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Find Subgroup with Point Addition
This section provides a tutorial example on how to find the subgroup of a given point on an elliptic curve using a loop of point additions with tinyec Python library. 2022-10-01, 364🔥, 0💬

Create EC Public Key File
This section provides a tutorial example on how to extract the public key out of an EC private key file with the 'openssl ec -pubout' command. 2022-10-01, 360🔥, 0💬

EC Cryptography in Java
This chapter provides tutorial notes on generating EC (Elliptic Curve) keys with Java technology. Topics covered include using 'keytool' command to generate EC private-public key pairs; selecting different name elliptic curves or key sizes; writing Java program to generate EC keys. 2022-10-01, 357🔥, 0💬

Same Point Addition on an Elliptic Curve
This section describes how to perform the addition operation of a point P to the same point P on an elliptic curve. In this case, we will draw a straight line that passes P and tangent to the curve to find -R. 2022-10-01, 342🔥, 0💬

Abelian Group and Elliptic Curves
This chapter provides an introduction to Abelian Group, which can be expressed in the multiplicative notation or the additive notation. An Abelian Group and be defined on an elliptic curve using the 'rule of chord' operation. 2022-10-01, 316🔥, 0💬

Examples of Discrete Logarithm Problem (DLP)
This section describes the Discrete Logarithm Problem (DLP) in several Abelian Group examples, including elliptic curve groups. 2022-10-01, 299🔥, 0💬

Algebraic Introduction to Elliptic Curves
This chapter provides an algebraic introduction of addition operations on elliptic curves. Algebraic solutions are provided to calculate addition operations by dividing the problem into 4 cases: two symmetric points, one infinity point, two identical points, and two distinct points. 2022-10-01, 293🔥, 0💬

"sect283r1" - For 256-Bit ECC Keys
This section describes 'sect283r1' elliptic curve domain parameters for generating 256-Bit ECC Keys as specified by secg.org. 2022-10-01, 292🔥, 0💬

Addition Operation on an Elliptic Curve
This section describes the addition operation on an elliptic curve geometrically. The addition of points P and Q on an elliptic curve is a point R on the curve, which is the symmetrical point of -R, which is the third intersection of the curve and the straight line passing through P and Q. 2022-10-01, 286🔥, 0💬

Modular Addition of 10 - Abelian Group
This section provides an Abelian Group using the modular arithmetic addition of 10 (integer addition operation followed by a modular reduction of 10). 2022-10-01, 276🔥, 0💬

Is EC Private Key Secure
This section discusses the question of: how secure is the EC (Elliptic Curve) key pair? Or how hard is it someone to figure out the private key from a given EC public key? 2022-10-01, 271🔥, 0💬

Discrete Logarithm Problem (DLP)
This chapter provides an introduction to the Discrete Logarithm Problem (DLP), which is the reverse operation of the exponentiation operation in Abelian Groups in multiplicative notation, or the scalar multiplication in additive notation. The DLP in many Abelian Groups is easy to solve. But the DLP ... 2022-10-01, 262🔥, 0💬

Finite Fields
This chapter provides an introduction to Finite Fields. Topics covered include definition of finite fields; examples of finite fields: prime fields GF(p), binary fields GF(2^n) and polynomial fields GF(p^n); field order as the number of elements; field characteristic p is the least positive integer ... 2022-10-01, 262🔥, 0💬

Elliptic Curve Subgroups
This chapter provides notes on subgroup generation from reduced elliptic curve groups, Ep(a,b). Python programs are provided to perform point addition, scalar multiplication, and subgroup generation. 2022-10-01, 261🔥, 0💬

Elliptic Curves with Singularities
This section describes elliptic curves with singularities where curves are not smooth. 2022-10-01, 256🔥, 0💬

Reduced Elliptic Curve Groups
This chapter provides notes and tutorials on reduced elliptic curve groups. Topics include elliptic curve on in integer space; elliptic curves and the addition operation reduced by modular arithmetic; elliptic curve groups and examples. 2022-10-01, 249🔥, 0💬

"Legacy SunEC curve disabled" Error
This section provides a tutorial example on how to resolve the 'Legacy SunEC curve disabled' error and get short and insecure EC private-public key pairs. 2022-10-01, 246🔥, 0💬

Associativity of Elliptic Curve Operations
This section describes the associativity of the addition operation on an elliptic curve. P + (Q + S) = (P + Q) + S is true. 2022-10-01, 244🔥, 0💬

Scalar Multiplication on Elliptic Curve as Trapdoor Function
This section confirms that Scalar Multiplication on Elliptic Curve is a good Trapdoor Function by the comparing difficulty level against its reverse operation, which is the DLP. 2022-10-01, 244🔥, 0💬

Reduced Elliptic Curve Group - E23(1,4)
This section provides an example of a reduced Elliptic Curve group E23(1,4). A detailed calculation of reduced point doubling operation on (0,2) is also provided. 2022-10-01, 244🔥, 0💬

Niels Henrik Abel and Abelian Group
Abelian Groups are named after early 19th century mathematician Niels Henrik Abel. 2022-10-01, 242🔥, 0💬

What Is Order of Element
This section describes the order of a given element in a finite Abelian Group, which is defined as the least positive integer n, such that the scalar multiplication of n and P is 0, where 0 is the identity element. 2022-10-01, 242🔥, 0💬

What Is Cyclic Group
This section describes Cyclic Group, which is a finite Abelian group that can be generated by a single element using the scalar multiplication operation in additive notation (or exponentiation operation in multiplicative notation). 2022-10-01, 242🔥, 0💬

References
List of reference materials used in this book. 2022-10-01, 239🔥, 0💬

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Android ASP Astrology Big5 Bitcoin Blowfish Built-in C# Calendar CD-DVD Cheminformatics Chinese Cryptography DES Email Encoding Ethereum Flash GB2312 General H Herong History HTML Java JavaScript JDBC JSP JVM Linux Molecule Movie Music MySQL Neural Network Perl PHP Physics PKI Publication Python Reference REST SOAP Sorting Swing TV Series UML Unicode VBScript Web Web-Services Windows WSDL XML XMLPad XSD XSL-FO