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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, ∼364🔥, 0💬

What Is Discrete Logarithm Problem (DLP)
This section describes what is Discrete Logarithm Problem (DLP), which is the reverse operation of an exponentiation (or scalar multiplication) operation in an Abelian group. 2022-10-01, ∼356🔥, 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, ∼355🔥, 0💬

Order of Subgroup and Lagrange Theorem
This section describes Lagrange Theorem which states that the order of any subgroup in an finite Abelian group divides the order of the parent group. 2022-10-01, ∼355🔥, 0💬

What Are Standard Elliptic Curves
This section provides a list of standard elliptic curves selected and recommended by different organizations to generate secure EC private-pubic key pairs. 2022-10-01, ∼353🔥, 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, ∼351🔥, 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, ∼341🔥, 0💬

Additive Notation of Abelian Group
This section describes the Additive notation of an Abelian Group. The addition sign, +, is used as the operator. Number 0 is used as the identity element. 2022-10-01, ∼338🔥, 0💬

Scalar Multiplication or Exponentiation
This section describes what is Scalar Multiplication or Exponentiation in Abelian Groups. They are used represent the process of performing Abelian Group operations consecutively n times with the same element. 2022-10-01, ∼336🔥, 0💬

Algebraic Solution for Point Doubling
This section provides an algebraic solution for calculating the addition operation of two points at the same location on an elliptic curve. 2022-10-01, ∼334🔥, 0💬

Terminology
List of terms used in this book. 2022-10-01, ∼334🔥, 0💬

References
List of reference materials used in this book. 2022-10-01, ∼334🔥, 0💬

Set Subgroup Order to Higher Value
This section provides a tutorial example on how to set the subgroup order a value greater than the order of the entire group, like 2 times of the modulo, to ensure correct result of scalar multiplications. 2022-10-01, ∼331🔥, 0💬

What Is Subgroup in Abelian Group
This section describes Subgroups in a Abelian Group. A subgroup in a Abelian Group is a subset of the Abelian Group that itself is an Abelian Group. The subgroup and its parent group are using the same operation. 2022-10-01, ∼330🔥, 0💬

Elliptic Curve Operation Summary
This section provides a summary of elliptic curve operations and their properties discussed in this chapter. 2022-10-01, ∼324🔥, 0💬

Modular Multiplication of 11 - Abelian Group
This section provides an Abelian Group using the modular arithmetic multiplication of 11 (integer multiplication operation followed by a modular reduction of 11). 2022-10-01, ∼320🔥, 0💬

Converting Elliptic Curve Groups
This section describes steps on how to convert real number elliptic curve groups to cyclic subgroups of integer elliptic curve groups. 2022-10-01, ∼317🔥, 0💬

What Is Abelian Group
This section describes Abelian Group, which a set of elements with a binary operation satisfing 5 conditions. 2022-10-01, ∼312🔥, 0💬

Integer Points of First Region as Element Set
This section describes how to use all integer points from the first region on a reduced elliptic curve as the element set to try to construct an Abelian group. 2022-10-01, ∼310🔥, 0💬

Elliptic Curves in Integer Space
This section describes the fact that elliptic equations in 2-dimensional integer space can not be used to construct Abelian groups. 2022-10-01, ∼304🔥, 0💬

Prove of Elliptic Curve Addition Operation
This section describes how to prove that the addition operation on an elliptic curve can be successfully performed geometrically. 2022-10-01, ∼303🔥, 0💬

Reduced Elliptic Curve Group - E97(-1,1)
This section provides an example of a reduced Elliptic Curve group E97(-1,1). Some example points on the curve are is also provided. 2022-10-01, ∼303🔥, 0💬

EC Curves Supported by Java
This section provides is a list of named curves that are supported or not supported by 'keytool' and Java default library. 2022-10-01, ∼303🔥, 0💬

Doubling or Squaring in Abelian Group
This section describes what is Doubling or Squaring in Abelian Groups. When performing the Abelian Group operation on two identical elements, this operation is called Squaring in multiplicative notation or Doubling in additive notation. 2022-10-01, ∼301🔥, 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 REST SOAP Sorting Swing TV Series UML Unicode VBScript Web Web-Services Windows WSDL XML XMLPad XSD XSL-FO