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Niels Henrik Abel and Abelian Group
Abelian Groups are named after early 19th century mathematician Niels Henrik Abel. 2022-10-01, ∼384🔥, 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, ∼384🔥, 0💬

What Is Subgroup Generator in Abelian Group
This section describes subgroup generator in a Abelian Group. A subgroup generator is an element in an Abelian Group that can be used to generator a subgroup using a series of scalar multiplication operations. 2022-10-01, ∼382🔥, 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, ∼382🔥, 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, ∼380🔥, 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, ∼379🔥, 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, ∼373🔥, 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, ∼371🔥, 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, ∼370🔥, 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, ∼368🔥, 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, ∼366🔥, 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, ∼357🔥, 0💬

Terminology
List of terms used in this book. 2022-10-01, ∼357🔥, 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, ∼352🔥, 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, ∼350🔥, 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, ∼349🔥, 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, ∼348🔥, 0💬

Elliptic Curve Operation Summary
This section provides a summary of elliptic curve operations and their properties discussed in this chapter. 2022-10-01, ∼345🔥, 0💬

References
List of reference materials used in this book. 2022-10-01, ∼343🔥, 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, ∼337🔥, 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, ∼334🔥, 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, ∼328🔥, 0💬

Finite Elliptic Curve Group, Eq(a,b), q = p^n
This section describes finite elliptic curve groups constructed with modular arithmetic reduction of prime power numbers, p^n. 2022-10-01, ∼322🔥, 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, ∼321🔥, 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