e + rdA)G,
= kG.
kG equates to the same point from Algorithm 1, step 2. This is the same point as only the x co-
ordinate was transmitted, with different equations, therefore v = r, demonstrating the signatures are
equal, verifying the transaction, concluding how we arrive at the algorithm ECDSA.
The decentralised and transparent system is further reinforced through the fact that any user on the
blockchain can verify whether the signature is authentic as only the public key is required.
8 Conclusion
In recent times, bitcoin and the block chain technology have been prevalent in the major financial
institutions in which financial institutions are exploring the option to streamline systems, potentially
slashing infrastructure costs [12]. This project was used to explore the mathematics behind these
secure transactions, gaining an understanding of the fundamentals behind the algorithm. ECDSA is
one of the adaptions of ECC, suggested as a substitute to other public cryptosystems such as RSA.
This project is served as an introduction to ECDSA and has looked under the bonnet of the algorithm.
With this said, hopefully the reader will have an enhanced knowledge of this.
References
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